Sindbad~EG File Manager
/* M. Beeson, for MathXpert.
Hints associated with menu choices.
Text between dollar signs need not be translated. It represents
mathematical formulas. Some of the text between dollar signs may
look unintelligible if the file is viewed in ISO-Latin1
character set, but DO NOT ALTER IT.
(It is correct in the OEM character set.)
The rest of the text strings (text betwen double quotes)
should be viewed and translated in the ISO-Latin1 character set.
8.20.94 Original date
11.24.98 modified
12.29.98 added seven more lines in binomial_theorem
1.14.99 added last entry in advanced_sigma_notation
2.21.99 moved two menus out to hints.c because of Borland compiler limits.
6.8.99 corrected รณ to = in the last item in absolute_value_ineq1
1.10.00 modified advanced_factoring and numerical_calculation1 arrays.
4.2.00 added four new lines in absolute_value_ineq2
3.9.01 modified for signed_fractions
9.12.04 four more lines in complex_numbers
1.23.06 two more lines in advanced_sigma_notation
8.9.07 Corrected the hint containing Vieta
5.3.13 changed names of exported functions
5.24.13 two more in the first menu
6.3.13 added one more in signed_fractions
6.4.13 two more in fractional_exponents
6.5.13 two more in log_ineq4
change history of hints2.c:
Original date 5.24.95 (extracted from hints.c)
Sent to translator 8.12.98
8.13.98, two new operations added in improper_integrals
8.17.98, logarithmic_limits operations added.
8.19-21.98, new operations in series4
1.6.99, new operations in series2, changed wording on first line of series3
1.7.99, new operations in series1.
1.12.99 Now there are 12 series menus with new entries.
1.28.99 last modified
2.21.99 added four new lines under complex_hyperbolic and one under
more_infinities.
Moved two more menus in from hints.c
sent to German translator
3.23.99 modified limits_of_quotients choice 8
6.10.99 reconciled to French translation
6.10.99 changed 'cos' to 'cosh' in last item in hyper_limits
6.10.99 reconciled to French version
1.4.00 added four new lines in complex_hyperbolic
2.27.00 removed extra text at the end that didn't belong there
2.27.00-3.4.00 added series_convergence2 text
1.27.06 new operations under sg_function2
1.14.11 six new operations under inverse_hyperbolic.
5.3.13 changed names of exported functions
5.24.13 added series_bernoulli
5.28.13 corrected typo in sg_function2
6.11.13 four more under series_bernoulli
6.13.13 corrected a typo
12.2.14 "Finish the divergence test" was out of order, corrected the order.
11.19.24 hints2.c folded into hints.c
*/
#include "mtext.h" /* MAXLENGTH */
#include "english1.h"
static char arithhint[] = "There is some arithmetic to be performed.";
static char dummystring[] = "dummy";
static char *hintstrings1[][MAXLENGTH] =
{
{ /* numerical_calculation1 */
arithhint,
"Perform decimal arithmetic.",
"Calculate the decimal value of a root.",
"Calculate the decimal value of a power.",
"Calculate the decimal value.",
"It may help to factor an integer, for example under a root or square root sign.",
"Evaluate numerically at a point.", /* Not used in auto mode */
"Compute the decimal value.", /* decimal value of pi_term, not used in auto mode. */
"Compute the decimal value.", /* decimal value of e, not used in auto mode. */
"Compute a numerical value of a function.",
"You can find the roots of a polynomial numerically, and hence find its factors, approximately. Choose 'factor polynomial numerically' to do that.",
"Evaluate a Bernoulli number to a rational number",
"Evaluate an Euler number to a rational number"
},
{ /* numerical_calculation2 */
"Convert a decimal to a fraction.", /* Not used in auto mode. */
"Express a number as a square",
"Express a number as a cube",
"Express a number as an $n$-th power for a suitable $n$.",
"Express a number as a power of a specified base.",
"Express an integer as a power, for example write $9$ as $3^2$.",
"Express an integer as a sum, using $x = ? + (x-?)$",
},
{ /* complex_arithmetic */
"Use the definition of the complex number $i$, namely $i^2 = -1$.",
"Integer powers of the complex number $i$ can be simplified.",
"Integer powers of the complex number $i$ can be simplified.",
"Integer powers of the complex number $i$ can be simplified.",
"Integer powers of the complex number $i$ can be simplified.",
"There is some complex arithmetic to be done.",
"There is a power of a complex number that could be evaluated.",
"There is some complex arithmetic to be done.",
"Perform complex decimal arithmetic",
"It may help to factor an integer.",
"Sometimes an integer can be factored into complex factors, like $5 = (2-i)(2+i)$.",
"Factor an expression $n+mi$ into complex factors. For example, $7-5i = (2-i)(3-i)$.",
"Compute the decimal value.", /* decimal value of root, not used in auto mode. */
"Compute the decimal value.", /* decimal value of x^n, not used in auto mode. */
"Compute the decimal value.", /* decimal value of a function, not used in auto mode. */
"Evaluate numerically at a point." /* not used in auto mode */
},
{ /* simplify_sums */
"Get rid of the double minus sign.",
"Push the minus sign into the sum.",
"Bring the minus signs out of the sum.", /* never done in auto mode anyway */
arithhint,
"When you have a sum containing a sum, you can regroup the terms to remove the extra parentheses.",
"Put the terms in a sum in proper order.",
"You can drop a zero summand using the law $x+0 = x$.",
"There are terms that will cancel out.",
"Collect like terms.",
"Collect like terms.",
"Use the commutative law of addition.",
"Pull a minus sign out using $a(b-c) = -a(c-b)$.",
"-ab = a(-b)",
"-abc = ab(-c)",
"a(-b)c = ab(-c)"
},
{ /*simplify_products */
"Zero times any number is zero.",
"You can drop a factor of one.",
"Bring out the minus sign using $a(-b) = -ab$",
"Bring out the minus sign using $a(-b-c) = -a(b+c)$",
"Bring out the minus sign using $(-a-b)c = -(a+b)c$",
"Regroup factors to get rid of the extra parentheses, using the associative law of multiplication.",
"When more than one number appears in a product, collect them together at the beginning of the product.",
"Put the factors of a product in standard order.",
"Collect powers, that is, combine terms with the same base into a single term.",
"Multiply out using the distributive law, $a(b+c)=ab+ac$.",
"Use the law for making $(a-b)(a+b)$ into a difference of squares.",
"Expand the square of a sum using a standard formula.",
"Expand the square of a difference using a standard formula.",
"Do you recognize a difference of cubes in its factored form?",
"Do you recognize a sum of cubes in its factored form?",
"Use the commutative law of multiplication."
},
{ /* expand_menu */
"A product of sums, or a power of a sum, can always be multiplied out to get a single sum. Sometimes this leads to further simplifications.",
"Maybe if you multiply out the numerator, things will get simpler.",
"Maybe if you multiply out the denominator, things will get simpler.",
"Use the operation $na = a + ... + a$." /* never used in auto mode anyway */
},
{ /* fractions */
"Get rid of the fraction with 0 in the numerator.",
"Get rid of the 1 in the denominator.",
"You have something times its reciprocal here--that makes 1",
"Multiply the fractions to get a single fraction",
"Use the law $a(b/c) = ab/c$ to get a single fraction",
"Cancel a common factor from numerator and denominator.",
"Add fractions with the same denominator.",
"Break a fraction with a sum in the numerator apart into two fractions.",
"Break a fraction with a sum in the numerator apart into two fractions, one of which will simplify by cancellation.",
"Use polynomial division to simplify a fraction, when the degree of the numerator is more than the degree of the denominator.",
"You may be able to cancel by polynomial division.",
"Combine numbers in the numerator and denominator into a single rational number using the law au/bv=(a/b)(u/v).",
"Make the denominator into a coefficient using the law $a/b = (1/b) a$",
"Pull out the real factors from numerator and denominator using $au/b = (a/b)u$.",
"Break a fraction apart using $ab/cd = (a/c)(b/d)$.",
"Combine the numerical parts of numerator and denominator into a single coefficient using the law $ab/c = (a/c)b$"
},
{ /* signed_fractions */
"Cancel the minus signs in numerator and denominator.",
"Push the minus sign into the numerator using the law $-(a/b) = (-a)/b$.",
"Push the minus sign into the denominator using the law $-(a/b) = a/(-b)$.",
"Pull that minus sign out of the numerator so it applies to the fraction as a whole.",
"Pull that minus sign out of the denominator so it applies to the fraction as a whole.",
"Pull the minus signs out of the numerator using the law $(-a-b)/c = -(a+b)/c$.",
"Pull the minus signs out of the denominator using the law $a/(-b-c) = -a/(b+c)$.",
"Adjust the sign of the denominator using the law $a/(b-c) = -a/(c-b)$.",
"Pull the minus signs out of the denominator using the law $-a/(-b-c) = a/(b+c)$.",
"Adjust the sign using the law $-a/(b-c) = a/(c-b)$",
"Pull the minus signs out of the numerator using the law $-(-a-b)/c = (a+b)/c$.",
"Change the order of terms in both numerator and denominator. Select the entire fraction to do this.",
"ab/c = a(b/c)",
"Break a fraction apart using $a/bc = (1/b) (a/c)$."
},
{ /* compound_fractions */
"When numerator and denominator are both fractions with the same denominator, you can use the law $(a/c)/(b/c) = a/b$ to get rid of compound fractions.",
"When the denominator is a fraction itself, you invert it and multiply, using the law $a/(b/c)=ac/b$",
"The reciprocal of a fraction is simplified using the law $1/(a/b) = b/a$.",
"When the numerator is a fraction, you can apply the law $(a/b)/c = a/(bc)$ to get rid of the compound fraction.",
"Use $(a/b)/c = (a/b)(1/c)$", /* never suggested in auto mode */
"When the numerator is a product containing a fraction, you can apply the law $(a/b)c/d = ac/bd$",
"Sometimes it helps to factor the denominator.",
"If you have a sum of fractions in the numerator or denominator, you have to first use common denominator to convert that sum to a single fraction. Then you can proceed to reduce the resulting compound fraction."
},
{ /* common_denominators */
"First factor the denominator, so the true common denominator will be exposed.",
"The denominators aren't the same. Therefore you must find a common denominator.",
"The denominators aren't the same. Therefore you must fine a common denominator. But only add the fractions.",
"You've got a product of fractions, not yet combined into a single fraction. Multiply your fractions together.",
"You've got something multiplied by a fraction. Multiply it in, to get a single fraction.",
"It's good housekeeping to keep your factors in the proper order. It helps to recognize like terms and notice cancellations.",
"Now you have fractions with the same denominator. They can easily be added to make a single fraction.",
"You have fractions to put over a common denominator.",
"You have fractions to put over a common denominator.",
"You have fractions to put over a common denominator.",
"You have fractions to put over a common denominator.",
"Multiply numerator and denominator by something."
},
{ /* exponents */
"You have an exponent of zero. Get rid of it.",
"You have an exponent of one. Get rid of it.",
"Zero to any (nonzero) power is zero.",
"One to any power is one.",
"Minus one raised to an integer power can be evaluated: it is 1 for even powers, and -1 for odd powers.",
"You have a power raised to a power. There is a law for combining such a thing into a single power.",
"You can pull a minus sign out of a power using $(-a)^n = (-1)^na^n$.",
"It might help to push the exponent into the numerator and denominator using $(a/b)^n = a^n/b^n$.",
"You have a power of a product. It would simplify the expression to push the exponent in using $(ab)^n = a^nb^n$.",
"You can expand the square of a sum using $(a+b)^2 = a^2+2ab+b^2$.",
"The binomial theorem might be fruitful here.",
"You have two or more powers of the same base multiplied together. Collect those powers.",
"You have a power of a sum; transform it to a product of powers.",
"You have a fraction of the form $a^n/b^n$. Pull the exponent outside the fraction like this: $(a/b)^n$.",
"You have powers of the same base in both numerator and denominator. Combine them into a single power in the numerator.",
"You have powers of the same base in both numerator and denominator. Combine them into a single power in the denominator."
},
{ /* expand_powers */
"Expand a square.", /* Never used in auto mode */
"Expand a cube.",
"Expand a power.",
"Break a power into a product of smaller powers",
"Expand a square of a sum.",
"Expand a cube of a sum.",
"Expand a cube of a difference.",
"Use the law $a^(bc) = (a^b)^c$ if $a>0$ or $c\\in Z$.",
"Use the law $a^(bc) = (a^c)^b$ if $a>0$ or $c\\in Z$.",
"Use the law $a^(bc) = (a^b)^c$, entering the value of $c$.",
"Bring an exponent out of the denominator using $1/a^n = (1/a)^n$"
},
{ /* negative_exponents */
"Use the definition of a negative exponent, $a^(-n) = 1/a^n$.",
"Negative exponents in the numerator convert to positive exponents in the denominator.",
"Use the definition of a exponent of $-1$, $a^(-1) = 1/a$.",
"Use the definition of a negative exponent, $a^(-n) = 1/a^n$.",
"Negative exponents in the numerator convert to positive exponents in the denominator.",
"Negative exponents in the denominator convert to positive exponents in the numerator.",
"Positive exponents in the denominator convert to negative exponents in the numerator.",
"You can always get rid of a fraction by converting the denominator to a factor with an exponent of -1.",
"A fraction to a negative exponent can be written with a positive exponent after inverting.",
"You have powers of the same base in both numerator and denominator. Combine them into a single power in the numerator.",
"You have powers of the same base in both numerator and denominator. Combine them into a single power in the denominator.",
"Use the law $a^(b-c) = a^b/a^c$"
},
{ /* square_roots */
"Combine your product of square roots into a single square root.",
"Make your square root into a product of square roots.",
"You have a squared factor under the square root sign. Pull it out--but be careful about the sign.",
"The square root of $x^2$ is $x$, at least for positive $x$; but if $x$ is negative, you have to make it the absolute value of $x$.",
"The square root of $x^2$ is $x$, at least for positive $x$; but if $x$ is negative, you have to make it the absolute value of $x$.",
"To simplify the square root of an integer, you begin by factoring the integer.",
"The square root of a fraction can be written as a fraction of square roots, using $\\sqrt (x/y) = \\sqrt x/\\sqrt y$",
"The square root of a fraction can be written as a fraction of square roots, using $\\sqrt (x/y) = \\sqrt |x|/\\sqrt |y|$. The absolute value signs are needed if the signs of $x$ and $y$ are unknown.",
"You have a quotient of square roots. Try to change this into a single square root.",
"Remember that $\\sqrt x$ times $\\sqrt x$ is $x$. Therefore $x/\\sqrt x$ simplifies to $\\sqrt x$.",
"Remember that $\\sqrt x$ times $\\sqrt x$ is $x$. Therefore $\\sqrt x/x$ simplifies to $/\\sqrt x$.",
"An even power of a square root can be simplified using $(\\sqrt x)^2^n = x^n$, at least for nonnegative $x$",
"An odd power of a square root can be simplified using $(\\sqrt x)^(2n+1) = x^n\\sqrt x$.",
"Perhaps the square root can be evaluated exactly?",
"Evaluate the square root using decimal numbers",
arithhint
},
{ /* advanced_square_roots */
"Do the numerator and denominator have a common factor under the square root sign?",
"Factor the polynomial under the square root sign.",
"Rationalize the denominator. That means to multiply numerator and denominator by the same thing, chosen in order to get rid of square roots in the denominator.",
"Rationalize the numerator. That means to multiply numerator and denominator by the same thing, chosen in order to get rid of square roots in the numerator.",
"A square root of an even power can be simplified using absolute value",
"There is a common factor under the square roots in numerator and denominator. Cancel the common square root.",
"Multiply out under the square root sign.",
"It may help to think of $b$ as the square of $\\sqrt b$, so $a^2-b = (a-\\sqrt b)(a+\\sqrt b)$.",
"A root with index 2 should be converted to a square root.",
"Express a square root as a root of a power, for example $\\sqrt 2 = ^4\\sqrt 4$",
"Express a square root as a power of a root, for example $\\sqrt 3 = (^4\\sqrt 3)^2$",
"An even power is a square, so you have a square under the square root sign.",
"You have a power more than two under the square root sign; bring some powers outside the square root.",
"Push something under the square root using $a\\sqrt b = \\sqrt (a^2b)$.",
"Rationalize the denominator and simplify."
},
{ /* fractional_exponents */
"An exponent of $\\onehalf $ can be converted to a square root.",
"A fraction in the exponent with denominator 2 can be converted to a square root, using $$a^(n/2) = sqrt (a^n)$$.",
"A fraction in the exponent with denominator $n$ can be converted to an $n$-th root, using $$a^(b/n) = root(n,a^b)$$.",
"A square root can be converted to an exponent of $\\onehalf $",
"An $n$-th root can be converted to an exponent of $1/n$",
"Eliminate roots of powers by changing to fractional exponents.",
"Eliminate powers of roots by changing to fractional exponents.",
"Eliminate powers of square roots by changing to fractional exponents.",
"An $n$-th root in the denominator can be converted to a negative exponent of $1/n$",
"Express a square root in the denominator using a negative fractional exponent.",
"Powers of $-1$ can be explicitly evaluated",
"Factor an integer which is raised to a fractional exponent",
"Get the fractional exponent out of the denominator.",
"Get the fractional exponent out of the numerator.",
"Make the fractional exponent into a power of a square root",
"Make the fractional exponent into a power of a root"
},
{ /*nth_roots */
"Combine the product of roots into a single root.",
"Split the root of a product into a product of roots.",
"Bring the exponent outside the root so everything is a function of the same root.",
"You have an $n$-th power under an $n$-th root. Pull it out.",
"An $n$-th root of an $n$-th power can be simplified, but be careful: $^n\\sqrt (x^n) = x$ isn't always true!",
"You can simplify the root: for example the cube root of $x^6$ is $x^2$",
"Sometimes you can lower the index of a root. For example, the 6-th root of $x^3$ is $\\sqrt x$.",
"Sometimes you can lower the index of a root. For example, the 6-th root of $x^2$ is the cube root of $x$.",
"Remember the definition of the $n$-th root of $x$: when you raise it to the $n$-th power, you get $x$.",
"You have a power of a root. Bring the exponent under the root, as in $(^n\\sqrt x)^2 = ^n\\sqrt (x^2)$.",
"You have a power of a $n$-th root, say of $x$. Pull out some factors of $x^n$ until the power is less than $n$. Example: $(^3\\sqrt 2)^7 = 2^2 ^3\\sqrt 2$.",
"Factor the integer under the root sign.",
"You have an odd root of a negative expression; bring the minus sign out from under the root.",
"Maybe the root can be evaluated exactly.",
"Factor the polynomial under the root sign.",
"Multiply out under the root sign."
},
{ /* roots_of_roots */
"A square root of a square root can be expressed as a fourth root.",
"A square root of an n-th root can be expressed as a 2n-th root.",
"An n-th root of a square root can be expressed as a 2n-th root.",
"A root of a root can be expressed as a single root. For example, a cube root of a fourth root is a 12-th root."
},
{ /* roots_and_fractions */
"Turn your root of a fraction into a fraction of roots.",
"Turn a quotient of two roots into a single root.",
"Combine the roots in numerator and denominator, getting a single root.",
"Combine the roots in numerator and denominator, getting a single root.",
"Cancel a factor under the root sign. Select the whole fraction.",
"Cancel a root from numerator and denominator. Select the whole fraction.",
"The numerator and denominator have a common factor under the root sign. Select the whole fraction.",
"Push something under the root using $a\\sqrt b = \\sqrt (a^2b)$.",
"Push something under the root using $a\\sqrt b = \\sqrt (a^2b)$.",
"Push a minus sign under a root.",
"Bring the whole fraction under the root.",
"Bring the whole fraction under the root.",
"Bring the whole fraction under the square root.",
"Bring the whole fraction under the square root.",
"A power of a root can be simplified, making it a root with a lower index",
"A power of a root can be simplified, creating a square root."
},
{ /* complex_numbers */
"You know that $i^2$ is $-1$. It follows that $1/i$ is $-i$.",
"Since $1/i$ is $-i$, $i$ can be brought up from the denominator to the numerator if you change the sign of the fraction.",
"Since $1/i$ is $-i$, $i$ can be brought up from the denominator to the numerator if you change the sign of the fraction.",
"By definition, the square root of $-1$ can be written as $i$.",
"The square root of a negative number can be expressed in terms of $i$, using the law $\\sqrt (-a) = i\\sqrt a$.",
"You can clear $i$ out of the the denominator entirely, by multiplying numerator and denominator by the complex conjugate of the denominator.",
"A complex number times its conjugate simplifies according to $(a-bi)(a+bi) = a^2+b^2$.",
"You can factor a sum of squares using complex numbers, according to $a^2+b^2 = (a-bi)(a+bi)$.",
"By the Pythagorean theorem, we have $|u + vi|^2 = u^2 + v^2$",
"By the Pythagorean theorem, we have $|u + vi| = \\sqrt (u^2+v^2)$",
"Express the quotient as a single complex number, using $(u+vi)/w = u/w + (v/w)i$.",
"Write complex numbers in the form $u+vi$",
"Express a complex square root to the form $u+vi$",
"Express a complex square root to the form $u+vi$",
"Express a complex square root to the form $u+vi$",
"Express a complex square root to the form $u+vi$"
},
{ /* factoring */
"Factor out a number.",
"Clear your numerical denominators, so you can see better what is going on.",
"There is a common factor that you could pull out using the distributive law, $ab+ac = a(b+c)$",
"Factor out the highest common power.",
"Do you see the perfect square of a sum? Remember $a^2+2ab+b^2 = (a+b)^2$.",
"Do you see the perfect square of a difference? Remember $a^2-2ab+b^2 = (a-b)^2$.",
"A difference of squares can be factored using $a^2-b^2 = (a-b)(a+b)$.",
"This doesn't seem to fit any of the simpler patterns, but it is a quadratic trinomial, so maybe it can be factored.",
"If it won't factor any other way, you could always use the quadratic formula on it.",
"An even power can be written as a square, using $a^2^n = (a^n)^2$. Then maybe you can use factoring patterns involving squares.",
"Try combining powers using the law $a^nb^n = (ab)^n$",
"It may help to factor the coefficients of your polynomial.",
"Factor that integer.",
"It may help to make a substitution.",
"Now eliminate your defined variable.",
"Regard a variable as constant."
},
{ /* advanced_factoring */
"This is too complicated to factor directly, but if you write it as a function of some subexpression, correctly, you may make progress.",
"This is too complicated to factor directly, but if you write it as a function of some subexpression, correctly, you may make progress.",
"Express a higher power as a cube using the formula $a^(3n) = (a^n)^3$",
"Express a power using the formula $a^(mn) = (a^m)^n$.",
"There is a formula for factoring the difference of cubes.",
"There is a formula for factoring the sum of cubes.",
"There is a formula for factoring $a^n-b^n$.",
"There is a formula for factoring $a^n-b^n$.",
"There is a formula for factoring $a^n+b^n$.",
"There are formulae for factoring the sum of fourth powers.",
"Some fourth-degree polynomials can be factored by special formulas.",
"Try making a substitution. Select the term to be replaced by a new variable.",
"Guess a factor.", /* guess a factor isn't used in auto mode */
"If all else fails, you can search for a linear factor systematically",
"Try to factor by grouping",
"Write it as a polynomial in some variable or expression. Select the variable or expression."
},
{ /* solve_equations */
"Switch sides, in order to get the unknown on the left.",
"Change the signs of both sides.",
"Add something to both sides of your equation.",
"Subtract something from both sides of your equation.",
"Transfer an appropriate term from left to right.",
"Transfer an appropriate term from right to left.",
"Multiply both sides of your equation by something.",
"Divide both sides of your equation by something.",
"Square both sides of your equation.",
"Cancel a term from both sides of your equation.",
"Cancel a common factor of the two sides of your equation.",
"Subtract to put in form $u=0$.",
"When an equation reduces to an identity, any number (for which the sides are defined) is a solution. The equation reduces to the logical expression 'true'.",
"When the two sides of an equation have opposite signs, the only way the equation can have a solution at all is if both sides are zero. That is, $a = -b$ becomes $a^2 = -b^2$, provided $a$ and $b$ are both non-negative. This way of writing the equation will often save you from generating false solutions that have to be rejected in the end.",
"When the two sides of an equation have opposite signs, the only way the equation can have a solution at all is if both sides are zero. That is, $a = -b$ becomes $a=0$, provided $a$ and $b$ are both non-negative. At the end, you check the solution, and if $b$ was not also zero, the solution will be rejected.",
"When the two sides of an equation have opposite signs, the only way the equation can have a solution at all is if both sides are zero. That is, $a = -b$ becomes $b=0$, provided $a$ and $b$ are both non-negative. At the end, you check the solution, and if $a$ was not also zero, the solution will be rejected."
},
{ /* quadratic_equations */
"You have a product equal to zero. Split that into two (or more) equations setting each factor to zero, using the law: if ab=0 then a=0 or b=0.",
"You can always use the quadratic formula on any quadratic equation.",
"You can always use the quadratic formula on any quadratic equation.",
"Complete the square.", /* I don't think this is used in automode except in calculus */
"Take the square root of both sides.",
"You have an equation of fractions, with no obvious simplifications, so the thing to do is cross multiply.",
"If the discriminant is negative, a quadratic has no real roots.",
"You have two equations of the form $u^2 = a$ and $u^2 = -a$. They can be combined into $u^2 = |a|$.",
arithhint
},
{ /* numerical_equations */
"Evaluate numerically at a point.", /* Never used in auto mode */
"You could choose 'solve numerically' to let the computer find solutions using an iterative approximation method."
},
{ /* advanced_equations */
"You have an equation of fractions, with no obvious simplifications, so the thing to do is cross multiply.",
"You could raise both sides to a power, using the law, if $u=v$ then $u^n=v^n$.",
"In order to get at the unknown under the square root, take the square root of both sides.",
"In order to get at the unknown under the root, take the $n$-th root of both sides.",
"In order to get at the unknown, apply a suitable function to both sides.",
"Put your fractions over a common denominator.",
"Split your equation into two or more equations using the law: if ab=0 then a=0 or b=0",
"Split your equation into two or more equations using the law: if ab=ac then a=0 or b=c",
"Select one equation.", /* Not used in auto mode */
"Show all your equations again, you're finished with the one that is visible.",
"Collect multiple solutions.",
"Maybe you could make a helpful substitution. Select the expression to be replaced by a new variable.",
"Now eliminate your defined variable.",
"One of your equations is impossible--reject it.",
"Don't forget to check the roots in the original equation.",
"You could solve that linear equation at once."
},
{ /* cubic_equations */
"Make an appropriate substitution to eliminate the quadratic term.",
"The discriminant determines whether there are 3 real roots or only 1, and you have to compute it first to know which cubic formula to apply.",
"You must show the cubic equation again in order to continue working on it.",
"As Vieta discovered in 1592, when solving $cx^3 + ax + b = 0$, you can substitute $x = y - a/(3cy)$, which will produce an equation quadratic in $y^3$. Select the whole equation to see that choice.",
"Your cubic has only one real root, because the discriminant is positive.",
"Your cubic has three real roots, because the discriminant is negative.",
"Your cubic has only one real root, because the discriminant is positive.",
"Make a substitution $x = f(u)$ where $x$ is an old variable and $u$ is new.",
"Now it is time to get rid of the defined variable.",
"These two expressions will be the same if you change one of the integer variables. Select one of the integer variables and make a substitution. After that one equation will drop out. At present each equation stands for three roots, so there are apparently six roots, but really there are only three.",
"Evaluate the expressions for the roots to get the exact answers.",
"The best you can do is find approximate decimal values for the roots",
"Simplify"
},
{ /* logarithmic_equations */
"Try to get the logarithm in the exponent using the law: if $u=v$ then $a^u = a^v$.",
"Get rid of the logarithm on the left side using: if $ln u = v$ then $u = e^v$.",
"Get rid of the logarithm on the left side using: if $log u = v$ then $u = 10^v$.",
"Get rid of the logarithm on the left side using: if $log(b,u) = v$ then $u = b^v$.",
"Since both sides are powers, and the bases are the same, the exponents must be equal too.",
"Take the log of both sides.",
"Take the ln of both sides.",
"One of your equations is impossible--remember logarithms of negative numbers aren't defined."
},
{ /* cramers_rule */
"Use Cramer's rule",
"Evaluate the determinant. MathXpert will do that for you in one easy step."
},
{ /* several_linear_equations*/
"First get the variables on the left side and the constants on the right side.",
"Collect like terms, so that you have only one term in each variable.",
"Line up the variables nicely, so you can compare the coefficients in different equations easily.",
"Add two equations.",
"Subtract two equations.",
"Multiply an equation by a constant.",
"Divide an equation by a constant.",
"Add a multiple of an equation to another equation.",
"Subtract a multiple of an equation from another equation.",
"Swap two equations.",
"Put the solved equations in order.",
"Drop an identity.",
"Regard a variable as constant, so as to solve for the rest.",
"Can these equations actually be solved? It seems you may have a contradiction on your hands."
},
{ dummystring, /* selection_mode_only, these operators */
dummystring, /* are not used in automode so need no hints */
dummystring,
dummystring
},
{ /* linear_equations_by_selection */
"Add two equations",
"Subtract two equations",
"Multiply an equation by a constant.",
"Divide an equation by a constant.",
"Add a multiple of an equation to another equation.",
"Subtract a multiple of an equation from another equation.",
"Swap two equations.",
"Solve one of the unsolved equations for one variable in terms of the rest.",
"Add two rows.",
"Subtract one row from another.",
"Multiply some row by a constant.",
"Divide some row by a constant.",
"Add a multiple of one row to another row.",
"Subtract a multiple of one row from another row.",
"Swap two rows.",
"Write a matrix $A$ as the product $IA$, where $I$ is the matrix identity. Then when you perform row operations, the inverse of $A$ will develop where $I$ is."
},
{ /* linear_equations_by_substitution */
"Collect like terms, so that you have only one term in each variable.",
"Solve one of the unsolved equations for one variable in terms of the rest.",
"Simplify one or more of your equations.",
"Cancel a term that occurs on both sides of one of your equations.",
"Add something to both sides of one of your equations.",
"Subtract something from both sides of one of your equations.",
"Divide one of your equations by a constant to isolate a variable.",
"After you have expressed one variable in terms of the rest, use that equation to eliminate that variable from the other equations, by substituting for that variable.",
"Your equations are contradictory."
},
{ /* matrix_methods */
"To begin with, write your equations in matrix form.",
"Multiply the right side by the matrix identity $I$.",
"Swap two rows.",
"Add two rows.",
"Subtract one row from another.",
"Multiply some row by a constant.",
"Divide some row by a constant.",
"Add a multiple of one row to another row.",
"Subtract a multiple of one row from another row.",
"Multiply matrices.",
"A column entirely consisting of zeroes can be dropped.",
"A row consisting entirely of zeroes can be dropped.",
"A duplicate row can be dropped.",
"Your equations are contradictory.",
"A matrix equation can be converted to a system of ordinary equations."
},
{ /* advanced_matrix_methods */
"Multiply matrices.",
"Solve using a symbol for the matrix inverse: $AX = B => X = A^(-1)B$",
"There is an explicit formula for a 2 by 2 matrix inverse.",
"Ask MathXpert to compute the exact matrix inverse. Select the matrix inverse you want to compute.",
"You could ask MathXpert to compute the decimal matrix inverse. Select the matrix inverse you want to compute.",
},
{ /* absolute_value */
"For nonnegative $u$, you can get rid of absolute value signs using $|u| = u$.",
"You could always assume $u\\ge 0$ and set $|u| = u$.",
"For negative $u$, you can get rid of absolute value signs using $|u| = -u$.",
"You can pull a nonnegative quantity out of absolute value using $|cu| = c|u|$.",
"You can get a positive denominator out of absolute value using $|u/c| = |u|/c$.",
"You can simplify a product of absolute values using $|u||v| = |uv|$.",
"If it helps, you can break up an absolute value using $|uv| = |u||v|$.",
"Push absolute values into numerator and denominator using $|u/v| = |u| / |v|$.",
"Get absolute values out of your fraction using $|u| / |v| = |u/v|$",
"Even powers of absolute value can be simplified using $|u|^2^n=u^2^n$ if $u$ is real.",
"Absolute values of a power obey the law $|u^n|=|u|^n$ if $n$ is real.",
"Absolute values of square roots obey the law $|\\sqrt u| = \\sqrt |u|$.",
"Absolute values of roots obey the law $|^n\\sqrt u| = ^n\\sqrt |u|$.",
"You can cancel under the absolute value signs using the law $|ab|/|ac|=|b|/|c|$",
"You can cancel under the absolute value signs using the law $|ab|/|a|=|b|$",
"Maybe there is a common factor in what is inside the absolute values in numerator and denominator. If so it would be helpful to show it explicitly.",
},
{ /* absolute_value_ineq1 */
"If $c\\ge 0$, you can always split an equation $|u|=c$ into the two equations $u=c$, $u = -c$.",
"The equation $|u|/u=c$ has real solutions $u$ only when $c$ is 1 or $-1$, and then the solutions are $u = 1$, $u = -1$.",
"For $v\\ge 0$, $|u| < v$ iff $u$ is (strictly) between $-v$ and $v$",
"For $v\\ge 0$, $|u| \\le v$ iff $u$ is between $-v$ and $v$",
"$u < |v|$ iff $v < -u$ or $u < v$",
"$u \\le |v|$ iff $v \\le -u$ or $u \\le v$",
"An equation $|u| = u$ can be converted to an inequality $0 \\le u$, eliminating the absolute value sign.",
"An equation $|u| = -u$ can be converted to an inequality $u \\le 0$, eliminating the absolute value sign.",
"An absolute value can't be negative: $0 \\le |u|$ is always true.",
"An absolute value can't be negative: $|u| < 0$ is always false.",
"An absolute value can't be negative: $-c \\le |u|$ is always true provided $c$ is nonnegative.",
"An absolute value can't be negative: $-c < |u|$ is always true provided $c$ is positive.",
"An absolute value can't be negative: $|u| < -c$ is false, provided $c$ is nonnegative",
"An absolute value can't be negative: $|u| \\le -c$ is false, provided $c$ is positive",
"If $c \\ge 0$, the inequality $|u| \\le -c$ is possible only if $u$ and $c$ are both zero. In MathXpert, you handle this situation by using $|u| \\le -c$ iff $u=0$ assuming $c=0$. The assumption $c=0$ will be made. If it eventually contradicts $u=0$ there will be no solution. Otherwise you will find the solution by solving $u=0$.",
"If $c \\ge 0$, the equation $|u| = -c$ is possible only if $u$ and $c$ are both zero. In MathXpert, you handle this situation by using $|u| = -c$ iff $u=0$ assuming $c=0$. The assumption $c=0$ will be made. If it eventually contradicts $u=0$ there will be no solution. Otherwise you will find the solution by solving $u=0$."
},
{ /* absolute_value_ineq2 */
"$v>|u|$ iff $u$ is (strictly) between $-v$ and $v$",
"$v\\ge |u|$ iff $u$ is between $-v$ and $v$",
"$|v|>u$ iff $-u>v$ or $v>u$",
"$|v|\\ge u$ iff $-u\\ge v$ or $v\\ge u$",
"Absolute values are always nonnegative.",
"An absolute value cannot be negative.",
"An absolute value cannot be negative.",
"An absolute value cannot be negative.",
"If $c \\ge 0$, the inequality $-c \\ge |u|$ is possible only if $u$ and $c$ are both zero. In MathXpert, you handle this situation by using $|u| \\le -c$ iff $u=0$ assuming $c=0$. The assumption $c=0$ will be made. If it eventually contradicts $u=0$ there will be no solution. Otherwise you will find the solution by solving $u=0$.",
"An absolute value cannot be negative.",
"An absolute value cannot be negative.",
"For $v\\ge 0$, $|u| \\le v$ iff $u$ is between $-v$ and $v$",
"$u < |v|$ iff $v < -u$ or $u < v$",
"You can write an even power as a power of an absolute value",
"Absolute values of a power obey the law $|u|^n = |u^n|$ if $n$ is real."
},
{ /* less_than */
"$u < v$ means the same as $v > u$",
"Add a suitable term to both sides of your inequality.",
"Subtract a suitable term from both sides of your inequality.",
"Change the signs of both sides, but remember that will change the direction of the inequality: -u < -v => v < u",
"You can change the signs of both sides, but you must change $<$ to $>$ at the same time.",
"You can multiply both sides of an inequality by the same quantity $c$. But the sign of $c$ must be known, and if you only know $0 \\le c$ you will give up $<$ for $\\le $.",
"If you would like to multiply both sides by something, but you don't know whether it is positive or negative, you can always multiply by its square instead, since that is always non-negative.",
"You can divide both sides of an inequality by the same quantity $c$. But the sign of $c$ must be known.",
"When both sides are numbers, you can just evaluate the inequality numerically.",
"A square, or any even power, is always non-negative.",
"A square, or any even power, can never be negative.",
"Square both sides, which is legal since both sides are nonnegative.",
"Square both sides. Since the smaller side isn't obviously non-negative, you will get an extra inequality to account for the possibility that it is negative.",
"You have an inequality $u < v$ and the corresponding equation $u = v$; combine them.",
"Two of your solutions define overlapping intervals. Combine those intervals.",
"You have one or more solutions that do not satisfy the original inequality. Such solutions can be introduced by squaring an inequality or cancelling an expression. Use the assumptions to reject or correct these solutions.",
},
{ /* greater_than */
"$u > v$ means the same as $v < u$",
"You can change the signs of both sides, but you must change $>$ to $<$ at the same time.",
"You can change the signs of both sides and keep the same inequality sign by changing $-u > -v$ to $v > u$.",
"A square, or any even power, is always non-negative",
"A square, or any even power, can never be negative.",
"Square both sides. Since the smaller side isn't obviously non-negative, you will get an extra inequality to account for the possibility that it is negative.",
"You have an inequality $u > v$ and the corresponding equation $u = v$; combine them."
},
{ /* less_than_or_equal */
"$x \\le y$ means the same as $y \\ge x$",
"Add a suitable term to both sides of your inequality.",
"Subtract a suitable term from both sides of your inequality.",
"Change the signs of both sides, but remember that will change the direction of the inequality.",
"You can change the signs of both sides and keep the same inequality sign by changing $-u \\le -v$ to $v \\ge u$.",
"You can multiply both sides of an inequality by the same quantity, but you must know the sign, because $\\le $ must change to $\\ge $ when you multiply by a negative quantity.",
"If you would like to multiply both sides by something, but you don't know whether it is positive or negative, you can always multiply by its square instead, since that is always non-negative.",
"You can divide both sides of an inequality by the same quantity, but you must know the sign, because $<$ must change to $>$ when you divide by a negative quantity.",
"When both sides are numbers, you can just evaluate the inequality numerically.",
"A square, or any even power, is always non-negative",
"A square, or any even power, can never be negative.",
"Square both sides, which is legal since both sides are nonnegative.",
"Square both sides. Since the smaller side isn't obviously non-negative, you will get an extra inequality to account for the possibility that it is negative.",
"Two of your solutions define overlapping intervals. Combine those intervals.",
"You have one or more solutions that do not satisfy the original inequality. Such solutions can be introduced by squaring an inequality or cancelling an expression. Use the assumptions to reject or correct these solutions.",
},
{ /* greater_than_or_equal */
"$x \\ge y$ means the same as $y \\le x$",
"You can change the signs of both sides, but you must change $\\ge $ to $\\le $ at the same time.",
"You can change the signs of both sides and the same inequality sign by changing $-u \\ge -v$ to $v \\ge u$.",
"A square, or any even power, is always non-negative",
"A square, or any even power, can never be negative.",
"Square both sides. Since the smaller side isn't obviously non-negative, you will get an extra inequality to account for the possibility that it is negative."
},
{ /* square_ineq1 */
"You can take the square root of both sides, but you have to be careful: $u^2 < a => |u| < \\sqrt a$. Don't forget the absolute value.",
"Take the square root of both sides; you should get an interval between the two square roots of the constant side.",
"You can take the square root of both sides, but you have to be careful: $0 \\le u < v^2 => \\sqrt u < |v|$",
"When you take the square root of this inequality, you will get two inequalities, corresponding to the positive and negative square roots.",
"When you take the square root of this inequality, you will get two inequalities, corresponding to the positive and negative square roots.",
"Squares are always non-negative, so the first inequality can be dropped. Select the whole inequality to do this.",
"Squares are always non-negative, so the first inequality can be dropped. Select the whole inequality to do this.",
"Get rid of a square root or absolute value by squaring both sides of your inequality.",
"Get rid of a square root or absolute value by squaring both sides of your inequality.",
"Get rid of a square root or absolute value by squaring both sides of your inequality.",
"You can take the square root of both sides of an inequality if you know everything is nonnegative: $0 \\le u < v => \\sqrt u < \\sqrt v$",
"Squares are always non-negative.",
"Squares are always non-negative.",
"Square roots are always non-negative, but if you square a square root, don't forget that what is under the root must be nonnegative."
},
{ /* square_ineq2 */
"You can take the square root of both sides, but you have to be careful: $u^2 < a => |u| < \\sqrt a$. Don't forget the absolute value.",
"Take the square root of both sides; you should get an interval between the two square roots of the constant side.",
"You can take the square root of both sides, but you have to be careful: $0 \\le u < v^2 => \\sqrt u < |v|$",
"When you take the square root of this inequality, you will get two inequalities, corresponding to the positive and negative square roots.",
"When you take the square root of this inequality, you will get two inequalities, corresponding to the positive and negative square roots.",
"Squares are always non-negative, so the first inequality can be dropped. Select the whole inequality to do this.",
"Squares are always non-negative, so the first inequality can be dropped. Select the whole inequality to do this.",
"You have a square root. Get rid of it by squaring both sides of your inequality",
"You have a square root. Get rid of it by squaring both sides of your inequality",
"You have a square root. Get rid of it by squaring both sides of your inequality",
"You can take the square root of both sides of an inequality if you know everything is nonnegative: $0 \\le u < v => \\sqrt u < \\sqrt v$",
"Squares are always non-negative.",
"Squares are always non-negative.",
"Square roots are always non-negative, but if you square a square root, don't forget that what is under the root must be nonnegative."
},
{ /* recip_ineq1 */
"Take the reciprocal of both sides",
"Take the reciprocal of both sides",
"Take the reciprocal of both sides",
"Take the reciprocal of both sides",
"Take the reciprocal to get the unknown out of the denominator.",
"Take the reciprocal to get the unknown out of the denominator.",
"Take the reciprocal to get the unknown out of the denominator.",
"Take the reciprocal to get the unknown out of the denominator.",
"Take the reciprocal, but be careful when the interval includes zero!",
"Take the reciprocal, but be careful when the interval includes zero!"
},
{ /* recip_ineq2 */
"Take the reciprocal of both sides",
"Take the reciprocal of both sides",
"Take the reciprocal of both sides",
"Take the reciprocal of both sides",
"Take the reciprocal to get the unknown out of the denominator.",
"Take the reciprocal to get the unknown out of the denominator.",
"Take the reciprocal to get the unknown out of the denominator.",
"Take the reciprocal to get the unknown out of the denominator.",
"Take the reciprocal, but be careful when the interval includes zero!",
"Take the reciprocal, but be careful when the interval includes zero!"
},
{ /* root_ineq1 */
"You can take odd roots of both sides of any inequality.",
"You can take even roots of both sides, but you have to be careful: $u^2^n < a => |u| < ^2^n\\sqrt a$.",
"You can take even roots of both sides, but you will get a part corresponding to the negative root: $u^2^n < a$ iff $-^2^n\\sqrt a < u < ^2^n\\sqrt a$.",
"You can take even roots of both sides, but you have to be careful: $0 \\le a < u^2^n => ^2^n\\sqrt a < |u|$.",
"You can teke even roots of both sides, but you will get a part corresponding to the negative root: $a < u^2^n$ iff $v < -^2^n\\sqrt a$ or $^2^n\\sqrt a < u$.",
"You can take an even root of all three terms, but you will get an extra interval corresponding to the negative roots.",
"You have an $n$-th root. Get rid of it by raising both sides to the $n$-th power. But remember that even roots of negative numbers are not defined, so you must explicitly keep that condition. For example, $^4\\sqrt x < 16$ becomes $0 \\le x < 2$.",
"You have an $n$-th root. Get rid of it by raising both sides to the $n$-th power.",
"You have an $n$-th root. Get rid of it by raising both sides to the $n$-th power.",
"You have an $n$-th root. Get rid of it by raising both sides to the $n$-th power.",
"You can always raise both sides of any inequality to a positive odd power.",
"You can raise both sides of an inequality to any positive power, if both sides are known to be nonnegative.",
"Even-index roots are always non-negative, but if you raise such a root to a power, don't forget that what is under the root must be nonnegative."
},
{ /* root_ineq2 */
"You can take odd roots of both sides of any inequality.",
"You can take even roots of both sides, but you have to be careful: $u^2^n \\le a$ iff $|u| < ^2^n\\sqrt a$.",
"You can take even roots of both sides, but you will get a part corresponding to the negative root: $u^2^n \\le a$ iff $-^2^n\\sqrt a \\le u \\le ^2^n\\sqrt a$",
"You can take even roots of both sides, but you have to be careful: $0 \\le a \\le u^2^n$ iff $^2^n\\sqrt a \\le |u|$",
"You can teke even roots of both sides, but you will get a part corresponding to the negative root: $a \\le u^2^n$ iff $v \\le -^2^n\\sqrt a$ or $^2^n\\sqrt a \\le u$.",
"You can take an even root of all three terms, but you will get an extra interval corresponding to the negative roots.",
"You have an $n$-th root. Get rid of it by raising both sides to the $n$-th power. But remember that even roots of negative numbers are not defined, so you must explicitly keep that condition. For example, $^4\\sqrt x \\le 16$ becomes $0 \\le x \\le 2$.",
"You have an $n$-th root. Get rid of it by raising both sides to the $n$-th power.",
"You have an $n$-th root. Get rid of it by raising both sides to the $n$-th power.",
"You have an $n$-th root. Get rid of it by raising both sides to the $n$-th power.",
"You can always raise both sides of any inequality to a positive odd power.",
"You can raise both sides of an inequality to any positive power, if both sides are known to be nonnegative.",
"Even-index roots are always non-negative, but if you raise such a root to a power, don't forget that what is under the root must be nonnegative."
},
{ /* zero_ineq1 */
"You should drop any positive factors.",
"The numerator is positive, so the fraction is positive if and only if the denominator is positive.",
"In $0 < u/\\sqrt v$, multiply by $v\\sqrt v$, not just $\\sqrt v$, or you will lose domain information. Note that $v\\sqrt v$ is positive. The square roots will cancel out.",
"$u/v$ is positive if and only if $u$ and $v$ have the same sign. That's the same condition for $uv$ to be positive, and $0 < uv$ may be easier to work with than $0 < u/v$.",
"In $u/\\sqrt v < 0$, multiply by $v\\sqrt v$, not just $\\sqrt v$, or you will lose domain information. Note that $v\\sqrt v$ is positive. The square roots will cancel out.",
"$u/v$ is negative if and only if $u$ and $v$ have opposite signs. That's the same condition for $uv$ to be negative, and $uv < 0$ may be easier to work with than $u/v < 0$.",
"In solving a linear inequality, it may help to factor out the coefficient of the unknown: $ax \\pm b < 0$ iff $a(x\\pm b/a) < 0$.",
"$u < v$ means the same as $v > u$",
"When you have an inequality of the form $(x-a)(x-b) < 0$, the solution set is the interval between the zeroes of the quadratic, that is, $a < x < b$, if $a < b$.",
"When you have an inequality of the form $0 < (x-a)(x-b)$, say with $a < b$, the solution set is composed of all values not between the two roots, that is, $x < a$ or $b < x$."
},
{ /* zero_ineq2 */
"You should drop any positive factors.",
"The numerator is positive, so the fraction is non-negative if and only if the denominator is non-negative.",
"In $0 \\le u/\\sqrt v$, multiply by $v\\sqrt v$, not just $\\sqrt v$, or you will lose domain information. Note that $v\\sqrt v$ is positive. The square roots will cancel out.",
"$u/v$ is positive if and only if $u$ and $v$ have the same sign. That's the same condition for $uv$ to be positive, and $0 \\le uv$ may be easier to work with than $0 \\le u/v$.",
"In $u/\\sqrt v \\le 0$, multiply by $v\\sqrt v$, not just $\\sqrt v$, or you will lose domain information. Note that $v\\sqrt v$ is positive. The square roots will cancel out.",
"$u/v$ is negative if and only if $u$ and $v$ have opposite signs. That's the same condition for $uv$ to be negative, and $uv \\le 0$ may be easier to work with than $u/v \\le 0$.",
"In solving a linear inequality, it may help to factor out the coefficient of the unknown: $ax \\pm b < 0$ iff $a(x\\pm b/a) < 0$.",
"$u \\le v => v \\ge u$",
"When you have an inequality of the form $(x-a)(x-b) \\le 0$, the solution set is the interval between the zeroes of the quadratic, that is, $a \\le x \\le b$, if $a < b$.",
"When you have an inequality of the form $0 \\le (x-a)(x-b)$, say with $a < b$, the solution set is composed of all values not between the two roots, that is, $x \\le a$ or $b \\le x$."
},
{ /* square_ineq3 */
"You can take the square root of both sides, but you have to be careful: $a > u^2$ becomes $\\sqrt a > |u|$. Don't forget the absolute value.",
"Take the square root of both sides; you should get an interval between the two square roots of the constant side.",
"You can take the square root of both sides, but you have to be careful: $v^2 > a$ becomes $|v| > \\sqrt a$ provided $a > 0$.",
"When you take the square root of this inequality, you will get two inequalities, corresponding to the positive and negative square roots.",
"You have a square root. Get rid of it by squaring both sides of your inequality",
"You have a square root. Get rid of it by squaring both sides of your inequality",
"You have a square root. Get rid of it by squaring both sides of your inequality",
"You can take the square root of both sides of an inequality if you know everything is nonnegative: $0 \\le u < v$ implies $\\sqrt u < \\sqrt v$",
"Squares are always non-negative.",
"Squares are always non-negative.",
"Square roots are always non-negative, but if you square a square root, don't forget that what is under the root must be nonnegative."
},
{ /* square_ineq4 */
"You can take the square root of both sides, but you have to be careful: $a \\ge u^2$ becomes $\\sqrt a \\ge |u|$. Don't forget the absolute value.",
"Take the square root of both sides; you should get an interval between the two square roots of the constant side.",
"You can take the square root of both sides, but you have to be careful: $0 \\le u < v^2$ becomes $\\sqrt u < |v|$",
"When you take the square root of this inequality, you will get two inequalities, corresponding to the positive and negative square roots.",
"You have a square root. Get rid of it by squaring both sides of your inequality",
"You have a square root. Get rid of it by squaring both sides of your inequality",
"You have a square root. Get rid of it by squaring both sides of your inequality",
"You can take the square root of both sides of an inequality if you know everything is nonnegative: $0 \\le u < v => \\sqrt u < \\sqrt v$",
"Squares are always non-negative.",
"Squares are always non-negative.",
"Square roots are always non-negative, but if you square a square root, don't forget that what is under the root must be nonnegative."
},
{ /* recip_ineq3 */
"Take the reciprocal of both sides",
"Take the reciprocal of both sides",
"Take the reciprocal of both sides",
"Take the reciprocal of both sides"
},
{ /* recip_ineq4 */
"Take the reciprocal of both sides",
"Take the reciprocal of both sides",
"Take the reciprocal of both sides",
"Take the reciprocal of both sides"
},
{ /* root_ineq3 */
"You can take odd roots of both sides of any inequality.",
"You can take even roots of both sides, but you have to be careful: $a > u^2^n$ becomes $ ^2^n\\sqrt a > |u|$.",
"You can take even roots of both sides, but you will get a part corresponding to the negative root: $ a > u^2^n$ iff $-^2^n\\sqrt a < u < ^2^n\\sqrt a$.",
"You can take even roots of both sides, but you have to be careful: $0 \\le a < u^2^n$ becomes $^2^n\\sqrt a < |u|$.",
"You can teke even roots of both sides, but you will get a part corresponding to the negative root: $a < u^2^n$ iff $v < -^2^n\\sqrt a$ or $^2^n\\sqrt a < u$.",
"You can take an even root of all three terms, but you will get an extra interval corresponding to the negative roots.",
"You have an $n$-th root. Get rid of it by raising both sides to the $n$-th power.",
"You have an $n$-th root. Get rid of it by raising both sides to the $n$-th power.",
"You have an $n$-th root. Get rid of it by raising both sides to the $n$-th power.",
"You can always raise both sides of any inequality to a positive odd power.",
"You can raise both sides of an inequality to any positive power, if both sides are known to be nonnegative.",
"Even-index roots are always non-negative, but if you raise such a root to a power, don't forget that what is under the root must be nonnegative."
},
{ /* root_ineq4 */
"You can take odd roots of both sides of any inequality.",
"You can take even roots of both sides, but you have to be careful: $u^2^n \\le a iff |u| < ^2^n\\sqrt a$.",
"You can take even roots of both sides, but you will get a part corresponding to the negative root: $u^2^n \\le a$ iff $-^2^n\\sqrt a \\le u \\le ^2^n\\sqrt a$.",
"You can take even roots of both sides, but you have to be careful: $0 \\le a \\le u^2^n $iff $^2^n\\sqrt a \\le |u|$.",
"You can teke even roots of both sides, but you will get a part corresponding to the negative root: $a \\le u^2^n$ iff $ v \\le -^2^n\\sqrt a$ or $^2^n\\sqrt a \\le u$.",
"You can take an even root of all three terms, but you will get an extra interval corresponding to the negative roots.",
"You have an $n$-th root. Get rid of it by raising both sides to the $n$-th power.",
"You have an $n$-th root. Get rid of it by raising both sides to the $n$-th power.",
"You have an $n$-th root. Get rid of it by raising both sides to the $n$-th power.",
"You can always raise both sides of any inequality to a positive odd power.",
"You can raise both sides of an inequality to any positive power, if both sides are known to be nonnegative.",
"Even-index roots are always non-negative, but if you raise such a root to a power, don't forget that what is under the root must be nonnegative."
},
{ /* zero_ineq3 */
"The numerator is positive, so the fraction is positive if and only if the denominator is positive.",
"In $0 < u/\\sqrt v$, multiply by $v\\sqrt v$, not just $\\sqrt v$, or you will lose domain information. Note that $v\\sqrt v$ is positive. The square roots will cancel out.",
"$u/v$ is positive if and only if $u$ and $v$ have the same sign. That's the same condition for $uv$ to be positive, and $0 < uv$ may be easier to work with than $0 < u/v$.",
"In $u/\\sqrt v < 0$, multiply by $v\\sqrt v$, not just $\\sqrt v$, or you will lose domain information. Note that $v\\sqrt v$ is positive. The square roots will cancel out.",
"$u/v$ is negative if and only if $u$ and $v$ have opposite signs. That's the same condition for $uv$ to be negative, and $uv < 0$ may be easier to work with than $u/v < 0$.",
"In solving a linear inequality, it may help to factor out the coefficient of the unknown: $ax \\pm b < 0$ iff $a(x\\pm b/a) < 0$.",
"When you have an inequality of the form $(x-a)(x-b) < 0$, the solution set is the interval between the zeroes of the quadratic, that is, $a < x < b$, if $a < b$.",
"When you have an inequality of the form $0 < (x-a)(x-b)$, say with $a < b$, the solution set is composed of all values not between the two roots, that is, $x < a$ or $b < x$."
},
{ /* zero_ineq4 */
"The numerator is positive, so the fraction is non-negative if and only if the denominator is non-negative.",
"In $0 \\le u/\\sqrt v$, multiply by $v\\sqrt v$, not just $\\sqrt v$, or you will lose domain information. Note that $v\\sqrt v$ is positive. The square roots will cancel out.",
"$u/v$ is positive if and only if $u$ and $v$ have the same sign. That's the same condition for $uv$ to be positive, and $0 \\le uv$ may be easier to work with than $0 \\le u/v$.",
"In $u/\\sqrt v \\le 0$, multiply by $v\\sqrt v$, not just $\\sqrt v$, or you will lose domain information. Note that $v\\sqrt v $is positive. The square roots will cancel out.",
"$u/v$ is negative if and only if $u$ and $v$ have opposite signs. That's the same condition for $uv$ to be negative, and $uv \\le 0$ may be easier to work with than $u/v \\le 0$.",
"In solving a linear inequality, it may help to factor out the coefficient of the unknown: $ax \\pm b < 0$ iff $a(x\\pm b/a) < 0$.",
"When you have an inequality of the form $(x-a)(x-b) \\le 0$, the solution set is the interval between the zeroes of the quadratic, that is, $a \\le x \\le b$, if $a < b$.",
"When you have an inequality of the form $0 \\le (x-a)(x-b)$, say with $a < b$, the solution set is composed of all values not between the two roots, that is, $x \\le a$ or $b \\le x$."
},
{ /* binomial_theorem */
"Expand the power, using the binomial theorem.",
"Use the binomial theorem in the form with the binomial coefficients $(n k)$.",
"Express the binomial coefficients in terms of factorials, using $(n k) = n!/((n-k)!k!)$.",
"Use the definition of factorial, $n! = n(n-1)(n-2)...1$.",
"Compute the factorials explicitly.",
arithhint,
"Evaluate the binomial coefficients (n k).",
"Expand the $\\sum $ notation to an ordinary sum.",
"Evaluate the sum written in $\\sum $ notation to a rational number.",
"Use the recursion equation for the factorial function, $n! = n(n-1)$.",
"$n!$ is divisible by $n$, with quotient $(n-1)!$.",
"$n!$ is divisible by $(n-1)!$, with quotient $n$.",
"$n!$ is divisible by $k!$ when $k$ is less than $n$.",
"$n!$ is divisible by $n$, with quotient $(n-1)!$.",
"$n!$ is divisible by $(n-1)!$, with quotient $n$.",
"$n!$ is divisible by $k!$ when $k$ is less than $n$."
},
{ /* factor_expansion */
"Do you recognize the cube of a sum? Factor it.",
"Do you recognize the cube of a difference? Factor it.",
"Do you recognize the fourth power of a sum? Factor it.",
"Do you recognize the fourth power of a difference? Factor it.",
"Do you recognize a power of a sum? Factor it.",
"Do you recognize a power of a difference? Factor it."
},
{ /* sigma_notation */
"The summand doesn't depend on the index variable, so the sum is just the summand times the number of terms.",
"Try to get the minus sign outside the $\\sum $ sign.",
"Pull constants outside the $\\sum $ sign",
"Break the sum into two or more sums using $\\sum (u+v) = \\sum u + \\sum v$",
"Break the sum into two sums using $\\sum (u-v) = \\sum u - \\sum v$",
"Expand the sum written using $\\sum $ as an ordinary sum, written with $+$.",
"There is a formula for the sum of the first $n$ integers.",
"There is a formula for the sum of the first $n$ squares.",
"There is an explicit formula for the sum $1+x+..+x^n$.",
"Show the first few terms.", /* Not used in auto mode */
"Evaluate the sum written in $\\sum $ notation to a rational.",
"Evaluate to decimal.", /* Not used in auto mode */
"Evaluate the sum written in $\\sum $ notation to a rational.",
"Evaluate to decimal.", /* Not used in auto mode */
"Express the summand as a polynomial in the index variable.",
"This is a telescoping sum: part of each term cancels with the next term."
},
{ /* advanced_sigma_notation */
"Shift the summation index; that is, add something to both lower and upper limits and change the sum accordingly so it still represents the sum of the same terms.",
"Rename the index variable.",
"A product of two sums converts to a double sum: $(\\sum u)(\\sum v) = \\sum \\sum uv$",
"Split off the last term of the sum, so as to be able to use the induction hypothesis.",
"There is a formula for the sum of the first $n$ cubes.",
"There is a formula for the sum of the first $n$ fourth powers.",
"You can differentiate term by term. That is, the derivative of a sum is the sum of the derivatives.",
"Pull the derivative out of the sum. Select the whole sum to see this choice.",
"You can integrate term by term. The integral of an indexed sum is the sum of the integrals.",
"Pull the integral out of the sum. Select the whole sum to see this choice.",
"Push a constant into a sum.",
"If the lower index of summation were zero, you would be able to solve this.",
"If the lower index of summation were different, you would be able to solve this."
},
{ /* prove_by_induction */
"Select the induction variable.",
"Start with the basis case.",
"Start your induction step.",
"Now use your induction hypothesis.",
"You've got all the pieces. Just draw your final conclusion!"
},
{ /* trig_ineq */
"Remember that the sin function takes values between $-1$ and 1: $|sin u| \\le 1$",
"Remember that the cos function takes values between $-1$ and 1: $|cos u| \\le 1$",
"$sin u \\le u$ if $u\\ge 0$",
"$1 - u^2/2 \\le cos u$",
"By definition of the arctan function, we have $|arctan u| \\le \\pi /2$",
"$arctan u \\le u$ if $u\\ge 0$",
"$u \\le tan u$ if $u\\ge 0$"
},
{ /* log_ineq1 */
"You can take the ln of any inequality (if the sides are positive).",
"You can take the log of any inequality (if the sides are positive).",
"Try to eliminate logarithms by taking powers.",
"Try to eliminate logarithms by taking powers.",
"Try to eliminate logarithms by taking powers.",
"Try to eliminate logarithms by taking powers.",
"Try to eliminate logarithms by taking powers.",
},
{ /* log_ineq2 */
"You can take the ln of any inequality (if the sides are positive).",
"You can take the log of any inequality (if the sides are positive).",
"Try to eliminate logarithms by taking powers.",
"Try to eliminate logarithms by taking powers.",
"Try to eliminate logarithms by taking powers.",
"Try to eliminate logarithms by taking powers.",
"Try to eliminate logarithms by taking powers.",
},
{ /* log_ineq3 */
"You can take the ln of any inequality (if the sides are positive).",
"You can take the log of any inequality (if the sides are positive).",
"Try to eliminate logarithms by taking powers.",
"Try to eliminate logarithms by taking powers.",
"Try to eliminate logarithms by taking powers.",
"Try to eliminate logarithms by taking powers.",
"Try to eliminate logarithms by taking powers.",
},
{ /* log_ineq4 */
"You can take the ln of any inequality (if the sides are positive).",
"You can take the log of any inequality (if the sides are positive).",
"Try to eliminate logarithms by taking powers.",
"Try to eliminate logarithms by taking powers.",
"Try to eliminate logarithms by taking powers.",
"Try to eliminate logarithms by taking powers.",
"Try to eliminate logarithms by taking powers.",
"exponentials dominate polynomials",
"algebraic functions dominate logarithms"
},
{ /* logarithms_base10 */
"Remember that log $a$ is the number such that $$10^log a = a$$.",
"A log in the exponent can be simplified using the law: $$10^(n log a) = a^n$$",
"Remember $log 10^n = n$, at least for $n$ real.",
"Remember the logarithm of 1 is 0.",
"Remember log 10 is 1.",
"Express log in terms of ln using the conversion formula: $log a = (ln a)/(ln 10)$.",
"Any power $u^v$ can expressed using logarithms as $$10^(v log u)$$",
"If you factor a number, you can then break up its logarithm.",
"You can simplify a logarithm by factoring out powers of 10.",
"log(a/b) = -log(b/a)",
"log(b,a/c) = -log(b,c/a)"
},
{ /* logarithms */
"Break logs of powers up using $log a^n = n log a$.",
"To multiply, add logarithms: $log ab = log a + log b$",
"The log of the reciprocal is the negative log: $log 1/a = -log a$",
"To divide, subtract logarithms: $log a/b = log a - log b$",
"To multiply, add logarithms: $log a + log b = log ab$",
"To divide, subtract logarithms: $log a - log b = log a/b$",
"To multiply or divide, add or subtract logs: $log a + log b - log c =log ab/c$",
"You can push a factor inside the log using: $n log a = log a^n (n real)$",
"logs of square roots simplify according to: $log \\sqrt a = \\onehalf log a$",
"logs of roots simplify according to: $log ^n\\sqrt a = (1/n) log a$",
"The log of 1 is 0.",
"Factor a number completely to help simplify its logarithm.",
"Factor out powers of 10 to help simplify the logarithm.",
"Try writing $log(u)$ as $1/a log u^a$",
"You could evaluate logs numerically.",
"Express log in terms of ln using the conversion formula: $log a = (ln a)/(ln 10)$."
},
{ /* logarithms_base_e */
"a logarithm in the exponent can be simplified by the law: $e^ln a = a$",
"ln e = 1",
"ln 1 = 0",
"$ln e^n = n$ ($n$ real)",
"You can write any power $u^v$ in the form $$e^(v ln u)$$.",
"a logarithm in the exponent can be simplified by the law: $$e^((ln c) a) = c^a$$"
},
{ /* natural_logarithms */
"$ln a^n = n ln a$.",
"To multiply, add logarithms: $ln ab = ln a + ln b$.",
"The ln of a reciprical is the negative ln: $ln 1/a = -ln a$.",
"To divide, subtract lns: $ln a/b = ln a - ln b$.",
"ln 1 = 0",
"Factor a number completely.",
"Sums of natural logarithms combine according to: $ln a + ln b = ln ab$.",
"Differences of natural logarithms combine according to: $ln a - ln b = ln a/b$.",
"To multiply or divide, add or subtract natural logarithms: $ln a + ln b - ln c = ln (ab/c)$.",
"$n ln a = ln a^n$ ($n$ real)",
"natural logarithms of square roots simplify by: $ln \\sqrt a = \\onehalf ln a$.",
"natural logarithms of roots simplify by: $ln ^n\\sqrt a = (1/n) ln a$.",
"Try writing $ln(1+v)$ as $v ln((1+v)^(1/v))$, and then use the limit definition of $e$",
"Evaluate numerically.",
"ln(a/b) = -ln(b/a)"
},
{ /* reverse_trig */
"Use the formula for the sine of a sum in reverse.",
"Use the formula for the sine of a difference in reverse.",
"Use the formula for the cosine of a sum in reverse.",
"Use the formula for the cosine of a difference in reverse.",
"Use one of the formulas for the tangent of a half angle in reverse.",
"Use one of the formulas for the tangent of a half angle in reverse.",
"Use one of the formulas for the cotangent of a half angle in reverse.",
"Use one of the formulas for the cotangent of a half angle in reverse.",
"Use the formula for the tangent of a sum in reverse.",
"Use the formula for the tangent of a difference in reverse.",
"Use the formula for the cotangent of a sum in reverse.",
"Use the formula for the cotangent of a difference in reverse.",
"Express $1 - cos \\theta $ as $2 sin^2(\\theta /2)$"
},
{ /* complex_polar_form */
"Express the complex number in polar form",
"Express the complex exponential using $sin$ and $cos$",
"The complex exponential represents a point on the unit circle, which therefore has absolute value 1.",
"The complex exponential represents a point on the unit circle, which therefore has absolute value 1.",
"The complex exponential represents a point on the unit circle, which therefore has absolute value 1.",
"The minus sign must be eliminated using $-a = ae^(i\\pi )$.",
"$^n\\sqrt (-a)$ does not equal $-^n\\sqrt a$ when complex numbers are in use. Instead, a complex factor appears: $$sqrt (-a) = e^(pi i/n) root(n,a)$$.",
"Complex exponents should be brought to the numerator.",
"Use de Moivre's theorem, which gives a formula for the $n$ complex $n$-th roots of a number.",
"Substitute specific integers for the integer parameter to obtain a complete list of specific solutions."
},
{ /* logs_to_any_base */
"Use the definition of logarithms: $$b^(log(b,a)) = a$$",
"A logarithm in the exponent can be simplified by the law: $$b^(n log(b,a)) = a^n$$",
"$$log(b,b) = 1$$",
"$$log(b,b^n) = n$$",
"A log of a product can be simplified using the law: $log xy = log x + log y$",
"The log of a reciprocal can be simplified using the law: $log (1/x) = -log x$",
"To divide, subtract logarithms: $log x/y = log x-log y$",
"$$log(b,1) = 0$$",
"Factor the base of logarithms",
"$$log(b^n,x) = (1/n) log (b,x)$$",
"$log x^n = n log x$",
"Factor out powers of the base of logarithms.",
"$log x + log y = log xy$",
"$log x - log y = log x/y$",
"$log x + log y - log z =log xy/z$",
"$n log x = log x^n$ ($n$ real)"
},
{ /* change_base */
"Change the logarithms to natural logarithms.",
"Change the logarithms to base 10.",
"Change the base of the logarithms.",
"Change the logs to a common base, using the law $$log(b^n,x) = (1/n) log (b,x)$$",
"Base 10 logarithms can be written as log",
"Base $e$ logarithms are written as ln",
"Change log to ln",
"Change ln to log",
"Express the power with the variable in the exponent, using $$u^v = b^(v log(b,u))$$."
},
{ /* evaluate_trig_functions */
"The sine function is zero at zero.",
"The cosine function is one at zero.",
"The tangent function is zero at zero.",
"The zeroes of the sin function are at multiples of $\\pi $",
"cos takes the value 1 at even multiples of $\\pi $",
"The zeros of the tangent function are at multiples of $\\pi $",
"Since the trig functions are periodic, you should find a coterminal angle less than $360\\deg $. Select a trig function with an argument in the wrong range.",
"Since the trig functions are periodic, you should find a coterminal angle less than $2\\pi $. Select a trig function with an argument in the wrong range.",
"The values of the trig functions when the angle is a multiple of $90\\deg $ are known.",
"Use the relationships in a $1-2-\\sqrt 3$ triangle.",
"Use the relationships in a $1-1-\\sqrt 2$ triangle.",
"Change radians to degrees.",
"Change degrees to radians.",
"Express the angle in the form $a 30\\deg + b 45\\deg $; then you can use sum formulas to break it up. ",
"Evaluate numerically"
},
{ /* basic_trig */
"Express tan in terms of sin and cos",
"Express cot in terms of tan",
"Express cot in terms of cos and sin",
"Express sec in terms of cos",
"Express csc in terms of sin",
"Combine sin and cos into tan",
"Combine cos and sin int cot"
},
{ /* trig_reciprocals */
"Change $1 / sin$ to csc",
"Change $1 / cos$ to sec",
"Change $1 / tan$ to cot",
"Change $1 / tan$ to $cos / sin$",
"Change $1 / cot$ to tan",
"Change $1 / cot$ to $sin / cos$",
"Change $1 / sec$ to cos",
"Change $1 / csc$ to sin",
"Express sin in terms of csc",
"Express cos in terms of sec",
"Express tan in terms of cot"
},
{ /* trig_squares */
"Use the law $sin^2 u + cos^2 u = 1$.",
"Notice an expression matching the pattern $1 - sin^2 u$.",
"Notice an expression matching the pattern $1 - cos^2 u$",
"Try rewriting $sin^2$ as $1 - cos^2$",
"Try rewriting $cos^2$ as $1 - sin^2$",
"Use the law $sec^2 u - tan^2 u = 1$.",
"Notice an expression matching the pattern $tan^2 u + 1$.",
"Notice an expression matching the pattern $sec^2 u - 1$.",
"Try rewriting $sec^2$ as $tan^2 + 1$",
"Try rewriting $tan^2$ as $sec^2 u - 1$",
"Get rid of all powers of $sin$ using $sin^(2n+1) u = sin u (1-cos^2 u)^n$",
"Get rid of all powers of $cos$ using $cos^(2n+1) u = cos u (1-sin^2 u)^n$",
"Get rid of all powers of $tan$ using $tan^(2n+1) u = tan u (sec^2 u-1)^n$",
"Get rid of all powers of $sec$ using $sec^(2n+1) u = sec u (tan^2 u+1)^n$",
"Combine powers of $(1-cos t)$ and powers of $(1+cos t)$ to a power of $sin^2 t$",
"Combine powers of $(1-sin t)$ and powers of $(1+sin t)$ to a power of $cos^2 t$"
},
{ /* csc_and_cot_identities */
"Notice an expression matching the pattern $csc^2 u - cot^2 u$",
"Notice an expression matching the pattern $cot^2 u + 1$",
"Notice an expression matching the pattern $csc^2 u - 1$",
"Try rewriting $csc^2$ as $cot^2 + 1$",
"Try rewriting $cot^2$ as $csc^2 - 1$",
"Express $csc(\\pi /2-\\theta )$ in terms of $sec \\theta $",
"Express $cot(\\pi /2-\\theta )$ in terms of $tan \\theta $",
"Get rid of all powers of $cot$ using $cot^(2n+1) u = cot u (csc^2 u-1)^n$",
"Get rid of all powers of $csc$ using $csc^(2n+1) u = csc u (cot^2 u+1)^n$"
},
{ /* trig_sum */
"Use the formula for $sin(u+v)$",
"Use the formula for $sin(u-v)$",
"Use the formula for $cos(u+v)$",
"Use the formula for $cos(u-v)$",
"Use the formula for $tan(u+v)$",
"Use the formula for $tan(u-v)$",
"Use the formula for $cot(u+v)$",
"Use the formula for $cot(u-v)$"
},
{ /* double_angle */
"Use the double-angle formula for sin",
"You have a formula of the form $cos(2\\theta )$. There are three different double-angle formulas beginning with $cos(2\\theta )$. Choose carefully, thinking about what will come next.",
"You have a formula of the form $cos(2\\theta )$. There are three different double-angle formulas beginning with $cos(2\\theta )$. Choose carefully, thinking about what will come next.",
"You have a formula of the form $cos(2\\theta )$. There are three different double-angle formulas beginning with $cos(2\\theta )$. Choose carefully, thinking about what will come next.",
"Select the sum containing $cos(2\\theta )+1$.",
"Select the sum containing $cos(2\\theta )-1$.",
"Use the double-angle formula for tan",
"Use the double-angle formula for cot",
"A product of sin times cos can be simplified to a single trig function using the law: $sin \\theta cos \\theta = \\onehalf sin 2\\theta $",
"A product of sin times cos can be simplified to a single trig function using the law: $2 sin \\theta cos \\theta = sin 2\\theta $",
"Combine some terms to get the cosine of a double angle.",
"Combine some terms to get the cosine of a double angle.",
"Combine some terms to get the cosine of a double angle."
},
{ /* multiple_angles */
"Expand a trig function by writing $n\\theta $ as $(n-1)\\theta + \\theta $ and using a sum formula.",
dummystring, /* not used in auto mode */
"There is a formula for expanding $sin(3\\theta )$.",
"There is a formula for expanding $cos(3\\theta )$.",
"You can expand $sin n\\theta $ as a polynomial in $sin \\theta $ and $cos \\theta $.",
"You can expand $cos n\\theta $ as a polynomial in $sin \\theta $ and $cos \\theta $."
},
{ /* verify_identities */
"You could cross multiply.",
"You might switch the sides.",
"Transfer a suitable term from left to right.",
"Transfer a suitable term from right to left.",
"Add something to both sides.",
"Subtract something from both sides.",
"Multiply both sides by something.",
"Cancel a term from both sides.",
"Raise both sides to the same power.",
"Take the square root of both sides.",
"Take the $n$-th root of both sides.",
"Apply a function to both sides.",
arithhint,
"Perhaps it isn't even a true identity. Check it numerically. If it's not an identity, you should soon find a number that makes the sides unequal.",
"Make a substitution."
},
{ /* solve_by_30_60_90 */
"When does $sin(u) = 1/2$ ?",
"When does $sin(u) = -1/2$ ?",
"When does $sin(u) = \\sqrt 3/2$ ?",
"When does $sin(u) = -\\sqrt 3/2$ ?",
"When does $cos(u) = \\sqrt 3/2$ ?",
"When does $cos(u) = -\\sqrt 3/2$ ?",
"When does $cos(u) = 1/2$ ?",
"When does $cos(u) = -1/2$ ?",
"When does $tan(u) = 1/\\sqrt 3$ ?",
"When does $tan(u) = -1/\\sqrt 3$ ?",
"When does $tan(u) = \\sqrt 3$ ?",
"When does $tan(u) = -\\sqrt 3$ ?"
},
{ /* solve_by_45_45_90 */
"When does $sin(u) = 1/\\sqrt 2$ ?",
"When does $sin(u) = -1/\\sqrt 2$ ?",
"When does $cos(u) = 1/\\sqrt 2$ ?",
"When does $cos(u) = -1/\\sqrt 2$ ?",
"When does $tan(u) = 1$ ?",
"When does $tan(u) = -1$ ?"
},
{ /* zeroes_of_trig_functions */
"When does $sin u = 0$ ?",
"When does $sin u = 1$ ?",
"When does $sin u = -1$ ?",
"When does $cos u = 0$ ?",
"When does $cos u = 1$ ?",
"When does $cos u = -1$ ?",
"When does $tan u = 0$ ?",
"When does $cot u = 0$ ?"
},
{ /* inverse_trig_functions */
"You can get rid of the sin by taking the arcsin, but there will be multiple solutions.",
"You can get rid of the sin by taking the arcsin, but there will be multiple solutions.",
"You can get rid of the cos by taking the arccos, but there will be multiple soltuions.",
"Try taking the arctan to get rid of the tangent.",
"Evaluate the arcsin exactly.",
"Evaluate the arccos exactly.",
"Evaluate the arctan exactly.",
"Get rid of the arccot, using the law $arccot x = arctan (1/x)$",
"Get rid of the arcsec, using the law $arcsec x = arccos (1/x)$",
"Get rid of the arccsc, using the law $arccsc x = arcsin (1/x)$",
"arcsin is an odd function.",
"Although arccos is neither an odd nor an even function, it does satisfy the law $arccos(-x) = \\pi -arccos x$",
"arctan is an odd function",
"Your solutions involve an integer parameter, so there are infinitely many of them. If the original equation is periodic with period $2\\pi $, you should rewrite your solutions so the solutions have the form $c + 2n\\pi $. Then you will only need to check the solutions in one period.",
"Remember the values of sin are all between $-1$ and 1.",
"Remember the values of cos are all between $-1$ and 1."
},
{ /* invsimp */
"$tan(arcsin x)$ is actually an algebraic function of $x$.",
"$tan(arccos x)$ is actually an algebraic function of $x$.",
"$tan(arctan x)$ is just $x$.",
"$sin(arcsin x)$ is just $x$.",
"$sin(arccos x)$ is actually an algebraic function of $x$.",
"$sin(arctan x)$ is actually an algebraic function of $x$.",
"$cos(arcsin x)$ is actually an algebraic function of $x$.",
"$cos(arccos x)$ is just $x$.",
"$cos(arctan x)$ is actually an algebraic function of $x$.",
"$sec(arcsin x)$ is actually an algebraic function of $x$.",
"$sec(arccos x)$ is just $1/x$.",
"$sec(arctan x)$ is actually an algebraic function of $x$.",
"$arctan(tan \\theta )$ is just $\\theta $, if $-\\pi /2\\le \\theta \\le \\pi /2$",
"$arcsin(sin \\theta )$ is just $\\theta $, if $-\\pi /2\\le \\theta \\le \\pi /2$",
"$arccos(cos \\theta )$ is just $\\theta $, if $0\\le \\theta \\le \\pi $",
"$arctan(tan x)$ in general is not equal to $x$, but it is $x$ minus a certain multiple of $pi$, so it can be espressed as $x + c1$ where $c1$ is constant on intervals where $tan x$ is defined."
},
{ /* adding_arctrig_functions */
"$arcsin x$ and $arccos x$ are complementary angles.",
"$arctan x$ and $arctan 1/x$ are complementary angles, but watch out for the signs if $x$ is negative."
},
{ /* complementary_trig */
"Remember cos means sin of the complement. So the cosine of the complement is the sin. That is, $cos(\\pi /2-\\theta ) = sin \\theta $.",
"Remember cos means sin of the complement. That is, $sin(\\pi /2-\\theta ) = cos \\theta $.",
"Remember cot means tan of the complement. So the cot of the complement is the tan. That is, $cot(\\pi /2-\\theta ) = tan \\theta $.",
"Remember cot means tan of the complement. That is, $tan(\\pi /2-\\theta ) = cot \\theta $.",
"Remember csc means sec of the complement. So the csc of the complement is the sec. That is, $csc(\\pi /2-\\theta ) = sec \\theta $.",
"Remember csc means sec of the complement. That is, $sec(\\pi /2-\\theta ) = csc \\theta $.",
"Rewrite the sine as cosine of the complement.",
"Rewrite the cosine as sine of the complement.",
"Rewrite the tangent as cot of the complement.",
"Rewrite the cotangent as tan of the complement.",
"Rewrite the secant as csc of the complement.",
"Rewrite the cosecant as sec of the complement."
},
{ /* complementary degrees */
"Remember cos means sin of the complement. So the cosine of the complement is the sin. That is, $cos(\\pi /2-\\theta ) = sin \\theta $.",
"Remember cos means sin of the complement. That is, $sin(90\\deg -\\theta ) = cos \\theta $.",
"Remember cot means tan of the complement. So the cot of the complement is the tan. That is, $cot(\\pi /2-\\theta ) = tan \\theta $.",
"Remember cot means tan of the complement. That is, $tan(90\\deg -\\theta ) = cot \\theta $.",
"Remember csc means sec of the complement. So the csc of the complement is the sec. That is, $csc(\\pi /2-\\theta ) = sec \\theta $.",
"Remember csc means sec of the complement. That is, $sec(90\\deg -\\theta ) = csc \\theta $.",
"Rewrite the sine as cosine of the complement.",
"Rewrite the cosine as sine of the complement.",
"Rewrite the tangent as cot of the complement.",
"Rewrite the cotangent as tan of the complement.",
"Rewrite the secant as csc of the complement.",
"Rewrite the cosecant as sec of the complement.",
"Combine the degrees into a single expression.",
"Combine the degrees into a single expression.",
"Combine the degrees into a single expression."
},
{ /* trig_odd_and_even */
"sin is an odd function.",
"cos is an even function.",
"tan is an odd function.",
"cot is an odd function.",
"sec is an even function.",
"csc is an odd function.",
"sin squared is an even function.",
"cos squared is an even function.",
"tan squared is an even function.",
"cot squared is an even function.",
"sec squared is an even function.",
"csc squared is an even function.",
},
{ /* trig_periodic */
"sin is periodic; use the formula expressing this fact.",
"cos is periodic; use the formula expressing this fact.",
"tan is periodic; use the formula expressing this fact.",
"sec is periodic; use the formula expressing this fact.",
"csc is periodic; use the formula expressing this fact.",
"cot is periodic; use the formula expressing this fact.",
"$sin^2$ is periodic withx period $\\pi $, even though the period of sin is $2\\pi .$",
"$cos^2$ is periodic with period $\\pi $, even though the period of cos is $2\\pi .$",
"$sec^2$ is periodic with period $\\pi $, even though the period of sec is $2\\pi .$",
"$csc^2$ is periodic with period $\\pi $, even though the period of csc is $2\\pi .$",
"Reduce the angle using $sin u = -sin(u-\\pi )$",
"Reduce the angle using $sin u = sin(\\pi -u)$",
"Reduce the angle using $cos u = -cos(u-\\pi )$",
"Reduce the angle using $cos u = -cos(\\pi -u)$"
},
{ /* half_angle_identities */
"Get rid of $sin^2$ using a half-angle identity.",
"Get rid of $cos^2$ using a half-angle identity.",
"Get rid of $sin^2$ using a half-angle identity.",
"Get rid of $cos^2$ using a half-angle identity.",
"A product of sin and cos can be simplified using the law: $sin \\theta cos \\theta = \\onehalf sin 2\\theta $",
"Use a half-angle identity",
"Use a half-angle identity",
"Use a half-angle identity",
"Use a half-angle identity",
"Use a half-angle identity",
"Use a half-angle identity",
"Use a half-angle identity",
"Use a half-angle identity",
"Write $\\theta $ as $2(\\theta /2)$; you can find this operation with the half-angle identities."
},
{ /* product_and_factor_identities */
"You can express $sin x cos x$ as $\\onehalf sin 2x$",
"You can write $sin x cos y$ as a sum of sines whose frequencies are the sum and difference of $x$ and $y$",
"You can write $cos x sin y$ as a difference of sines whose frequencies are the sum and difference of $x$ and $y$",
"You can write $sin x sin y$ as a difference of cosines whose frequencies are the sum and difference of $x$ and $y$",
"You can write $cos x cos y$ as a sum of cosines whose frequencies are the sum and difference of $x$ and $y$",
"You can write $sin x + sin y$ as a product of sines and cosines whose frequencies are the sum and difference of $x$ and $y$",
"You can write $sin x - sin y$ as a product of sines and cosines whose frequencies are the sum and difference of $x$ and $y$",
"You can write $cos x + cos y$ as a product of cosines whose frequencies are the sum and difference of $x$ and $y$",
"You can write $cos x - cos y$ as a product of sines whose frequencies are the sum and difference of $x$ and $y$",
"Substitute u,v for the expressions in the trig functions."
},
{ /* limits */
"Experiment numerically.", /* Not used in auto mode */
"The limit of a sum is the sum of the limits, at least if the limits exist.",
"The limit of a difference is the difference of the limits, at least if the limits exist.",
"The limit of a constant is that constant.",
"The limit of $x$ as $x$ goes to $c$ is just $c$ itself.",
"You can pull a constant through the limit.",
"You can pull a minus sign through the limit.",
"The limit of a product is the product of the limits, at least if the limits exist.",
"The limit of a (constant) power is the power of the limit.",
"The limit of $c^v$ is $c$ raised to the power $lim v$, when $c$ is constant.",
"$$lim(t->a, u^v)= lim(t->a, u)^lim(t->a, v)$$",
"The limit of a square root is the square root of the limit, provided it is positive.",
"The limit of an odd root is the root of the limit.",
"The limit of a root is the root of the limit, provided it is positive.",
"You can use MathXpert to calculate limits of polynomials in one step.",
"Push the limit inside the absolute value sign."
},
{ /* limits_of_quotients */
"You can pull a constant out from the numerator using $lim cu/v = c lim u/v$",
"The limit of a reciprocal is the reciprocal of the limit; more generally for $c$ constant we have $lim c/v = c/lim v$",
"The limit of a quotient is the quotient of the limits, at least if the limit in the denominator is nonzero.",
"Factor out powers of $(x-a)$ in a limit as $x$ approaches $a$.",
"You can use MathXpert to calculate the limit of a rational function in one step.",
"Sometimes it helps to write $a^n/b^n as (a/b)^n$.",
"Rationalize the fraction. Look for that operation with the limit of quotients operations.",
"Simplify your limit by pulling out a simple part of it which has a nonzero finite limits. This means to express $lim uv$ as $lim u lim v$, where $lim u$ is finite and nonzero. For example, you might pull out $sin(x)/x$ from the limit of $sin^2(x) /x$ as $x$ approaches 0.",
"Factor out a constant.",
"Multiply both numerator and denominator by something. The aim is to make the limit in the denominator nonzero.",
"Divide both numerator and denominator by something. The aim is to make the limit in the denominator nonzero.",
"Divide both numerator and denominator by something and then push the limit into numerator and denominator. Choose the quantity to divide by so that the denominator will have a nonzero limit.",
"With the limits of quotients operations you will find an algebraic formula which may be helpful: $$(ab+ac+d)/q = a(b+c)/q + d/q$$"
},
{ /* quotients_of_roots */
"You can bring the denominator inside the square root (squaring it).",
"You can bring the denominator under the square root (squaring it), but watch out for the sign.",
"You can bring the denominator under the radical.",
"You can bring the denominator under the radical, but watch out for the sign.",
"You can bring the numerator inside the square root (squaring it).",
"You can bring the numerator under the square root (squaring it), but watch out for the sign.",
"You can bring the numerator under the radical.",
"You can bring the numerator under the radical, but watch out for the sign."
},
{ /* lhopitalmenu */
"Use L'Hospital's rule.",
"You can ask MathXpert to evaluate the derivative in one step",
"Put everything but the logarithm in the denominator, and then use L'Hospital's rule. Select the whole limit term to find the right operation.",
"Put everything but the logarithm in the denominator, and then use L'Hospital's rule. Select the whole limit term to find the right operation.",
"Put the negative exponent in the denominator as a positive exponent, and then use L'Hospital's rule.",
"Move the exponential function to the denominator, and then use L'Hospital's rule.",
"Move a trig function to the denominator (using a trig identity), and then use L'Hospital's rule.",
"Convert the product to a fraction by moving one or more factors to the denominator, creating a compound fraction.",
"Put the fractions over a common denominator and simplify."
},
{ /* special_limits */
"There is a special limit formula involving $(sin t)/t$",
"There is a special limit formula involving $(tan t)/t$",
"There is a special limit formula involving $(1-cos t)/t$",
"There is a special limit formula involving $(1-cos t)/t^2$",
"There is a special limit formula involving $(1+t)^(1/t)$",
"There is a special limit formula involving $(ln(1+t))/t$",
"There is a special limit formula involving $(e^t-1)/t$",
"There is a special limit formula involving $(e^(-t)-1)/t$",
"The singularity of $ln x$ at the origin is so weak that any positive power of $t$ will kill it. MathXpert has an operation for handling such a limit in one step, or you can put the power in the denominator and use L'Hospital's rule.",
"The function $cos(1/t)$ makes infinitely many oscillations between -1 and 1 as $t$ approaches 0.",
"The function $sin(1/t)$ makes infinitely many oscillations between -1 and 1 as $t$ approaches 0.",
"The function $tan(1/t)$ behaves quite wildly as $t$ approaches 0.",
"The function $cos t$ makes infinitely many oscillations between -1 and 1 as $t$ approaches infinity.",
"The function $sin t$ makes infinitely many oscillations between -1 and 1 as $t$ approaches infinity.",
"The function $tan t$ takes on all real values for arbitrarily large $t$, so it can't approach any limit as $t$ approaches infinity."
},
{ /* hyper_limits */
"There is a special limit formula involving $(sinh t)/t$",
"There is a special limit formula involving $(tanh t)/t$",
"There is a special limit formula involving $(cosh t -1)/t$",
"There is a special limit formula involving $(cosh t - 1)/t^2$"
},
{ /* advanced_limits */
"The limit of a ln is the ln of the limit, at least if it's positive.",
"Limits of continuous functions are calculated by $lim f(u)=f(lim u)$. In fact, this is the definition of continuity.",
"You can change the limit variable using the formula for composition of functions. Namely, $$lim(t->a,f(g(t)))=lim(u->g(a),f(u))$$",
"You can ask MathXpert to evaluate a simple limit in one step.",
"To calculate the limit of a non-constant power, first make the base be constant, using the law $$lim(t->a, u^v) = lim(t->a, e^(v ln u))$$.",
"If the limit of a product seems to be indeterminate, you can try the law: $lim uv = lim v/(1/u)$. Sometimes the resulting limit of a quotient can be evaluated.",
"A limit is undefined if the function whose limit is being taken limit fails to be defined in a suitable neighborhood of the limit point.",
"Try the law: $$lim(t->a, u) = e^(lim(t->a, ln u))$$",
"Maybe you can remove a troublesome term, perhaps an oscillatory factor, using the squeeze theorem.",
"You can try something similar to rationalizing the numerator, even though there is no numerator: $$lim(t->a, sqrt(u)-v)=lim(t->a, (sqrt(u)-v)(sqrt(u)+v)/(sqrt(u)+v))$$",
"You can neglect all but the leading terms in numerator and denominator.",
"A complicated limit can be replaced by the limit of the leading term.",
"You can replace a sum by its leading term in a limit under certain conditions, but not always. You must take care that the leading terms don't cancel out to zero, causing you to lose the real answer among the terms you neglected.",
"An expression with undefined parts is itself undefined",
"$$lim(t->a,e^u) = e^(lim(t->a, u))$$",
"$lim(ln u) = ln(lim u)$"
},
{ /* logarithmic_limits */
"The singularity of $ln x$ at the origin is so weak that any positive power of $t$ will kill it. MathXpert has an operation for handling such a limit in one step, or you can put the power in the denominator and use L'Hospital's rule.",
"The singularity of $ln x$ at the origin is so weak that any positive power of $t$ will kill it. MathXpert has an operation for handling such a limit in one step, or you can put the power in the denominator and use L'Hospital's rule.",
"The singularity of $ln x$ at the origin is so weak that any positive power of $t$ will kill it. MathXpert has an operation for handling such a limit in one step, or you can put the power in the denominator and use L'Hospital's rule.",
"The singularity of $ln x$ at the origin is so weak that any positive power of $t$ will kill it. MathXpert has an operation for handling such a limit in one step, or you can put the power in the denominator and use L'Hospital's rule.",
"An algebraic function always dominates a logarithm.",
"An algebraic function always dominates a logarithm.",
"An algebraic function always dominates a logarithm.",
"An algebraic function always dominates a logarithm.",
"An algebraic function always dominates a logarithm.",
"An algebraic function always dominates a logarithm.",
"An algebraic function always dominates a logarithm.",
"An algebraic function always dominates a logarithm.",
},
{ /* limits_at_infinity */
"For $t$ large, $t^n$ is large too, so $1/t^n$ is small.",
"For $t$ large, $t^n$ is large too.",
"For $t$ large, $e^t$ is large too.",
"For $t$ large and negative, $e^t$ is very small.",
"For $t$ large, $ln t$ is large too.",
"For $t$ large, $\\sqrt t$ is large, too.",
"For $t$ large, $^n\\sqrt t$ is large, too.",
"For $|t|$ large, $arctan t$ is close to $\\pi/2$ or $-\\pi/2$",
"The arccot of a large positive number is close to zero.",
"The arccot of a large negative number is close to $\\pi$",
"For $|t|$ large, $tanh t$ is close to 1 or -1.",
"$lim \\sqrt u-v=lim (\\sqrt u-v)(\\sqrt u+v)/\\sqrt u+v)$",
"$lim(sin u) = sin(lim u)$ if the limit is finite.",
"$lim(cos u) = cos(lim u)$ if the limit is finite",
"Limits at infinity can be transformed to limits at zero if $f(t)$ is replaced by $f(1/t)$.",
"You can neglect all but the leading terms in numerator and denominator."
},
{ /* infinite_limits */
"For $u$ small, $1/u^2^n$ is large.",
"For $u$ small, $1/u^n$ is large, but if $n$ is odd, it has opposite signs for $u$ positive and $u$ negative, which makes trouble for the two-sided limit as $u$ approaches zero.",
"For $u$ small and positive, $1/u^n$ is large.",
"For $u$ small and negative, $1/u^n$ is large and (if $n$ is odd) negative.",
"If the denominator goes to zero and the numerator does not, then the limit is undefined.",
"For $t$ small and positive, $ln t$ is large and negative.",
"tan $t$ has singularities at odd multiples of $\\pi /2$. But it approaches the singularities with different signs from the left and right.",
"cot $t$ has singularities at multiples of $\\pi $. But it approaches the singularities with different signs from the left and right.",
"sec $t$ has singularities at odd multiples of $\\pi /2$. But it approaches the singularities with different signs from the left and right.",
"csc $t$ has singularities at multiples of $\\pi $. But it approaches the singularities with different signs from the left and right.",
"Multiply one factor and divide the other by something chosen to make it possible to calculate the limits.",
"Multiply one factor and divide the other by something chosen to make it possible to calculate the limits.",
},
{ /* infinities */
"$\\pm \\infty /$positive = $\\pm \\infty $",
"nonzero$/\\pm \\infty = 0$",
"positive$\\times \\pm \\infty = \\pm \\infty $",
"$\\pm \\infty \\times \\infty = \\pm \\infty $",
"$\\pm \\infty +$ finite$ = \\pm \\infty $",
"$\\infty + \\infty = \\infty $",
"$u^\\infty = \\infty $ if $u > 1$",
"$u^\\infty = 0$ if $0 < u < 1$",
"$$u^(-infinity ) = 0$$ if $u > 1$",
"$$u^(-infinity ) = infinity$$ if $0 < u < 1$",
"$\\infty ^n = \\infty $ if $n > 0$",
"You have a sum containing infinities of different signs; such a sum is undefined."
},
{ /* zero_denom */
"$a/0+ = \\infty $ if $a>0$",
"$a/0- = -\\infty $ if $a>0$",
"$a/0 =$ undefined",
"$\\infty /0+ = \\infty $",
"$\\infty /0- = -\\infty $",
"$\\infty /0 = $ undefined",
"$\\infty /0^2 = \\infty $",
"$\\infty /0^2^n = \\infty $",
"$a/0^2 = \\infty if a > 0$",
"$a/0^2 = -\\infty if a < 0$",
"$a/0^2^n = \\infty if a > 0$",
"$a/0^2^n = -\\infty if a < 0$"
},
{ /* more_infinities */
"$ln \\infty = log \\infty = \\infty $",
"$\\sqrt \\infty = \\infty $",
"$^n\\sqrt \\infty = \\infty $",
"$arctan \\pm \\infty = \\pm \\pi /2$",
"$arccot \\infty = 0$",
"$arccot -\\infty = \\pi $",
"$arcsec \\pm \\infty = \\pi /2$",
"$arccsc \\pm \\infty = 0$",
"trig limits at $\\infty $ are undefined, because the trig function oscillate (or worse)",
"$cosh \\pm \\infty = \\infty $",
"$sinh \\pm \\infty = \\pm \\infty $",
"$tanh \\pm \\infty = \\pm 1$",
"$ln 0 = -\\infty $"
},
{ /* polynomial_derivs */
"The derivative of a constant is zero. Here a 'constant' means anything that doesn't depend on the variable with respect to which you are differentiating.",
"You have an expression $dx/dx$. This should evaluate to 1.",
"The derivative of a sum is the sum of the derivatives.",
"You can pull a minus sign out through the derivative sign",
"You can pull a constant out through the derivative sign",
"Use the 'power law' for differentiating a power.",
"You can use MathXpert to differentiate a polynomial in one step.",
"By definition, $f'(x) = d/dx f(x)$."
},
{ /* derivatives */
"Use the formula that defines a derivative as a certain limit. It's with the other operations for derivatives.",
"You can ask MathXpert to differentiate a polynomial in one step.",
"The derivative of a sum (or difference) is the sum (or difference) of the derivatives.",
"You can pull a minus sign out through the derivative sign",
"You can pull a constant out through the derivative sign",
"You have a constant in the denominator. Pull it out using: $$diff(u/c,x)=(1/c)diff(u,x)$$. Any constants in the numerator will also come out.",
"Use the 'power law' for differentiating a power.",
"Use the 'product rule' for derivatives",
"There is a simple formula for the derivative of a reciprocal: $$diff(1/v,x) = -diff(v,x)/v^2$$ It's well worth memorizing this special case of the quotient rule.",
"Use the 'quotient rule' for derivatives",
"There is a formula for the derivative of a square root. Often it is much simpler to differentiate a square root directly, rather than convert it to a fractional exponent and use the power law.",
"To differentiate a root, first convert it to fractional exponent form.",
"To differentiate a power in the denominator, you don't have to first convert it to a negative exponent as so many students do. You can use the power law directly in the form: $$diff(c/x^n,x) = -nc/x^(n+1)$$",
"There is a simple formula for differentiating absolute values: $d/dx |x| = x/|x|$. If your textbook omits this formula, check it yourself by considering separately the cases when $x$ is positive and negative. Of course, both sides of the formula are undefined when $x=0$.",
"By definition, $f'(x) = d/dx f(x)$"
},
{ /* dif_trig */
"The derivative of sin is cos",
"The derivative of cos is $-sin$",
"The derivative of tan is $sec^2$",
"The derivative of sec is sec tan",
"The derivative of cot is $-csc^2$",
"The derivative of csc is - csc cot"
},
{ /* dif_explog */
"$e^x$ is its own derivative",
"Exponential functions are their own derivatives, except for a constant:$ d/dx c^x = (ln c) c^x$",
"To differentiate a power with a non-constant exponent, make the base constant by using the law: $$ diff(u^v,x) = diff(e^(v ln u),x)$$",
"The derivative of $ln x is 1/x$",
"The derivative of $ln |x| = 1/x$",
"Try rewriting $dy/dx$ as $y (d/dx) ln y$",
"Use the formula: $d/dx e^u = e^u du/dx$",
"To differentiate a power with constant base, use the formula: $$diff(c^u,x)=(ln c)c^u diff(u,x)$$",
"To differentiate a logarithm, use the formula: $$diff(ln u,x) = (1/u)(diff(u,x))$$",
"Use the formula: $$diff(ln abs(u),x) = (1/u) diff(u,x)$$",
"There is a formula for differentiating $ln(cos x)$ in one step.",
"There is a formula for differentiating $ln(sin x)$ in one step."
},
{ /* dif_inverse_trig */
"$d/dx arctan x = 1/(1+x^2)$",
"$d/dx arcsin x = 1/\\sqrt (1-x^2)$",
"$d/dx arccos x = -1/\\sqrt (1-x^2)$",
"$d/dx arccot x = -1/(1+x^2)$",
"$d/dx arcsec x = 1/(|x|\\sqrt (x^2-1))$",
"$d/dx arccsc x = -1/(|x|\\sqrt (x^2-1))$",
"$d/dx arctan u = (du/dx)/(1+u^2)$",
"$d/dx arcsin u = (du/dx)/\\sqrt (1-x^2)$",
"$d/dx arccos u = -(du/dx)/\\sqrt (1-x^2)$",
"$d/dx arccot u = -(du/dx)/(1+u^2)$",
"$d/dx arcsec u=(du/dx)/(|u|\\sqrt (u^2-1))$",
"$d/dx arccsc u=-(du/dx)/(|u|\\sqrt (u^2-1))$"
},
{ /* chain_rule */
"Use the chain rule form of the power rule: $$diff(u^n,x) = nu^(n-1) diff(u,x)$$",
"Use the chain rule with the rule for differentiating square roots: $$diff(sqrt(u),x) = diff(u,x)/(2 sqrt(u))$$",
"Use the chain rule with the formula for the derivative of sin",
"Use the chain rule with the formula for the derivative of cos",
"Use the chain rule with the formula for the derivative of tan",
"Use the chain rule with the formula for the derivative of sec",
"Use the chain rule with the formula for the derivative of cot",
"Use the chain rule with the formula for the derivative of csc",
"Use the chain rule with the formula for the derivative of absolute value",
"Use the chain rule in the form $$diff(f(u),x) = f'(u) diff(u,x)$$",
"Make a substitution.",
"Now eliminate your defined variable."
},
{ /* maxima_and_minima */
"Experiment numerically.", /* Not used in auto mode */
"Consider points where $f'(x)=0$",
"Consider endpoints of interval",
"Are there points where $f'(x)$ is undefined?",
"Consider the limits at open ends of the interval.",
"Reject a point outside interval",
"Make a table of decimal $y$-values",
"Make a table of exact $y$-values",
"Choose the maximum value(s) from your table.",
"Choose the minimum value(s) from your table.",
"You can ask MathXpert to evaluate a derivative in one step.",
"Now solve the equation.",
"You can ask MathXpert to evaluate a simple limit in one step.",
"Get rid of the integer parameter.",
"This function is constant, so the max equals the min equals the value."
},
{ /* implicit_diff */
"Evaluate the derivative.",
"Simplify the expression.",
"Solve the equation."
},
{ /* related_rates */
"Differentiate the equation.",
"Evaluate the derivative.",
"Eliminate the derivative of a variable by substituting for it.",
"Solve the equation."
},
{ /* simplify */
"Simplify the expression.",
"Eliminate the compound fractions.",
"Put the fractions over a common denominator and simplify.",
"Factor out a common term.",
"Try to factor.",
"Multiply out and simplify.", /* meaning either collect or cancel or both */
"Is there a common factor in numerator and denominator?",
"Solve the equation.",
"Write it as a polynomial in some variable or expression.",
"Express some expression in polynomial form.",
"Make the leading coefficient of some polynomial 1.",
"Convert fractional exponents of 1/2 to square roots.",
"Convert fractional exponents to roots.",
"Eliminate roots and square roots in favor of fractional exponents."
},
{ /* higher_derivatives */
"Differentiate the identity using the law: $u=v => du/dx = dv/dx$.",
"Express the second derivative using $$diff(u,x,2) = diff(diff(u,x),x)$$",
"$$diff(u,x,n) = diff(diff(u,x,n-1),x)$$",
"The derivative of the derivative is the second derivative.",
"Differentiating an $n$-th derivative produces an $n+1$-st derivative.",
"You can ask MathXpert to evaluate a derivative in one step.",
"Evaluate numerically at a point."
},
{ /* basic_integration */
"$\\int 1 dt = t$",
"There is a constant integrand, so use the law $$integral(c,t) = ct$$",
"$\\int t dt = t^2/2$",
"$\\int cu dt = c\\int u dt (c constant)$",
"Bring the minus sign out of the integral using $$integral(-u,t) = -integral(u,t)$$",
"The integrand is a sum, so you can use the property known as linearity of the integral: $$integral(u+v,t) = integral(u,t) + integral(v,t) $$",
"The integrand is a difference, so you can use the property known as linearity of the integral: $$integral(u-v,t) = integral(u,t) - integral(v,t) $$",
"The integrand is a sum or difference, so you can use the property known as linearity of the integral: $$integral(au+bv,t) = a integral(u,t) + b integral(v,t) $$ This property also works with a minus sign, or with a mixture of plus and minus signs.",
"$\\int t^n dt=t^(n+1)/(n+1) (n \\ne -1)$",
"$\\int 1/t^(n+1) dt= -1/(nt^n) (n \\ne 0)$",
"The integrand is a polynomial. You can ask MathXpert to integrate it in one step.",
"$\\int (1/t) dt = ln |t|$",
"$\\int 1/(t\\pm a) dt = ln |t\\pm a|$",
"Multiply out the integrand, obtaining a sum of simpler terms.",
"expand $(a+b)^n$ in integrand",
"$\\int |t| dt = t|t|/2$",
},
{ /* trig_integration */
"Integrate the sine.",
"Integrate the cosine.",
"Integrate the tangent.",
"Integrate the cotangent.",
"Integrate the secant.",
"Integrate the cosecant.",
"Integrate the square of the secant.",
"Integrate the square of the cosecant.",
"There is a formula for the integral of $tan^2 t$, or you can do it by parts.",
"There is a formula for the integral of $cot^2 t$, or you can do it by parts.",
"$sec t tan t$ can be directly integrated, since it is the derivative of $sec t$.",
"$csc t cot t$ can be directly integrated, since it is the derivative of $csc t$."
},
{ /* trig_integration2 */
"Integrate the sine.",
"Integrate the cosine.",
"Integrate the tangent.",
"Integrate the cotangent.",
"Integrate the secant.",
"Integrate the cosecant.",
"Integrate the square of the secant.",
"Integrate the square of the cosecant.",
"There is a formula for the integral of $tan^2 t$, or you can do it by parts.",
"There is a formula for the integral of $cot^2 t$, or you can do it by parts.",
"$sec t tan t$ can be directly integrated, since it is the derivative of $sec t$.",
"$csc t cot t$ can be directly integrated, since it is the derivative of $csc t$."
},
{ /* integrate_exp */
"The exponential function is its own integral: $$integral(e^t,t) = e^t$$",
"An exponential function is its own integral, but if the exponent contains a constant the integral has a corresponding factor: $\\int e^at dt =(1/a) e^at$",
"$\\int e^(-t)dt = -e^(-t)$",
"$\\int e^(-at)dt = -(1/a) e^(-at)$",
"$$integral( e^(t/a),t) = a e^(t/a)$$",
"An exponential function is its own integral, except that if the base is not $e$, then a constant factor must be thrown in.",
"$$integral( u^v, t) = integral(e^(v ln u),t)$$",
"$\\int ln t = t ln t - t$",
"$$integral( e^(-t^2),t) = (sqrt pi) /2 Erf(t)$$"
},
{ /* integrate_by_substitution */
"Try integration by substitution",
"Try integration by substitution",
"compute $du/dx$",
"Evaluate the derivative",
"Get back your original integral with 'show integral again'",
"Express the integrand as a function of the new variable, by choosing: integrand = $f(u) \\times du/dx$",
"Eliminate the original variable of integration entirely now.",
"Now eliminate your defined variable.",
"Integrate by substitution.",
"Integrate by substitution.", /* autointsub, not used in auto mode anyway */
},
{ /* integrate_by_parts */
"Try integration by parts.",
"Try integration by parts.", /* autointegratebyparts not used in auto mode */
"Set the current line equal to the original problem, getting an equation.",
"Isolate the original integral on the left side of the equation.",
"Evaluate the derivative.",
"Integrate by substitution.",
"Integrate by substitution", /* autointsub, not used in auto mode anyway */
"You can ask MathXpert to evaluate a simple integral in one step."
},
{
"Use the fundamental theorem of calculus",
"Use the fundamental theorem of calculus"
},
{ /* definite_integration */
"Get rid of the bar for function evaluation.",
"Get rid of the bar for function evaluation.",
"Switch the limits of integration, introducing a minus sign.",
"Combine two definite integrals with the same integrand into one integral, if they represent integration over different parts of the same interval.",
"It may help to break a definite integral up into two (or more) integrals, introducing an intermediate point (or points) as a new limit of integration.",
"Break up the integral into two or more integrals whose endpoints are at the zeroes of the integrand. Then you will be able to get rid of absolute value.",
"You can ask MathXpert to calculate the numerical value of an integral, if the integral has a numerical value.",
"You can ask MathXpert to calculate the numerical value of an integral, if the integral has a numerical value.",
"Notice that the upper and lower limits of integration are the same."
},
{ /* improper_integrals */
"Express an improper integral as a limit of proper integrals.",
"Express an improper integral as a limit of proper integrals.",
"Express an improper integral as a limit of proper integrals.",
"Express an improper integral as a limit of proper integrals.",
"If the integrand does not tend to zero at $\\infty $, an improper integral diverges.",
"If the integrand does not tend to zero at $-\\infty $, an improper integral diverges."
},
{ /* oddandeven */
"The integral of an odd function over an interval whose midpoint is the origin has to be zero.",
"The integral of an even function over an interval whose midpoint is the origin is twice the integral over the positive half of the interval."
},
{ /* trig_substitutions */
"Use a trig substitution",
"Use a trig substitution",
"Use a trig substitution",
"Use a trig substitution",
"Use a trig substitution",
"Use a trig substitution",
"Use an inverse substitution", /* define your own substitution (not used in auto mode) */
"Evaluate the derivative.",
"You can ask MathXpert to evaluate a simple integral in one step."
},
{ /* trigonometric_integrals */
"Get rid of the $sin^2$ term in the integrand using: $sin^2 t = (1-cos 2t)/2$ in integral. You can find this formula with the trigonometric integral formulas as well as with the trig formulas.",
"Get rid of the $cos^2$ term in the integrand using: $cos^2 t = (1+cos 2t)/2$ in integral. You can find this formula with the trigonometric integral formulas as well as with the trig formulas.",
"Make a substitution $u=cos x$ after using $sin^2=1-cos^2$. Select the whole integral to see this choice.",
"Make a substitution $u=sin x$ after using $cos^2=1-sin^2$. Select the whole integral to see this choice.",
"Make a substitution $u=tan x$ after using $sec^2=1+tan^2$. Select the whole integral to see this choice.",
"Make a substitution $u=cot x$ after using $csc^2=1+cot^2$. Select the whole integral to see this choice.",
"Make a substitution $u=sec x$ after using $tan^2=sec^2-1$. Select the whole integral to see this choice.",
"Make a substitution $u=csc x$ after using $cot^2=csc^2-1$. Select the whole integral to see this choice.",
"Use the identity $tan^2 x = sec^2 x - 1$ in the integrand. Select the whole integral to see this choice.",
"Use the identity $cot^2 x = csc^2 x - 1$ in the integrand. Select the whole integral to see this choice.",
"Use a reduction formula to reduce this to another similar integral, but with a lower power of sec.",
"Use a reduction formula to reduce this to another similar integral, but with a lower power of csc.",
"Use the Weierstrass substitution: $u = tan(x/2)$. Select the whole integral to see this choice.",
},
{ /* trigrationalize */
"Multiply both numerator and denominator by $1+cos x$.",
"Multiply both numerator and denominator by $1-cos x$.",
"Multiply both numerator and denominator by $1+sin x$.",
"Multiply both numerator and denominator by $1-sin x$.",
"Multiply both numerator and denominator by $sin x + cos x$.",
"Multiply both numerator and denominator by $cos x - sin x$.",
},
{ /* integrate_rational*/
"Use polynomial division to reduce to the case in which the numerator is of lower degree than the denominator",
"Factor the denominator if you can.",
"Is there any common factor in the numerator and denominator?",
"You can ask MathXpert to perform 'square-free factorization', which will find any repeated factors. This operation uses an algorithm not usually taught in textbooks.",
"You can use MathXpert to factor a polynomial numerically. Close decimal approximations to the roots will be used.",
"Expand the integrand in partial fractions.",
"Complete the square in the denominator.",
"A reciprocal of a linear function integrates to a logarithm.",
"A reciprocal of a power of a linear function integrates to another such function. You could reduce the integral by substitution to a power of the variable, but you might as well do it in one step.",
"A reciprocal of a sum of squares integrates to an arctan.",
"A reciprocal of a difference of squares integrates to an arccoth, an arctanh, or a logarithm.",
"A reciprocal of a difference of squares integrates to an arccoth, an arctanh, or a logarithm.",
"A reciprocal of a difference of squares integrates to an arccoth, an arctanh, or a logarithm.",
"A reciprocal of a difference of squares integrates to an arccoth, an arctanh, or a logarithm.",
},
{ /* integrate_sqrtdenom */
"Complete the square in the denominator",
"A reciprocal of a square root of a difference of squares integrates to an arcsin.",
"A reciprocal of a square root of a sum of squares integrates to a logarithm.",
"Look on the menu for integrating square roots in the denominator.",
"Make a rationalizing substitution."
},
{ /* integrate_arctrig */
"There is an integration formula for arcsin",
"There is an integration formula for arccos",
"There is an integration formula for arctan",
"There is an integration formula for arccot",
"There are two integration formulas for arccsc--be careful.",
"There are two integration formulas for arccsc--be careful.",
"There are two integration formulas for arcsec--be careful.",
"There are two integration formulas for arcsec--be careful."
},
{ /* simplify_calculus */
"Simplify the expression.",
"Eliminate compound fractions.",
"Put fractions over a common denominator and simplify.",
"Factor out a common term.",
"Try to factor",
"Multiply out and simplify.", /* meaning either collect or cancel or both */
"Is there a common factor in numerator and denominator?",
"Solve the equation.",
"Evaluate the derivative.",
"Evaluate the limit",
"Change the integral by substitution",
"You can ask MathXpert to evaluate a simple integral in one step.",
"Absorb numbers into the constant of integration."
},
{ /* integrate_hyperbolic */
"The integral of sinh is cosh.",
"The integral of cosh is sinh.",
"The integral of tanh is ln cosh.",
"The integral of coth is ln sinh.",
"The integral of csch is $ln tanh(u/2)$.",
"The integral of $sech u$ is $arctan (sinh u)$."
},
{ /* series_geom1 */
"Expand $1/(1-x)$ in a power series.",
"Expand $1/(1-x)$ in a power series.",
"Expand $1/(1-x)$ in a power series.",
"Expand $1/(1+x)$ in a power series.",
"Expand $1/(1+x)$ in a power series.",
"Expand $1/(1+x)$ in a power series.",
"Sum the series for $1/(1-x)$.",
"Sum the series for $1/(1-x)$.",
"Sum the series for $1/(1-x)$.",
"Sum the series for $1/(1+x)$.",
"Sum the series for $1/(1+x)$.",
"Sum the series for $1/(1+x)$."
},
{ /* series_geom2 */
"Expand $x/(1-x)$ in a power series.",
"Expand $x/(1-x)$ in a power series.",
"Expand $x/(1-x)$ in a power series.",
"Expand $x/(1+x)$ in a power series.",
"Expand $x/(1+x)$ in a power series.",
"Expand $x/(1+x)$ in a power series.",
"Sum the series for $x/(1-x)$.",
"Sum the series for $x/(1-x)$.",
"Sum the series for $x/(1-x)$.",
"Sum the series for $x/(1+x)$.",
"Sum the series for $x/(1+x)$.",
"Sum the series for $x/(1+x)$."
},
{ /* series_geom3 */
"Expand $1/(1-x^k)$ in a power series.",
"Expand $1/(1-x^k)$ in a power series.",
"Expand $1/(1-x^k)$ in a power series.",
"Expand $x^m/(1-x^k)$ in a power series.",
"Expand $x^m/(1-x^k)$ in a power series.",
"Expand $x^m/(1-x^k)$ in a power series.",
"Sum the series for $1/(1-x^k)$.",
"Sum the series for $1/(1-x^k)$.",
"Sum the series for $1/(1-x^k)$.",
"Sum the series for $x^m/(1-x^k)$.",
"Sum the series for $x^m/(1-x^k)$.",
"Sum the series for $x^m/(1-x^k)$."
},
{ /* series_geom4 */
"Expand $1/(1+x^k)$ in a power series.",
"Expand $1/(1+x^k)$ in a power series.",
"Expand $1/(1+x^k)$ in a power series.",
"Expand $x^m/(1+x^k)$ in a power series.",
"Expand $x^m/(1+x^k)$ in a power series.",
"Expand $x^m/(1+x^k)$ in a power series.",
"Sum the series for $1/(1+x^k)$.",
"Sum the series for $1/(1+x^k)$.",
"Sum the series for $1/(1+x^k)$.",
"Sum the series for $x^m/(1+x^k)$.",
"Sum the series for $x^m/(1+x^k)$.",
"Sum the series for $x^m/(1+x^k)$."
},
{ /* series_geom5 */
"You can expand $x^k/(1-x)$ as a geometric series",
"You can expand $x^k/(1-x)$ as a geometric series",
"You can expand $x^k/(1-x)$ as a geometric series",
"You can expand $x^k/(1+x)$ as a geometric series",
"You can expand $x^k/(1+x)$ as a geometric series",
"You can expand $x^k/(1+x)$ as a geometric series",
"Sum the geometric series.",
"Sum the geometric series.",
"Sum the geometric series.",
"Sum the geometric series.",
"Sum the geometric series.",
"Sum the geometric series."
},
{ /* series_ln */
"Expand $ln(1-x)$ in a power series.",
"Expand $ln(1-x)$ in a power series.",
"Expand $ln(1-x)$ in a power series.",
"Expand $ln(1+x)$ in a power series.",
"Expand $ln(1+x)$ in a power series.",
"Expand $ln(1+x)$ in a power series.",
"Sum the power series for $ln(1-x)$.",
"Sum the power series for $ln(1-x)$.",
"Sum the power series for $ln(1-x)$.",
"Sum the power series for $ln(1+x)$.",
"Sum the power series for $ln(1+x)$.",
"Sum the power series for $ln(1+x)$."
},
{ /* series_trig */
"Expand $sin x$ in a power series.",
"Expand $sin x$ in a power series.",
"Expand $sin x$ in a power series.",
"Expand $cos x$ in a power series.",
"Expand $cos x$ in a power series.",
"Expand $cos x$ in a power series.",
"Sum the series for $sin x$.",
"Sum the series for $sin x$.",
"Sum the series for $sin x$.",
"Sum the series for $cos x$.",
"Sum the series for $cos x$.",
"Sum the series for $cos x$."
},
{ /* series_exp */
"Expand $e^x$ in a power series.",
"Expand $e^x$ in a power series.",
"Expand $e^x$ in a power series.",
"Sum the series for $e^x$.",
"Sum the series for $e^x$.",
"Sum the series for $e^x$.",
"Expand $e^-x$ in a power series.",
"Expand $e^-x$ in a power series.",
"Expand $e^-x$ in a power series.",
"Sum the series for $e^-x$.",
"Sum the series for $e^-x$.",
"Sum the series for $e^-x$."
},
{ /* series_atan */
"Expand $arctan x$ in a power series.",
"Expand $arctan x$ in a power series.",
"Expand $arctan x$ in a power series.",
"Sum the series for arctan.",
"Sum the series for arctan.",
"Sum the series for arctan.",
"Use the binomial series to expand a power of a sum.",
"Use the binomial series to expand a power of a sum.",
"Use the binomial series to expand a power of a sum.",
"Sum the binomial series",
"Sum the binomial series",
"Sum the binomial series"
},
{ /* series_bernoulli */
"Expand $tan x$ in a power series.",
"Expand $tan x$ in a power series.",
"Expand $tan x$ in a power series.",
"Expand $cot x$ or $x cot x$ in a power series.",
"Expand $cot x$ or $x cot x$ in a power series.",
"Expand $cot x$ or $x cot x$ in a power series.",
"Expand $x/(e^x-1)$ in a power series.",
"Expand $x/(e^x-1)$ in a power series.",
"Expand $x/(e^x-1)$ in a power series.",
"Expand $sec x$ or $1/cos x$ in a power series.",
"Expand $sec x$ or $1/cos x$ in a power series.",
"Expand $sec x$ or $1/cos x$ in a power series.",
"Expand $\\zeta(s)$ in a power series.",
"Expand $\\zeta(s)$ in a power series.",
"Expand $\\zeta(s)$ in a power series.",
"The alternating harmonic series has a known sum."
},
{ /* series_appearance */
"You may want to express the series in the form $a_0 + a_1 + ... $",
"You may want to express the series in the form $a_0 + a_1 + a_2 + ... $",
"You may want to express the series using ... instead of sigma notation.",
"Express the series using sigma notation.",
"Show another term before ...",
"Show more terms before ...",
" ", /* these four appearance operations will not be used in auto mode */
" ",
" ",
" "
},
{ /* series_algebra */
"You have a telescoping series.",
"Multiply series",
"Two power series can be multiplied to produce a new power series.",
"A power series can be divided by a polyomial, using a process like long division.",
"A polynomial can be divided by a power series , using a process like long division.",
"Two power series can be divided, using a process like long division.",
"The square of a series can be written as a double series.",
"The square of a power series can be written as another power series.",
"A power of a power series can be expressed as another power series.",
"Combine the sum of two series into a single series.",
"Combine the difference of two series into a single series."
},
{ /* series_manipulations */
"Split off the first few terms of an infinite series.",
"Perhaps by decreasing the lower limit of a series (subtracting the new terms) you can bring your series into a standard form.",
"Add something to the index variable to bring the series into a more manageable form.",
"Subtract something from the index variable to bring the series into a more manageable form.",
"Rename the index variable",
"Break a series $\\sum (a+b)$ into a sum of series $\\sum a + \\sum b$.",
"Differentiate term by term.",
"Pull a derivative out of the series.",
"Integrate term by term.",
"Pull an integral out of the series.",
"Calculate the first few terms.",
"Write the function as the integral of its derivative. Then expand the derivative in a series and integrate term-by-term.",
"Write the function as a definite integral of its derivative. Then expand the derivative in a series and integrate term-by-term.",
"Write the function as the derivative of its integral. Then expand the integral in a series and differentiate term-by-term.",
"Solve for the constant of integration in order to eliminate it.",
"Separate the terms with even and odd indices, getting two new series."
},
{ /* series_convergence_tests */
"You can show that a series is divergent by showing its general term does not tend to zero.",
"Use the integral test.",
"Use the ratio test.",
"Use the root test.",
"Use the comparison test to prove convergence. Find a convergent series with larger general term.",
"Use the comparison test to prove divergence. Find a divergent series with smaller general term.",
"Use the limit comparison test.",
"Use the condensation test.",
"Finish the divergence test.",
"Finish the integral test.",
"Finish the root test.",
"Finish the ratio test.",
"Finish the comparison test.", /* not a mistake to list this twice */
"Finish the comparison test.",
"Finish the limit comparison test.",
"Finish the condensation test."
},
{ /* series_convergence2 */
"You have finished showing the convergence of the comparison series. Now state the positive result about the convergence of the original series. To see this choice, select the entire current line.",
"You have finished showing the divergence of the comparison series. Now state the negative result about the convergence of the original series. To see this choice, select the entire current line.",
"The harmonic series $$sum(1/k,k,1,infinity)$$ is divergent, since its partial sum up to $n$ terms is approximately $ln n$.",
"There is a formula for $$sum(1/k^2,k,1,infinity)$$",
"The sum of the terms $1/k^s$ converges and is called $\\zeta(s)$.",
"The values of the $\\zeta$ function at even integers can be computed in terms of Bernoulli numbers."
},
{ /* complex_functions */
"Express a complex number in polar form to calculate its logarithm, using the law $$ln(u+iv) = ln(r e^(i theta))$$",
"Use the formula for complex logarithms: $$ln(re^(i theta))=ln r + i theta$$. There is a subtlety here: in applying this law, if $\\theta $ is not between $-\\pi $ and $\\pi $, it will be reduced to that range.",
"The natural logarithm of i is $i\\pi /2$, since $\\pi /2$ is the argument of i",
"The natural logarithm of -1 is $i\\pi $, since $-1 = e^(i\\pi )$",
"The natural logarithm of -a is $ln a + i\\pi $, since $-1 = e^(i\\pi )$. This formula assumes $a$ is positive.",
"Expand cos in terms of complex exponentials.",
"Expand sin in terms of complex exponentials.",
"To take a complex square root, you take the square root of the radius and half of the argument.",
"To take a complex $n$-th root, you take the $n$-th root of the radius, and divide the argument by $n$.",
"Expand the complex exponential using cos and sin",
"Expand the complex exponential using cos and sin",
"Use Euler's famous identity: $$e^(i pi) = -1 $$",
"Use Euler's famous identity: $$e^(-i pi) = -1 $$",
"$$e^(2n pi i) = 1$$, because as $\\theta $ varies, $e^(i\\theta)$ traces the unit circle.",
"As $\\theta $ varies, $e^i\\theta $ traces the unit circle. Therefore you can get rid of multiples of $2 pi i$ in the exponent.",
"Rewrite the complex exponential so it has base $e$, using the law $$u^v = e^(v ln u)$$"
},
{ /* complex_hyperbolic */
"$sin(it)$ can be expressed using the hyperbolic sine, instead of expanding in complex exponentials.",
"$cos(it)$ can be expressed using the hyperbolic cosine, instead of expanding in complex exponentials.",
"$sinh(it)$ can be expressed as $i sin t$, instead of expanding in exponentials.",
"$cosh(it)$ can be expressed as $cos t$, instead of expanding in exponentials.",
"$tan(it)$ can be expressed using the hyperbolic tangent,instead of expanding in complex exponentials.",
"$cot(it)$ can be expressed using the hyperbolic cotangent, instead of expanding in complex exponentials.",
"$tanh(it)$ can be expressed as $i tan t$, instead of expanding in exponentials.",
"$coth(it)$ can be expressed as $-i cot t$, instead of expanding in exponentials.",
"Use a complex exponential to express $cos t + i sin t$",
"Use a complex exponential to express $cos t - i sin t$",
"Simplify an expression in complex exponentials to a cosine.",
"Simplify an expression in complex exponentials to a sine.",
"Simplify an expression in complex exponentials to a cosine.",
"Simplify an expression in complex exponentials to a sine."
},
{ /* hyperbolic_functions */
"Use the definition of cosh",
"Combine exponentials into a cosh term",
"Use the definition of sinh",
"Combine exponentials into a sinh term",
"Combine exponentials into a cosh term",
"Combine exponentials into a sinh term",
"cosh is an even function",
"sinh is an odd function",
"Combine the cosh and sinh terms using: $cosh u + sinh u = e^u$",
"Combine the cosh and sinh terms using: $cosh u - sinh u = e^(-u)$",
"Remember $cosh 0 = 1$",
"Remember $sinh 0 = 0$",
"Express $e^x$ in terms of hyperbolic functions",
"Express $e^(-x)$ in terms of hyperbolic functions"
},
{ /* hyperbolic2 */
"Use the identity $sinh^2u + 1 = cosh^2 u$",
"Use the identity $cosh^2 u - 1 = sinh^2u $",
"Use the identity $cosh^2 u - sinh^2u = 1$",
"Use the identity $cosh^2 u = sinh^2u + 1$",
"Use the identity $sinh^2u = cosh^2 u - 1$",
"Use the identity $1 - tan^2u = sech^2u$",
"Use the identity $1 - sech^2u = tan^2u$"
},
{ /* more_hyperbolic */
"Express tanh in terms of sinh and cosh.",
"Combine sinh and cosh into tanh.",
"Express coth in terms of cosh and sinh",
"Combine cosh and sinh into coth",
"Express sech as the reciprocal of cosh",
"The reciprocal of cosh is sech",
"Express csch as the reciprocal of sinh",
"The reciprocal of sinh is csch",
"Use the formula $tanh^2 u + sech^2 u = 1$.",
"Use the formula $tanh^2 u = 1 - sech^2 u$.",
"Use the formula $sech^2 u = 1 - tanh^2 u$.",
"Use the formula for sinh of a sum or difference",
"Use the formula for cosh of a sum or difference",
"Use the double-angle formula: $sinh 2u = 2 sinh u cosh u$",
"Use the double-angle formula: $cosh 2u = cosh^2 u + sinh^2 u$",
"There is a formula to simplify $tanh(ln u)$."
},
{ /* inverse_hyperbolic */
"There is a formula to express arcsinh in terms of logarithms.",
"There is a formula to express arccosh in terms of logarithms.",
"There is a formula to express arctanh in terms of logarithms.",
"$sinh(arcsinh x)$ is just $x$.",
"$cosh(arccosh x)$ is just $x$.",
"$tanh(arctanh x)$ is just $x$.",
"$coth(arccoth x)$ is just $x$.",
"$sech(arcsech x)$ is just $x$.",
"$csch(arccsch x)$ is just $x$."
},
{ /* dif_hyperbolic */
"The derivative of sinh is cosh",
"The derivative of cosh is sinh",
"The derivative of tanh is $sech^2$",
"The derivative of coth is $-csch^2$",
"The derivative of sech is $- sech tanh$",
"The derivative of csch is $- csch coth$",
"The derivative of ln sinh is coth",
"The derivative of ln cosh is tanh"
},
{ /* dif_inversehyperbolic */
"The derivative of arcsinh is actually an algebraic function",
"The derivative of arccosh is actually an algebraic function",
"The derivative of arctanh is actually an algebraic function",
"The derivative of arccoth is actually an algebraic function",
"The derivative of arcsech is actually an algebraic function",
"The derivative of arccsch is actually an algebraic function"
},
{ /* sg_function1 */
"Eliminate the sgn function, since its argument is positive.",
"Eliminate the sgn function, since its argument is negative.",
"Eliminate the sgn function, since its argument is zero.",
"sgn is an odd function",
"sgn is an odd function",
"Express sgn in terms of absolute value",
"Express sgn in terms of absolute value",
"Express $|x|$ as $x sg(x)$",
"An even power is always positive",
"An odd power has the same sign as its base, so $sgn(x)$ to an odd power is $sgn(x)$",
"Bring sgn to the numerator using $1/sgn(x) = sgn(x)$",
"sgn(x) is constant when x is nonzero, in which case its derivative is zero.",
"sgn(x) can be integrated directly.",
"sgn(x) can be pulled through the integral sign if the integrand is nonzero.",
"sgn(x) is used to combine the cases of $x$ positive and $x$ negative, but sometimes they have to be treated separately.",
"sgn(x) is used to combine the cases of $x$ positive and $x$ negative, but sometimes they have to be treated separately."
},
{ /* sg_function2 */
"Drop positive factors inside the sgn function.",
"Drop negative factors inside the sgn function, adding a minus sign in front.",
"Drop positive factors inside the sgn function.",
"Drop negative factors inside the sgn function, adding a minus sign in front.",
"The sign of an odd power of $x$ is the same as the sign of $x$.",
"$1/x$ has the same sign as $x$.",
"$c/x$ has the same sign as $x$, if $c$ is positive.",
"Express $x sgn(x)$ as $|x|$.",
"Express $|x| sgn(x)$ as $x$."
},
{ /* bessel_functions */
"The derivative of $J0$ is $-J1$",
"$d/dx J1(x) = J0(x) - J1(x)/x$",
"$d/dx J(n,x)=J(n-1,x)-(n/x)J(n,x)$",
"The derivative of $Y0$ is $-Y1$",
"$d/dx Y1(x) = Y0(x) - Y1(x)/x$",
"$d/dx Y(n,x)=Y(n-1,x)-(n/x)Y(n,x)$"
},
{ /* modified_bessel_functions */
"The derivative of $I0$ is $-I1$",
"$d/dx I1(x) = I0(x) - I1(x)/x$",
"$d/dx I(n,x)=I(n-1,x)-(n/x)I(n,x)$",
"The derivative of $K0$ is $-K1$",
"$d/dx K1(x) = -K0(x) - K1(x)/x$",
"$d/dx K(n,x)= -K(n-1,x)-(n/x)K(n,x)$"
},
{ /* functions_menu */
"Use a defined function",
"Use a defined function",
"Use a defined function",
"Use a defined function",
"Use a defined function",
"Use a defined function",
"Use a defined function",
"Use a defined function",
"Use a defined function",
"Use a defined function",
"Use a defined function",
"Use a defined function",
"Use a defined function",
"Use a defined function",
"Use a defined function",
"Use a defined function"
},
{ /* automode_only */
"Multiply out products of sums and collect the resulting terms.", /* expand */
"Multiply out using $a(b+c) = ab+ac$, and then make a cancellation.", /* multiplyifcancels */
"Put factors in order.",
"The fractions need to be put over a common denominator before calculating the limit. Begin by factoring the denominators if necessary.",
"The fractions need to be put over a common denominator before calculating the limit.",
"The fractions need to be put over a common denominator before calculating the limit. Begin by eliminating negative exponents.",
"Express the square root using a fractional exponent.",
"Expand the cosine of a double angle.",
"Eliminate $sin^2 t$ by expressing it in terms of $cos^2 t$.",
"Eliminate $cos^2 t$ by expressing it in terms of $sin^2 t$.",
"Eliminate $tan^2 t$ by expressing it in terms of $sec^2 t$.",
"Eliminate $sec^2 t$ by expressing it in terms of $tan^2 t$.",
"Make a substitution.",
"Multiply coefficients",
"", /* no hints necessary for preparetocancel */
},
{ /* automode_only2 */
"Evaluate a simple square root.",
"Add or subtract something to both sides.",
"Add or subtract something to both sides.",
"Add or subtract something to both sides.",
"Add or subtract something to both sides.",
"Factor one of the summands to make a common factor explicit. After that you can factor out the common factor.",
"Make a substitution",
"Make a substitution",
"Multiply out using $a(b+c) = ab+ac$, and then make a cancellation.", /* distribandcancel */
"Multiply out and simplify.", /* difofpowers */
"Rewrite trig functions in terms of sin and cos so that common denominators can be found.", /* limsum4 */
"Use $ab+ac = a(b+c)$ to create the middle term of a quadratic expression.",
"Factor one or both sides of an identity if the result will permit a cancellation.",
"One side is a perfect square (or other power). Factor it."
},
{ /* automode_only3 */
"Get all the logarithms to have the same argument by using the law for logarithms of a power.",
"Get all the logarithms to have the same argument by using the law for logarithms of a power.",
"Get all the logarithms to have the same argument by using the law for logarithms of a product.",
"Get all the logarithms to have the same argument by using the law for logarithms of a product.",
"dummy",
"dummy"
}
};
/*_________________________________________________*/
const char *English_hints(int n, int m)
{ int nitems; /* number of menus represented in hintstrings1 */
nitems = sizeof(hintstrings1) / (MAXLENGTH * sizeof(char *));
if(n < nitems)
return hintstrings1[n][m];
else
return ""; // assert(0);
}
Sindbad File Manager Version 1.0, Coded By Sindbad EG ~ The Terrorists