Sindbad~EG File Manager
/* M. Beeson, for MathXpert.
status-line help for operations menus, in English
Translate text between quotes, except don't translate text
between dollar signs.
Original date 8.31.95
modified 7.30.98
12.29.98 added seven more lines in binomial_theorem
1.14.99 added one more line in advanced_sigma_notation
1.14.99 changed "Simplifies an inequality" to "Simplify an inequality"
in several lines.
2.19.99 added numerical_calculation2 menu.
6.25.99 changed e to $e$ near the beginning.
6.28.99 ensured that all " a " intended as variables are inside dollar signs
7.19.99 removed mistaken quotes around "arithhelp" at line 260.
4.2.00 four more lines in absolute_value_ineq2
3.9.01 modified for signed_fractions
6.21.04 modified for lnrecip, logrecip, logbrecip, minusintoproduct1, minusintoproduct1,
and minusintoproduct3.
6.24.04 four more lines in complex_numbers
5.3.13 changed names of exported functions.
5.17.13 added one more item to the first menu.
5.24.13 one more to the first menu
removed last entry of trig_sum, which didn't correspond to an operation or menu text
6.3.13 added one more to signed_fractions
6.4.13 one more under log_ineq4, and two more under fractional_exponents.
6.5.13 one more under log_ineq4
10.10.24 put in hundreds of $
11.14.24 in logs_to_any_base, corrected the order of entries
Change history of english_ophelp2.c:
8.13.98, two new operations added in improper_integrals
8.17.98, logarithmic_limits menu added.
1.12.99 Now there are 13 series menus with new entries.
1.13-30.99 series entries modified
2.21.99 four new lines under complex_hyperbolic and one under
more_infinities
3.3.99 more lines under geometric series menus
6.8.99 added some dollar signs and corrected 'become' to 'becomes' at line 162
corrected dollar sign problem at line 437
6.28.99 ensured that all " a " intended as variables are inside dollar signs
6.16.04 added a line for a new operator in definite_integration.
6.27.06 more operations under sg_function2
1.14.11 six new operations under inverse_hyperbolic, and corrections to the existing three.
5.3.13 changed names of exported functions
5.17.13 added text for series_bernoulli
5.24.13 modified for series_bernoulli
6.11.13 four more under series_bernoulli
6.13.13 two more under series_convergence2
10.23.23 changed \32 to ->
11.19.24 pasted english_ophelp2.c into english_ophelp1.c
11.24.24 modified English_ophelp accordingly
*/
#include <assert.h>
#include "mtext.h"
#include "operator.h"
#include "english1.h"
static const char arithhelp[] = "Evaluates expressions using exact rational arithmetic only.";
static const char *ophelp1[][MAXLENGTH] =
/* let the first dimension be calculated by the compiler from the
initialization. */
{
{ /* numerical_calculation1 */
arithhelp,
"Performs decimal arithmetic (which is not exact).",
"Example: $\\sqrt 2 = 1.414214$",
"Example: $2^(1/2) = 1.414214$",
"Example: $ln 2.0 = 0.69315$. Also evaluates sin, tan, etc.",
"Factor an integer (less than 4 billion). Example: $360 = 2^3\\times 3^2\\times 5$.",
"You will be prompted to enter a value of the variable (or variables)",
"Replace $\\pi $ by an approximate decimal value, 3.14159235...",
"Replace $e$ by an approximate decimal value, 2.718281828...",
"Compute a numerical value of a function using the definition of the function.",
"Example: $x^3-x+1 = (x+1.32472)(x^2 - 1.32472 x + 0.754878)$",
"Evaluate a Bernoulli number to a rational number",
"Evaluate an Euler number to a rational number"
},
{ /* numerical_calculation2 */
"Change some decimals to fractions. Use with caution on approximate values.",
"Example: $64 = 8^2$",
"Example: $1000 = 10^3$",
"Example: $256 = 4^4$. You will be prompted to enter the exponent.",
"Example: $256 = 4^4$. You will be prompted to enter the base.",
"Examples: $36 = 6^2$, or $256 = 2^8$.",
"Example: 3 is selected, you enter 2, the result is 2 + 1.",
},
{ /* complex_arithmetic */
"This is the most important property of the complex number i.",
"Examples: $i^4 = 1$, $i^8 = 1$, $i^12 = 1$",
"Examples: $i^5 = i$, $i^9 = i$, $i^(-3) = i$",
"Example: $i^6 = -1$",
"Example: $i^7 = -i$",
"Perform exact arithmetic (but not exponentiation) on complex numbers.",
"Example, $(1+i)^2 = \\sqrt 2 i$.",
"Perform exact arithmetic (including exponentiation) on complex numbers.",
"Perform approximate decimal-number arithmetic involving complex numbers.",
"Factor an integer (less than 4 billion). Example: $360 = 2^3\\times 3^2\\times 5$.",
"Factor an integer into Gaussian prime-power factors, e.g. $5 = (1+2i)(1-2i)$",
"Example: $-3+4i = (1+2i)^2$",
"Example: $\\sqrt i = 0.707168 + 0.707168 i$",
"Example, $i^(1/2) = 0.707168 + 0.707168 i$",
"Example, $cos i = 1.543080635$",
"Show the value of an expression after you enter values for the variables."
},
{ /* simplify_sums */
"Drop double minus signs.",
"Example: $-(x^2 - 2x + 1)$ becomes $x^2 + 2x - 1$",
"Example: $-x-5$ becomes $-(x+5)$",
arithhelp,
"Use the associative law. Example: $(a+b) + (c+d) = a+b+c+d$",
"Brings terms of a sum to standard order. Example: $y+x = x+y$",
"Example: $x^2 + 0 + 5 = x^2 + 5$",
"Example: $x^2 + x + sin x - x = x^2 + sin x$",
"Example: $x^2 + 3x + 2x = x^2 + 5x$",
"Example: $x^2 + 3x + 2x^2 + 2x = 3x^2 + 5x$",
"Commutative law: reverse the order of summation in the selected term.",
"Example: $5(1-x)$ becomes $-5(x-1)$",
"Example: $-5x$ becomes $5(-x)$",
"Example: $-5xy$ becomes $5x(-y)$",
"Example: $5x(-y)z$ becomes $5xy(-z)$"
},
{ /*simplify_products */
"Example: $2^100\\times 0$ becomes 0",
"Drop factors of 1.",
"Pull minus signs to the front of a product.",
"Pull minus signs to the front of a product.",
"Pull minus signs to the front of a product.",
"Use the associative law. Example: $(3x^2)(yz) = 3x^2yz$",
"Example: $2x\\times 3y = 6xy$",
"Put factors in a product into standard order. Example: $yx = xy$",
"Use the law $x^n x^m = x^(n+m)$. Example: $x^2x^3 = x^5$.",
"Distributive law. Example: $x(x^2 + 1) = x^3 + x$.",
"Example: $(x-2)(x+2) = x^2-4$",
"Example: $(x+3)^2 = x^2 + 6x + 9$",
"Example: $(x-3)^2 = x^2 - 6x + 9$",
"Example: $(x-1)(x^2+2x+1) = x^3-1$",
"Example: $(x+1)(x^2-2x+1) = x^3+1$",
"Commutative law: reverse the order of terms in a product"
},
{ /* expand_menu */
"Example: $(x+1)(x+2) = x^2 + 3x + 2$",
"Multiply out products of sums in the numerator, but not in the denominator.",
"Multiply out products of sums in the denominator, but not in the numerator.",
"Example: $3x = x + x + x$"
},
{ /* fractions */
"Zero divided by anything nonzero is zero.",
"Anything divided by 1 is unchanged.",
"Definition of reciprocal. Example, $2 \\times (1/2) = 1$",
"Example: $(3/4)(x/y) = 3x/(4y)$",
"Example: $3(x/2) = 3x/2$",
"Example: $x^2 y / x = xy$",
"Add fractions with the same denominator by adding the numerators.",
"Break a fraction whose numerator is a sum into two or more fractions.",
"Break $(a\\pm b)/c$ if one of the resulting fractions will cancel.",
"Example: $(x^2 + 2x + 2)/(x+1) = x+1 + 1/(x+1)$",
"Cancel the greatest common factor of numerator and denominator.",
"Example: $2x/3y = (2/3)(x/y)$",
"Example: $(x^2 + y^2)/\\sqrt 2 = (1/\\sqrt 2) x^2 + y^2$",
"Example: $3e^(it)/\\sqrt 2 = (3/\\sqrt 2) e^(it)$",
"Example: $ax/(2y) = (a/2)(x/y)$",
"Example: $\\sqrt 3x/2 = (\\sqrt 3/2)x$"
},
{ /* signed_fractions */
"Cancel a minus sign from numerator and denominator.",
"Push a minus sign into the numerator.",
"Push a minus sign into the denominator.",
"Pull a minus sign out of the numerator.",
"Pull a minus sign out of the denominator.",
"Pull minus signs out of a sum in the numerator.",
"Pull minus signs out of a sum in the denominator.",
"Change the order of terms in the denominator and adjust the sign.",
"Pull minus signs out of a sum in the denominator.",
"Pull minus signs out of a sum in the numerator.",
"Change the order of terms in the denominator and adjust the sign.",
"Example: $(1-x)/(3-x) = (x-1)/(x-3)$",
"Example: $2x/3 = 2(x/3)$",
"Example: $1/(x(1-x^2)) = (1/x)(1/(1-x^2)$"
},
{ /* compound_fractions */
"Example: $x/2 /(y/2) = x/y$",
"Example: $3/(2/x) = 3x/2$",
"Example: $1/(2/x) = x/2$",
"Example: $(3/2)/x = 3/(2x)$",
"Example: $(2/3)/x = (2/3)(1/x)$",
"Example: $(2/3)x/y = 2x/3y$",
"Example: $1/(x^2+2x+1) = 1/(x+1)^2$",
"Use common denominators on a sum of fractions inside a bigger fraction."
},
{ /* common_denominators */
"Example: $1/(x^2+2x+1) = 1/(x+1)^2$",
"Example: $1/x + 1/y = 1/x(y/y) + (1/y)(x/x)$",
"Same as find common denom, but ignores non-fractions in a sum.",
"Example: $(x/2)(y/3) = xy/6$",
"Example: $2(x/y) = 2x/y$",
"Put factors of a product in standard order. Example: $yx = xy$",
"Add fractions with the same denominator by adding the numerators.",
"Example: $1/x + 1/y + 1 = (y+x+xy)/(xy)$",
"Example: $1/x + 1/y + 1 = (y+x)/(xy) + 1$",
"Example: $y/x + x/y = (x^2+y^2)/xy$",
"Ignores non-fractions in the sum, working only on the fractions.",
"You specify what to multiply by. Example, $x/y = x^2/xy$ if you enter $x$."
},
{ /* exponents */
"Anything to the zero power is 1; except $0^0$ is undefined.",
"The first power of $x$ is just $x$.",
"Zero to any positive power is zero.",
"1 raised to any power is 1.",
"Examples: $(-1)^4 = 1$ and $(-1)^3 = -1$",
"$c\\in Z$ means that $c$ is an integer.",
"Here the number $a$ has to be positive.",
"Provided the new numerator and denominator are defined.",
"Example: $(2x)^2 = 4x^2$",
"Example: $(x+1)^2 = x^2+2x+1$",
"Example: $(x+1)^3 = x^3 + 3x^2 + 3x + 1$",
"Example: $x^2x^3 = x^5$",
"Example: $$3^(2+x) = 3^2 3^x$$",
"Example: $a^2/b^2 = (a/b)^2$",
"Example: $x^5/x^3 = x^2$",
"Example: $x^3/x^5 = 1/x^2$"
},
{ /* expand_powers */
"Example: $(x+1)^2 = (x+1)(x+1)$",
"Example: $(x+1)^3 = (x+1)(x+1)(x+1)$",
"Example: $(x+1)^4 = (x+1)(x+1)(x+1)(x+1)$",
"Example: $x^5 = x^2 x^3$. You enter the 2 when prompted.",
"Example: $(x+1)^2 = x^2 + 2x + 1$",
"Example: $(x+1)^3 = x^3 + 3x^2 + 3x + 1$",
"Example: $(x-1)^3 = x^3 - 3x^2 + 3x - 1$",
"Example: $2^(2n)=(2^2)^n$",
"Example: $2^(2n)=(2^n)^2$",
"Example: $2^(2nm) = (2^(2n))^m$",
"Example: $1/2^n = (1/2)^n$"
},
{ /* negative_exponents */
"Eliminate a constant negative exponent",
"Eliminate a constant negative exponent",
"Eliminate a negative exponent.",
"Eliminate a negative exponent. Example: $x^(-2) = 1/x^2$",
"Eliminate a negative exponent. Example: $x^(-2)/3 = 1/(3x^2)$",
"Eliminate a negative exponent in the denominator. Example: $1/x^(-2) = x^2$",
"Eliminate a negative exponent in the denominator. Example: $3/x^(-2) = 3x^2$",
"Example: $2/x = 2x^(-1)$",
"Example: $(2/x)^(-2) = (x/2)^2$",
"Example: $x^5/x^3 = x^2$",
"Example: $x^3/x^5 = 1/x^2$",
"Example: $x^(n-2) = x^n/x^2$"
},
{ /* square_roots */
"Provided both sides are defined. Example: $\\sqrt 2\\sqrt x = \\sqrt (2x)$",
"Provided both sides are defined. Example: $\\sqrt (2x) = \\sqrt 2\\sqrt x$",
"Example: $\\sqrt (4y) = 2\\sqrt y$",
"Square and square root are inverses, so long as $x$ is nonnegative.",
"If you don't know the sign of $x$, you need the absolute value sign.",
"Example: $\\sqrt 8 = \\sqrt 2^3$",
"Provided both sides are defined. Example: $\\sqrt (x/2) = \\sqrt x/\\sqrt 2$",
"When the signs of x and y are not known, you need the absolute value sign.",
"Provided both sides are defined. Example $\\sqrt x/\\sqrt 2 = \\sqrt (x/2)$",
"Since $\\sqrt x \\sqrt x = x$ by definition of $\\sqrt $. Of course, $x$ must be nonnegative.",
"Since $\\sqrt x \\sqrt x = x$ by definition of $\\sqrt $. Of course, $x$ must be nonnegative.",
"Example, $(\\sqrt x)^6 = x^3$",
"Example, $(\\sqrt x)^5 = x^2\\sqrt x$",
"Compute square roots if the value is a rational number. Example, $\\sqrt 16 = 4$",
"Compute approximate decimal values of square roots. Example, $\\sqrt 2 = 1.41416...$",
"Does not compute square roots or roots; performs (other) arithmetic."
},
{ /* advanced_square_roots */
"Example: $\\sqrt (x^2+2x+1)/\\sqrt (x^2-1) = \\sqrt (x+1)^2/\\sqrt (x-1)(x+1)$",
"Example: $\\sqrt (x^2+2x+1) = \\sqrt (x+1)^2$",
"Example: $1/(1-\\sqrt x) = (1+\\sqrt x)/((1-\\sqrt x)(1+\\sqrt x))$ and so later to $(1+\\sqrt x)/(1-x)$",
"Example: $(1-\\sqrt x)/(1+\\sqrt x) = (1-\\sqrt x)(1+\\sqrt x)/(1+\\sqrt x)^2$ and so later to $(1-x)/(1+\\sqrt x)^2$",
"If you don't know the sign of $x$, the absolute value sign is necessary.",
"Example: $\\sqrt (2x)/\\sqrt 2 = \\sqrt x$",
"Multiply out products of sums occurring inside a square root.",
"The operation $a^2-b^2 = (a-b)(a+b)$ will not create a new root; this one will.",
"$^2\\sqrt $ and $\\sqrt $ are two symbols with the same meaning.",
"Example: $\\sqrt x = ^4\\sqrt x^2$. You will be prompted to enter $n$.",
"Example: $\\sqrt x = (^4\\sqrt x)^2$. You will be prompted to enter $n$.",
"Example: $\\sqrt x^4 = x^2$",
"Example: $\\sqrt x^5 = x^2 \\sqrt x$",
"The factor outside the root must be nonnegative.",
"Example: $1/(1-\\sqrt x) = (1+\\sqrt x)/(1-x)$"
},
{ /* fractional_exponents */
"Express a fractional exponent of $\\onehalf $ as a square root.",
"Example: $a^(5/2) = \\sqrt (a^5)$",
"Example: $a^(5/3) = ^3\\sqrt (a^5)$",
"Express a square root using an exponent of $\\onehalf $",
"Express a root using a fractional exponent.",
"Example: $^3\\sqrt x^2 = x^(2/3)$",
"Example: $(^3\\sqrt x)^2 = x^(2/3)$",
"Example: $(\\sqrt x)^3 = x^(3/2)$",
"Express $1/\\sqrt x$ using a negative fractional exponent.",
"Express the reciprocal of a root using a negative fractional exponent",
"Example: $(-1)^(5/3) = -1$. Does not use complex roots.",
"Example: $8^(2/3) = (2^3)^(2/3)$",
"Example: $x/x^(1/3) = (x^3/x)^(1/3)$",
"Example: $x^(1/3)/x = (x/x^3)^(1/3)$",
"Example: $$x^(n/2) = (sqrt x)^n$$",
"Example: $$x^(n/3) = root(3,x)^n$$"
},
{ /*nth_roots */
"Example: $^3\\sqrt 5^3\\sqrt x = ^3\\sqrt (5x)$",
"Example: $^3\\sqrt (2x) = ^3\\sqrt 2 ^3\\sqrt x$",
"Example: $^3\\sqrt x^2 = (^3\\sqrt x)^2$",
"Example $^3\\sqrt x^5 = x ^3\\sqrt x^2$",
"Example: $^3\\sqrt (x^3) = x$", /* rootofpower */
"Example: $^3\\sqrt x^6 =x^2$",
"Example: $^6\\sqrt x^3 = \\sqrt x$", /* rootofpower2 */
"Example: $^9\\sqrt x^3) = ^3\\sqrt x$", /* rootofpower4 */
"Example: $(^3\\sqrt x)^3 = x$", /* powerofroot */
"Example: $(^3\\sqrt a)^2 = ^3\\sqrt (a^2)$", /* powerofroot2 */
"Example $(^3\\sqrt a)^8 = a^2 ^n\\sqrt a^2$", /* powerofroot3 */
"Example: $^3\\sqrt 12 = ^3\\sqrt (2^2\\times 3)$",
"Example: $^3\\sqrt (-a) = -^3\\sqrt a$, n odd",
"Perform arithmetic, evaluating roots to rational values if possible.",
"Example: $^3\\sqrt (x^3+3x^2+3x+1) = ^3\\sqrt (x+1)^3$",
"Multiply out sums of products under a root sign."
},
{ /* roots_of_roots */
"Example: $\\sqrt (\\sqrt 2) = ^4\\sqrt 2$",
"Example: $\\sqrt (^3\\sqrt 2) = ^6\\sqrt 2$",
"Example: $^3\\sqrt (\\sqrt 2) = ^6\\sqrt 2$",
"Example: $^3\\sqrt (^4\\sqrt 2) = ^(12)\\sqrt 2$"
},
{ /* roots_and_fractions */
"Write a root of a quotient as a quotient of roots",
"Write a quotient of roots as a root of quotients",
"Example: $x/^3\\sqrt x = (^3\\sqrt x)^2$",
"Example: $^3\\sqrt x/x = 1/(^3\\sqrt x)^2$",
"Example: $^3\\sqrt (2x)/^3\\sqrt (2y) = ^3\\sqrt x/^3\\sqrt y$",
"Example: $^n\\sqrt (2a)/^n\\sqrt a = ^n\\sqrt 2$",
"Find the greatest common divisor of u and v and factor it out of u and v",
"Example: $x^3\\sqrt y = ^3\\sqrt (x^3y)$",
"Example: $x^2(^4\\sqrt y) = ^4\\sqrt (x^8y)$",
"Example: $-^3\\sqrt 2 = ^3\\sqrt (-2)$",
"Example: $x/^3\\sqrt x = ^3\\sqrt (x^3/x)$",
"Example: $^3\\sqrt x/x = ^3\\sqrt (x/x^3)$",
"Example: $x^2/\\sqrt x = \\sqrt (x^4/x)$",
"Example: $\\sqrt x/x^2 = \\sqrt (x/x^4)$",
"Example: $(^6\\sqrt x)^2 = ^3\\sqrt x$",
"Example: $(^4\\sqrt x)^2 = \\sqrt x$"
},
{ /* complex_numbers */
"Since $i^2 = -1$, we have $1/i = -i$",
"Since $i^2 = -1$, we have $a/i = -ai$",
"Since $i^2 = -1$, we have $a/(bi) = -ai/b$",
"By definition, $i = \\sqrt (-1)$",
"Example: $\\sqrt (-3) = i\\sqrt 3$",
"Example: $1/i^3 = i$",
"Example: $(x-i)(x+i) = x^2+1$",
"Factor a sum of squares using complex factors.",
"This is really just the Pythagorean theorem.",
"This is the definition of absolute value of a complex number.",
"Example: $(3 + 5i)/2 = (3/2) + (5/2)i$",
"Bring a complex number to the standard form $u+vi$",
"Example: $\\sqrt i = \\sqrt(1/2) + \\sqrt(1/2) i$",
"Example: $\\sqrt(-i) = \\sqrt(1/2) - \\sqrt(1/2) i$",
"Example: $\\sqrt(3+4i) = \\sqrt((5+3)/2) + \\sqrt((5-3)/2) i$",
"Example: $\\sqrt(3-4i) = \\sqrt((5+3)/2) - \\sqrt((5-3)/2) i$"
},
{ /* factoring */
"Example: $2x^2 + 4x + 2 = 2(x^2 + 2x + 1)$",
"Example: $x^2 + x + 1/4 = (1/4) (4x^2+ 4x + 1)$",
"Example: $x^3y^2-x^3 = x^3(y^2-1)$",
"Example: $x^5 - x^3 = x^3(x^2-1)$",
"Example: $x^2+2x+1 = (x+1)^2$",
"Example: $x^2-2x+1 = (x-1)^2$",
"Example: $x^2-1 = (x-1)(x+1)$",
"Example: $x^2-3x+1 = (x-2)(x-1)$",
"Example: $x^2-x-1 = (x-1/2-\\sqrt 5/2)(x-1/2+\\sqrt 5/2)$",
"Example: $x^8 = (x^4)^2$",
"Example: $a^2b^2 = (ab)^2$",
"Example: $4x^2 + 6x + 9 = 2^2x^2 + 2\\times 3x + 3^2$",
"Factor an integer (less than 4 billion). Example: $360 = 2^3\\times 3^2\\times 5$",
"Introduce a new letter by a definition, to simplify the expression.",
"Replace a defined variable by its original definition throughout the line.",
"When solving equations, constants are treated differently than variables.",
},
{ /* advanced_factoring */
"No new variable will be used.",
"No new variable will be used.",
"Example: $x^12 = (x^4)^3$",
"Example: $x^12 = (x^3)^4$. You enter the 4 when prompted.",
"Factor a difference of cubes. Example: $x^3-1 = (x-1)(x^2+x+1)$",
"Factor a sum of cubes. Example: $x^3+1 = (x+1)(x^2-x+1)$",
"Example: $x^5-1 = (x-1)(x^4 + x^3 + x^2 + x + 1)$",
"Example: $x^4-1 = (x+1)(x^3 - x^2 + x - 1)$",
"Example: $x^5+1 = (x+1)(x^4 - x^3 + x^2 - x + 1)$",
"Example: $x^4+1 =(x^2-\\sqrt 2x+1)(x^2+\\sqrt 2x+1)$",
"Example (with $p=5$, $q=3$): $x^4+x^2+25=(x^2-3x+5)(x^2+3x+5)$",
"You do not select a term, but let MathXpert try to find a good substitution.",
"You enter a factor, and MathXpert gets the other factor by polynomial division.",
"Systematically tries all possible linear factors with integer coefficients.",
"Break the sum into two groups and factor out their greatest common divisor.",
"Write it as a polynomial in the selected term."
},
{ /* solve_equations */
"Example: $3 = x$ becomes $x = 3$",
"Example: $-x = -3$ becomes $x = 3$",
"Example: $x-3 = 2$ becomes $x = 5$",
"Example: $x+3 = 5$ becomes $x = 2$",
"Example: $x-3 = 5$ becomes $x = 8$",
"Example: $x^2 = x-1$ becomes $x^2-x+1 = 0$",
"Example: $x/2 = x + 1$ becomes $x = 2x + 2$",
"Example: $2x = 4$ becomes $x = 2$",
"Example: $\\sqrt x = 3$ becomes $x = 9$",
"Example: $x+y = 3+y$ becomes $x = 3$",
"Example: $2x^2 = 2$ becomes $x^2 = 1$",
"Example: $x^2 = x-1$ becomes $x^2-x+1 = 0$",
"Example: $3x = 3x$ becomes 'true'",
"Example: $\\sqrt x = -\\sqrt x$ becomes $x = -x$",
"Example: $\\sqrt x = -\\sqrt x$ becomes $\\sqrt x = 0$",
"Example: $-\\sqrt x = \\sqrt x$ becomes $\\sqrt x = 0$",
},
{ /* quadratic_equations */
"if $ab=0$ then $a=0$ or $b=0$",
"quadratic formula",
"$x = -b/2a \\pm \\sqrt (b^2-4ac)/2a$",
"complete the square",
"take square root of both sides",
"cross multiply",
"$b^2-4ac < 0$ => no real roots",
"Use this when the sign of $a$ cannot be determined.",
arithhelp
},
{ /* numerical_equations */
"Enter a value of the unknown and see the values of the two sides.",
"You will be asked for two values known to bracket a root.",
},
{ /* advanced_equations */
"Example: $x/3 = (x-1)/4$ becomes $4x = 3(x-1)$",
"Raise both sides to a power. The new equation can have extra roots.",
"Example: $x^2 = 9$ becomes $[x = 3, x = -3]$",
"Example: $x^3 = 8$ becomes $x = 2$",
"You will be asked what function to apply to both sides.",
"Put sums involving fractions over a common denominator.",
"Example: $(x^2-1)(x-2) = 0$ becomes $[x^2-1=0, x=2]$",
"Example: $ax^2=ax$ becomes $[a=0, x^2=x]$",
"The other equations will be hidden while you work on the selected one.",
"The equations you hid some time ago will be shown again.",
"Duplicate solutions can be combined.",
"It will work if the proposed substitution eliminates an old variable.",
"Replace a variable by its original definition throughout the line.",
"Example: $x = \\sqrt -3$ when seeking real solutions.",
"Some operations may have introduced extra roots which won't check.",
"Example: $3x-1 = x+1$ becomes $x=1$"
},
{ /* cubic_equations */
"This substitution will eliminate the quadratic term.",
"The discriminant of a cubic equation cx^3+ax+b is $D = b^2/4c + a^3/27c^3$.",
"Repeats the cubic equation so you can continue working on it.",
"This substitution will make the equation quadratic in $y^3$.",
"in $cx^3+ax+b=0$: $x=^3\\sqrt (-b/2c+\\sqrt D)+^3\\sqrt (-b/2c-\\sqrt D)$ where D = b^2/4c + a^3/27c^3.",
"in $cx^3-ax+b=0$: $x=[2\\sqrt (a/3)cos(t/3),2\\sqrt (a/3)cos(t+2pi/3),2\\sqrt (a/3)cos(t+4pi/3)]$ where $cos t = -b/(2c)\\sqrt (27/a^3)$.",
"in $cx^3+ax+b=0$: $x=[^3\\sqrt (-b/2c+\\sqrt D)+^3\\sqrt (-b/2c-\\sqrt D),(1/2)^3\\sqrt (-b/2c+\\sqrt D)+^3\\sqrt (-b/2c-\\sqrt D) \\pm (\\sqrt 3/2)(^3\\sqrt (-b/2c+\\sqrt D)-^3\\sqrt (-b/2c-\\sqrt D)]$",
"Make a substitution $x = f(u)$ where $x$ is an old variable and $u$ is new.",
"Eliminate a defined variable using its definition.",
"Example, change $n$ to $1-k$. Equivalent since $1-k$ takes all integer values.",
"Evaluate square and $n$-th roots if the answer is a rational number.",
"Evaluate numerical quantities using approximate decimal values.",
"Perform algebraic simplification."
},
{ /* logarithmic_equations */
"Example: $ln x = 2$ becomes $x = e^2$",
"Example: $ln x = 2$ becomes $x = e^2$",
"Example: $log x = 2$ becomes $x = 100$",
"Example: $log(3,x) = 2$ becomes $x = 9$",
"Example: $10^(x+1) = 10^(2x)$ becomes $x+1 = 2x$",
"Example: $10^x = 3$ becomes $x = log 3$",
"Example: $e^x = 3$ becomes $x = ln 3$",
"Logarithms of negative numbers are not defined.",
},
{ /* cramers_rule */
"Cramer's rule",
"Evaluate a numerical determinant, or a symbolic one of dimension 2 or 3.",
},
{ /* several_linear_equations*/
"Example: $x-1 = 2+y$ becomes $x - y = 1$",
"Example: $2x + 3 + x = 5$ becomes $3x + 3 = 5$",
"Align the terms in the same variable in the same column.",
"You will be asked for the numbers of the two equations.",
"You will be asked for the numbers of the two equations.",
"You will be asked for the equation number and what to multiply by.",
"You will be asked for the equation number and what to divide by.",
"You will be asked for the equation numbers and multiplier.",
"You will be asked for the equation numbers and multiplier.",
"You will be asked for the two equation numbers",
"Example: $y=1$, $x=2$ will be changed to $x=2$, $y=1$.",
"Eliminate an equation that has simplified to an identity, such as 2=2.",
"You will select a variable, which will subsequently be treated as constant.",
"Example: if you have derived $x = 5$, $x = 2$, the equations cannot be satisfied."
},
{ /* selection_mode_only */
"Push a nonnegative quantity inside absolute value.",
"Push a nonnegative denominator inside absolute value.",
"Push a nonnegative fraction inside absolute value.",
"Solve a linear equation for the selected variable."
},
{ /* linear_equations_by_selection */
"You will be asked for the number of the equation that will change.",
"You will be asked for the number of the equation that will change.",
"You will be asked what to multiply the selected equation by.",
"You will be asked what to divide the selected equation by.",
"You will be asked for the multiplier and the target equation.",
"You will be asked for the multiplier and the target equation.",
"You will be asked for the number of the other equation.",
"You will be asked to select a variable.",
"You will be asked for the number of the row that will change.",
"You will be asked for the number of the row that will change.",
"You will be asked for the multiplier.",
"You will be asked for the divisor.",
"You will be asked for the multiplier and the other row number.",
"You will be asked for the multiplier and the other row number.",
"You be asked for the number of the other row.",
"Insert an identity matrix on the right (for calculating the matrix inverse)."
},
{ /* linear_equations_by_substitution */
"Example: $2x + 3y + x = 5$ becomes $3x + 3y = 5$.",
"You will be asked to choose an equation number and then a variable.",
"Perform algebraic simplifications.",
"Example, $x + y = x + 2$ becomes $y = 2$",
"You will be asked to choose an equation and then to enter what to add.",
"You will be asked to choose an equation and then to enter what to subtract.",
"You will be asked to choose an equation and then to enter the divisor.",
"When one equation is solved, you can use it to substitute in other equations.",
"Example: if you have derived $x=2$ and $x=5$, the equations cannot be satisfied."
},
{ /* matrix_methods */
"write in matrix form",
"Insert an identity matrix on the right (for calculating the matrix inverse).",
"You will be asked which two rows to swap.",
"You will be asked for the numbers of the two rows.",
"You will be asked for the numbers of the two rows.",
"You will be asked for the number of the row and the multiplier.",
"You will be asked for the number of the row and the divisor.",
"You will be asked for two row numbers and the multiplier.",
"You will be asked for two row numbers and the multiplier.",
"Perform matrix multiplication.",
"Use this if you have all zeroes in one column.",
"Use this if you have all zeroes in one row.",
"Use this if two rows are exactly the same.",
"Use this if two rows are the same on the left, but not on the right.",
"Convert an equation of one-column matrices to a system of equations."
},
{ /* advanced_matrix_methods */
"Perform matrix multiplication",
"The matrix inverse will not be computed yet, just introduced symbolically.",
"Compute the matrix inverse of a 2 by 2 matrix.",
"Uses exact arithmetic and symbolic algebra. If it works the answer is exact.",
"Works on a numerical matrix, using decimal arithmetic with limited accuracy."
},
{ /* absolute_value */
"Drop absolute value signs around a nonnegative quantity.",
"Example: $|x-2| = x-2$, entering a new assumption $x\\ge 2$.",
"Example: $|-2| = 2$",
"Example: $|2u| = 2|u|$",
"Example: $|u/2| = |u|/2$",
"Example: $|x-1||x+1| = |(x-1)(x+1|$",
"Example: $|(x-1)(x+1)| = |x-1||x+1|$",
"Example: $|(x-1)/x| = |x-1| / |x|$",
"Example: $|x^2-1| / |x-1| = |(x^2-1)/(x-1)|$",
"Example: $|x|^4 =x^4$",
"Example: $|u^3|=|u|^3$",
"If $u$ is real, the absolute value on the right is unnecessary.",
"Example: $|^3\\sqrt u| = ^3\\sqrt |u|$",
"Cancel, disregarding absolute value signs.",
"Cancel, disregarding absolute value signs.",
"Factor out the greatest common divisor of numerator and denominator.",
},
{ /* absolute_value_ineq1 */
"Example: $|x|=2$ becomes $[x = 2, x = -2]$",
"Examples: $|x|/x = x-2$ becomes $[x-2 = 1, x-2 = -1]$",
"Example: $|x| < 2$ becomes $-2 < u < 2$",
"Example: $|x| \\le 2$ becomes $-2 \\le u \\le 2$",
"Example: $2 < |x|$ iff $x < -2$ or $2 < x$",
"Example: $2 \\le |x|$ iff $x \\le -2$ or $2 \\le x$",
"Example: $|x-1| = x-1$ becomes $0 \\le x-1$",
"Example: $|x-1| = 1-x$ becomes $x-1 \\le 0$",
"Example: $0 \\le |x^2+1|$ is always true.",
"Example: $-5 \\le |x^2+1|$ is always true.",
"Example: $-5 < |x^2+1|$ is always true.",
"Example: $|x^2+1| < 0$ has no solution.",
"Example: $|x| < -5$ has no solution.",
"Example: $|x| \\le -5$ has no solution.",
"Example: $|x^3-x| \\le -x^2$ becomes $x^3-x=0$, and $x=0$ will be assumed.",
"Example: $|x^3-x| = -x^2$ becomes $x^3-x=0$, and $x=0$ will be assumed."
},
{ /* absolute_value_ineq2 */
"Example: $2 > |x|$ becomes $-2 < x < 2$",
"Example: $2 \\ge |x|$ becomes $-2 \\le x \\le 2$",
"Example: $|x| > 2$ iff $-2 > x$ or $x > 2$",
"Example: $|x| \\ge 2$ iff $-2 \\ge x$ or $x \\ge 2$",
"Example: $|x^2-1| \\ge 0$ is true.",
"Example: $0 > |x^2-1|$ has no solution.",
"Example: $-5 > |x|$ has no solution.",
"Example: $-5 \\ge |x|$ has no solution.",
"Example: $-x^2 \\ge |x^3-x|$ becomes x^3-x = 0, and x=0 will be assumed.",
"Example: $|x| > -5$ is true",
"Example: $|x| \\ge -5$ is true",
"Example: $-2 \\le u \\le 2$ becomes $|x| \\le 2$",
"Example: $x < -2$ or $2 < x$ iff $2 < |x|$",
"Example: $x^4 = |x|^4$",
"Example: $|u|^3 = |u^3|$"
},
{ /* less_than */
"Example: $2 < x$ becomes $x > 2$",
"Example: $x-2 < 5$ becomes $x<7$. Select the 2.",
"Example: $x+2 < 5$ becomes $x=3$. Select the 2.",
"Example: $-2 < -x$ becomes $x < 2$.",
"Example: $-x < - 2$ becomes $x > 2$.",
"Example: $x/3 < 1$ becomes $x < 3$. Select the 3.",
"$x/(x-1) < 2$ becomes $x(x-1) < 2(x-1)^2$ when you select $x-1$.",
"Example: $5x < 10$ becomes $x < 2$. Select the 5.",
"Produces 'No solution' or 'true', when the equality involves only numbers.",
"Simplify an inequality of the form mentioned to 'true'.",
"Simplify an inequality of the form mentioned to 'No solution'.",
"$u < v$ becomes $u^2 < v^2$ provided $u$ is nonnegative. $0\\le v$ will be derived or assumed.",
"$u < v$ becomes $[u^2 < v^2, u<=0]$. Use this if $u$ can take negative values.",
"Example: $x<4$ or $x=4$ becomes $x\\le 4$. The \"or\" is implicit in bracket notation.",
"Example: $1<x$ or $2<x$ becomes $1<x$",
"Use assumptions to reject or improve solutions to satisfy the original inequality."
},
{ /* greater_than */
"Example: $2 > x$ becomes $x < 2$",
"Example: $-x > -2$ becomes $x < 2$",
"Example: $-2 > -x$ becomes $x > 2$",
"Example: $x^2 > -1$ is true",
"Example: $-1 > x^2$ is false",
"Example: $2 > x$ becomes $[4 > x^2, x < 0]$",
"Example: $[x > 2, x = 2]$ becomes $x \\ge 2$"
},
{ /* less_than_or_equal */
"Example: $x \\le 2$ becomes $2 \\ge x$",
"Example: $x-2 \\le 5$ becomes $x\\le 7$. Select the 2.",
"Example: $x+2 \\le 5$ becomes x=3. Select the 2.",
"Example: $-2 \\le -x$ becomes $x \\le 2$.",
"Example: $x \\le -2$ becomes $x \\ge 2$.",
"Example: $x/3 \\le 1$ becomes $x \\le 3$. Select the 3.",
"Example: $x/(x-1) \\le 2$ becomes $x(x-1) \\le 2(x-1)^2$. Select x-1",
"Example: $x/5 \\le 10$ becomes $x \\le 2$. Select the 5.",
"Produces 'No solution' or 'true', when the equality involves only numbers.",
"Simplify an inequality of the form mentioned to 'true'.",
"Simplify an inequality of the form mentioned to 'No solution'." ,
"$u \\le v$ becomes $u^2 \\le v^2$ provided u is nonnegative. $0\\le v$ will be derived or assumed.",
"$u \\le v$ becomes $u^2 \\le v^2$ or $u\\le 0$. Use this if u can take negative values.",
"Example: $1\\le x$ or $2\\le x$ becomes $1\\le x$",
"Use assumptions to reject or improve solutions to satisfy the original inequality."
},
{ /* greater_than_or_equal */
"Example: $2 \\ge x$ becomes $x \\le 2$",
"Example: $-x \\ge -2$ becomes $x \\le 2$",
"Example: $-2 \\ge -x$ becomes $x \\ge 2$",
"Example: $x^2 \\ge -1$ is true",
"Example: $-1 \\ge x^2$ is false",
"Example: $2 \\ge x$ becomes $[4 \\ge x^2, x \\le 0]$"
},
{ /* square_ineq1 */
"Example: $x^2 < 4$ becomes $|x| < 2$",
"Example: $x^2 < 4$ becomes $-2 < x < 2$",
"Example: $4 < x^2$ becomes $2 < |x|$",
"Example: $4 < x^2$ becomes $[x < -2, 2 < x]$",
"Example: $4 < x^2 < 9$ becomes $[-3 < x < -2, 2 < x < 3]$",
"Example: $-2 < x^2 < 9$ becomes $x^2 < 9$",
"Example: $-2 < x^2 \\le 9$ becomes $x^2 \\le 9$",
"Example: $\\sqrt x < 2$ becomes $0 \\le x < 4$",
"Example: $2\\sqrt x < 2$ becomes $0 \\le 4x < 4$",
"Example: $2 < \\sqrt x$ becomes 4 < x",
"Example: $x^2 < a => x < \\sqrt a$ if $0\\le x$ is already assumed.",
"Example: $-1 < x^2$ is always true.",
"Example: $x^2 < -1$ has no solution.",
"Example: $-1 < \\sqrt (x^2 - 1)$ becomes $0 \\le x^2 -1$"
},
{ /* square_ineq2 */
"Example: $x^2 \\le 4$ becomes $|x| \\le 2$",
"Example: $x^2 \\le 4$ becomes $-2 \\le x \\le 2$",
"Example: $4 \\le x^2$ becomes $2 \\le |x|$",
"Example: $4 \\le x^2$ becomes $[x \\le -2, 2 \\le x]$",
"Example: $4 \\le x^2 \\le 9$ becomes $[-3 \\le x \\le -2, 2 \\le x \\le 3]$",
"Example: $-2 \\le x^2 \\le 9$ becomes $x^2 \\le 9$",
"Example: $-2 \\le x^2 < 9$ becomes $x^2 < 9$",
"Example: $\\sqrt x \\le 2$ becomes $0 \\le x \\le 4$",
"Example: $2\\sqrt x \\le 2$ becomes $0 \\le 4x \\le 4$",
"Example: $2 \\le \\sqrt x$ becomes $4 \\le x$",
"Example: $x^2 \\le a => x \\le \\sqrt a$ if $0\\le x$ is already assumed.",
"Example: $-1 \\le x^2$ is always true.",
"Example: $x^2 \\le -1$ has no solution.",
"Example: $-1 \\le sqrt(x^2 - 1)$ becomes $0 \\le x^2 -1$"
},
{ /* recip_ineq1 */
"$1/x < a$ iff $x < 0$ or $1/a < x$, provided $a > 0$",
"$a < 1/x$ iff $0 < x < 1/a$ provided $a > 0$",
"$1/x < -a$ iff $-1/a < x < 0$ provided $a > 0$",
"$-a < 1/x$ iff $x < -1/a$ or $0 < x$ provided $a > 0$",
"Example: $1 < x < 2$ becomes $1/2 < x < 1$",
"Example: $1 < x \\le 2$ becomes $1/2 \\le x < 1$",
"Example: $-2 < 1/x < -1$ becomes $-1 < x < -1/2$",
"Example: $-2 < 1/x \\le -1$ becomes $-1 \\le x < -1/2$",
"Example: -2 < 1/x < 3 becomes [x < -1/2, 1/3 < x]",
"Example: $-2 < 1/x \\le 3$ becomes $[x < -1/2, 1/3 \\le x]$"
},
{ /* recip_ineq2 */
"$1/x \\le a$ iff x < 0 or $1/a \\le x$, provided $a > 0$",
"$a \\le 1/x$ iff $0 < x \\le 1/a$ provided $a > 0$",
"$1/x \\le -a$ iff $-1/a \\le x < 0$ provided $a > 0$",
"$-a \\le 1/x$ iff $x \\le -1/a$ or 0 < x provided $a > 0$",
"Example: $1 \\le 1/x < 2$ becomes $1/2 < x \\le 1$",
"Example: $1 \\le 1/x \\le 2$ becomes $1/2 \\le x \\le 1$",
"Example: $-2 \\le 1/x < -1$ becomes $-1 < x \\le -1/2$",
"Example: $-2 \\le 1/x \\le -1$ becomes $-1 \\le x \\le -1/2$",
"Example: $-2 \\le 1/x < 3$ becomes $[x \\le -1/2, 1/3 < x]$",
"Example: $-2 \\le 1/x \\le 3$ becomes $[x \\le -1/2, 1/3 \\le x]$"
},
{ /* root_ineq1 */
"Example: $x^3 < 27$ becomes $x < 3$",
"Example: $x^4 < 16$ becomes $|x| < 2$",
"Example: $x^4 < 16$ becomes $-2 < x < 2$",
"Example: $16 < x^4$ becomes $2 < |x|$",
"Example: $16 < x^4$ becomes $[x < -2, 2 < x]$",
"Example: $16 < x^4 < 81$ becomes $[-3 < x < -2, 2 < x < 3]$",
"Example: $^4\\sqrt x < 16$ becomes $0 \\le x < 2$",
"Example: $^3\\sqrt x < 2$ becomes $x < 8$",
"Example: $2 ^3\\sqrt x < 1$ becomes $8x < 1$",
"Example: $2 < ^3\\sqrt x$ becomes $8 < x$",
"Example: $^3\\sqrt x < 2$ becomes $x < 8$",
"Example: $x^4 < a$ becomes $x < ^4\\sqrt a$ if $0\\le x$ is already assumed.",
"Example: $-1 < ^4\\sqrt (x^2 - 1)$ becomes $0 \\le x^2 -1$"
},
{ /* root_ineq2 */
"Example: $x^3 \\le 27$ becomes $x \\le 3$",
"Example: $x^4 \\le 16$ becomes $|x| \\le 2$",
"Example: $x^4 \\le 16$ becomes $-2 \\le x \\le 2$",
"Example: $16 \\le x^4$ becomes $2 \\le |x|$",
"Example: $16 \\le x^4$ becomes $[x \\le -2, 2 \\le x]$",
"Example: $16 \\le x^4 < 81$ becomes $[-3 \\le x \\le -2, 2 \\le x \\le 3]$",
"Example: $^4\\sqrt x \\le 16$ iff $0 \\le x \\le 2$",
"Example: $^3\\sqrt x \\le 2$ becomes $x \\le 8$",
"Example: $2 ^3\\sqrt x \\le 1$ becomes $8x \\le 1$",
"Example: $2 \\le ^3\\sqrt x$ becomes $8 \\le x$",
"Example: $^3\\sqrt x \\le 2$ becomes $x \\le 8$",
"Example: $x^4 \\le a$ becomes $x \\le ^4\\sqrt a$ if $0\\le x$ is already assumed.",
"Example: $-1 \\le ^4\\sqrt (x^2 - 1)$ becomes $0 \\le x^2 -1$"
},
{ /* zero_ineq1 */
"Example: $0 < x(x^2+1)$ becomes $0 < x$",
"Example: $0 < 1/\\sqrt x$ becomes $0 < \\sqrt x$ ",
"Example: $0 < x/\\sqrt (x-1)$ becomes 0 < x(x-1)",
"Example: 0 < (x-1)/(x-2) becomes 0 < (x-1)(x-2)",
"Example: $1/\\sqrt x < 0$ becomes $\\sqrt x < 0$",
"Example: $x/\\sqrt (x-1) < 0$ becomes $x(x-1) < 0$",
"$ax \\pm b < 0$ iff $a(x\\pm b/a) < 0$",
"$u < v => v > u$",
"Example: $(x-1)(x+1) < 0$ iff $-1 < x < 1$. Also handles more factors.",
"Example: $0 < (x-1)(x+1)$ iff $x < -1$ or $1 < x$. Also handles more factors."
},
{ /* zero_ineq2 */
"Example: $0 \\le x(x^2+1)$ becomes $0 \\le x$",
"Example: $0 \\le 1/\\sqrt x$ becomes $0 \\le \\sqrt x$ ",
"Example: $0 \\le x/\\sqrt (x-1)$ becomes $0 \\le x(x-1)$",
"Example: $0 \\le (x-1)/(x-2)$ becomes $0 \\le (x-1)(x-2)$",
"Example: $1/\\sqrt x \\le 0$ becomes $\\sqrt x \\le 0$",
"Example: $x/\\sqrt (x-1) \\le 0 $becomes $x(x-1) \\le 0$",
"$ax \\pm b \\le 0$ iff $a(x\\pm b/a) \\le 0$",
"$u \\le v => v \\le u$",
"Example: $(x-1)(x+1) \\le 0$ iff $-1 \\le x \\le 1$. Also handles more factors.",
"Example: $0 \\le (x-1)(x+1)$ iff $x \\le -1 or 1 \\le x$. Also handles more factors."
},
{ /* square_ineq3 */
"Example: $4 > x^2$ becomes $2 > |x|$",
"Example: $4 > x^2$ becomes $-2 < x < 2$",
"Example: $x^2 > 4$ becomes $|x| > 2$",
"Example: $x^2 > 4$ becomes $[x < -2, x > 2]$",
"Example: $2 > \\sqrt x$ becomes $0 \\le x < 4$",
"Example: $2 > 2\\sqrt x < 2$ becomes $0 \\le 4x < 4$",
"Example: $\\sqrt x > 2$ becomes $x > 4$",
"Example: $4 > x^2$ becomes $2 > x$ if $0\\le x$ is already assumed.",
"Example: $x^2 > -1$ is always true.",
"Example: $-1 > x^2$ has no solution.",
"Example: $\\sqrt (x^2-1) > -1$ becomes $x^2-1 \\ge 0$"
},
{ /* square_ineq4 */
"Example: $4 \\ge x^2$ becomes $2 \\ge |x|$",
"Example: $4 \\ge x^2$ becomes $-2 \\le x \\le 2$",
"Example: $x^2 \\ge 4$ becomes $|x| \\ge 2$",
"Example: $x^2 \\ge 4$ becomes $[x \\le -2, 2 \\le x]$",
"Example: $2 \\ge \\sqrt x$ becomes $0 \\le x \\le 4$",
"Example: $2 \\ge 2\\sqrt x$ becomes $0 \\le 4x \\le 4$",
"Example: $\\sqrt x \\ge 2$ becomes $x \\ge 4$",
"Example: $4 \\ge x^2$ => $2 \\ge x$ if $0\\le x$ is already assumed.",
"Example: $x^2 \\ge -1$ is always true.",
"Example: $-1 \\ge x^2$ has no solution.",
"Example: $\\sqrt (x^2-1) \\ge -1$ becomes $x^2-1 \\ge 0$"
},
{ /* recip_ineq3 */
"$a > 1/x$ iff $x < 0$ or $x > 1/a$, provided $a > 0$",
"$1/x > a$ iff $0 < x < 1/a$, provided $a > 0$",
"$-a > 1/x$ iff $-1/a < x < 0$, provided $a > 0$ ",
"$1/x > -a$ iff $x < -1/a$ or $x > 0$, provided $a > 0$"
},
{ /* recip_ineq4 */
"$a \\ge 1/x$ iff $x < 0$ or $x \\ge 1/a$, provided $a > 0$",
"$1/x \\ge a$ iff $0 < x \\le 1/a$, provided $a > 0$",
"$-a \\ge 1/x$ iff $-1/a \\le x < 0$, provided $a > 0$",
"$1/x \\ge -a$ iff $x \\le -1/a$ or $x > 0$, provided $a > 0$"
},
{ /* root_ineq3 */
"Example: $27 > x^3$ becomes $3 > x$",
"Example: $16 > x^4$ becomes $2 > |x|$",
"Example: $16 > x^4$ becomes $-2 < x < 2$",
"Example: $x^4 > 16$ becomes $|x| > 2$",
"Example: $x^4 > 16$ becomes $[-2 > x, x > 2]$",
"Example: $16 < x^4$ < 81 becomes $[-3 < x < -2, 2 < x < 3]$",
"Example: $2 > ^3\\sqrt x$ becomes $8 > x$",
"Example: $1 > 2 ^3\\sqrt x$ becomes $1 > 8x$",
"Example: $^3\\sqrt x > 2$ becomes $x > 8$",
"Example: $2 > ^3\\sqrt x$ becomes $8 > x$ ",
"Example: $a > x^4$ becomes $^4\\sqrt a > x$ if $0\\le x$ is already assumed.",
"Example: $^4\\sqrt (x^2 - 1) > -1$ becomes $x^2 -1 \\ge 0$"
},
{ /* root_ineq4 */
"Example: $27 \\ge x^3$ becomes $3 \\ge x$",
"Example: $16 \\ge x^4$ becomes $2 \\ge |x|$",
"Example: $16 \\ge x^4$ becomes $-2 \\le x \\le 2$",
"Example: $x^4 \\ge 16$ becomes $|x| \\ge 2$",
"Example: $x^4 \\ge 16$ becomes $[-2 \\ge x, x \\ge 2]$",
"Example: $16 \\le x^4 < 81$ becomes $[-3 \\le x \\le -2, 2 \\le x \\le 3]$",
"Example: $2 \\ge ^3\\sqrt x$ becomes $8 \\ge x$",
"Example: $1 \\ge 2 ^3\\sqrt x$ becomes $1 \\ge 8x$",
"Example: $^3\\sqrt x \\ge 2$ becomes $x \\ge 8$",
"Example: $^3\\sqrt x \\le 2$ becomes $x \\le 8$",
"Example: $x^4 \\le a$ becomes $x \\le ^4\\sqrt a$ if $0\\le x$ is already assumed.",
"Example: $^4\\sqrt (x^2 - 1) \\ge -1$ becomes $x^2 -1 \\ge 0$"
},
{ /* zero_ineq3 */
"Example: $1/\\sqrt x > 0$ becomes $\\sqrt x > 0$",
"Example: $x/\\sqrt (x-1) > 0$ becomes $x(x-1) > 0$",
"Example: $(x-1)/(x-2) > 0$ becomes $(x-1)(x-2) > 0$",
"Example: $0 > 1/\\sqrt x$ becomes $0 > \\sqrt x$",
"Example: $0 > x/\\sqrt (x-1)$ becomes $0 > x(x-1)$",
"$0 > ax \\pm b$ iff $0 > a(x\\pm b/a)$",
"Example: $0 > (x-1)(x+1)$ iff $-1 < x < 1$. Also handles more factors.",
"Example: $(x-1)(x+1) > 0 iff $x < -1$ or $1 < x$. Also handles more factors."
},
{ /* zero_ineq4 */
"Example: $1/\\sqrt x \\ge 0$ becomes $\\sqrt x \\ge 0$",
"Example: $x/\\sqrt (x-1) \\ge 0$ becomes $x(x-1) \\ge 0$",
"Example: $(x-1)/(x-2) \\ge 0$ becomes $(x-1)(x-2) \\ge 0$",
"Example: $0 \\ge 1/\\sqrt x$ becomes $0 \\ge \\sqrt x$",
"Example: $0 \\ge x/\\sqrt (x-1)$ becomes $0 \\ge x(x-1)$",
"$0 \\ge ax \\pm b$ iff $0 \\ge a(x\\pm b/a)$",
"Example: $0 \\ge (x-1)(x+1)$ iff $-1 \\le x \\le 1$. Also handles more factors.",
"Example: $(x-1)(x+1) \\ge 0$ iff $x \\le -1$ or $1 \\le x$. Also handles more factors."
},
{ /* binomial_theorem */
"Expands all the way, does not use sigma notation. Can create terms.",
"Expands using sigma notation and binomial coefficients.",
"Express binomial coefficients using factorials.",
"Use the definition of factorial as a product. Does not multiply it out.",
"Compute the value of a factorial. Example: 6! = 720.",
arithhelp,
"Evaluate a specific binomial coefficient. Example: (4 2) = 6",
"Express $\\sum $ using +. The sum must have a constant number of terms.",
"If each term is a number, evaluate using exact rational arithmetic.",
"Example: $7! = 7\\times 6!$",
"Example: $7!/7 = 6!$",
"Example: $7!/6! = 7$",
"Example: $n!/(n-2)! = n(n-1)$",
"Example: $7/7! = 1/6!$",
"Example: $6!/7! = 1/7$",
"Example: $(n-2)!/n! = 1/(n(n-1))$"
},
{ /* factor_expansion */
"Factor the cube of a sum.",
"Factor the cube of a difference.",
"Factor the fourth power of a sum.",
"Factor the fourth power of a difference.",
"Factor a power of a sum.",
"Factor a power of a difference."
},
{ /* sigma_notation */
"Example: the sum of 1 from 1 to 10 is 10.",
"Pull a minus sign out from an indexed sum.",
"Pull a constant out of an indexed sum.",
"Break an indexed sum into two (or more) sums.",
"Break an indexed sum into two (or more) sums.",
"Express $\\sum $ using +. The sum must have a constant number of terms.",
"Example: the sum of $i$ for $i = 1$ to 100 is 100(101)/2 = 5050.",
"Formula for the sum of the first n perfect squares.",
"The sum of $x^i$ for $i=0$ to $n$ has this elegant closed form.",
"You will be asked how many terms to write out explicitly.",
"Specify a parameter value and evaluate using exact rational arithmetic.",
"Specify a parameter value and evaluate using (inexact) decimal arithmetic.",
"Evaluate a numerical sum using exact arithmetic. No parameters allowed.",
"Evaluate a numerical sum using decimal arithmetic. No parameters allowed.",
"Express the summand as a polynomial in the index variable, if possible.",
"Example: the sum of $1/(k+1) - 1/k$ from 1 to $n$ becomes $1/(n+1) - 1$"
},
{ /* advanced_sigma_notation */
"Example: change a sum from k=0 to n to a sum from k = 1 to n+1",
"Before multiplying out a product of sums you may need to rename a variable.",
"Convert a product of sums to a double sum using the distributive law.",
"Example: Change a sum from 1 to $n+1$ to a sum from 1 to $n$, plus the last term.",
"The formula for the sum of the first $n$ cubes",
"The formula for the sum of the first $n$ fourth powers",
"Push a derivative into an indexed sum",
"Pull a derivative out of an indexed sum",
"Push an integral into an indexed sum",
"Pull an integral out of an index sum",
"Push a constant into an indexed sum or series.",
"Write an indexed sum as a difference of two sums with zero as the starting index of summation.",
"Write an indexed sum as a difference of two sums with a new, specified starting index."
},
{ /* prove_by_induction */
"You will be asked to choose the induction variable.",
"You will be asked for the starting value of the induction variable.",
"Assume the induction hypothesis and state what is to be proved.",
"Use the induction hypothesis to simplify the current line.",
"Use this when the induction step is completed, to draw the final conclusion."
},
{ /* trig_ineq */
"Simplify an inequality of the stated form to true.",
"Simplify an inequality of the stated form to true.",
"Simplify an inequality of the stated form to true. Example: $sin x^2 \\le x^2$.",
"Simplify an inequality of the stated form to true.",
"Simplify an inequality of the stated form to true.",
"Simplify an inequality of the stated form to true.",
"Simplify an inequality of the stated form to true.",
},
{ /* log_ineq1 */
"$u < v$ iff $ln u < ln v$, provided $u > 0$.",
"$u < v$ iff $log u < log v$, provided $u > 0$.",
"Example: $2 < ln x$ becomes $e^2 < x$",
"Example: $ln x < 2$ becomes $x < e^2$",
"Example: $2 < log x$ becomes $10^2 < x$",
"Example: $log x < 2$ becomes $x < 10^2$",
"You will specify the number ? to use as the base of exponents."
},
{ /* log_ineq2 */
"$u \\le v$ iff $ln u \\le ln v$, provided u > 0.",
"$u \\le v$ iff $log u \\le log v$, provided u >0.",
"Example: $2 \\le ln x$ becomes $e^2 \\le x$",
"Example: $ln x \\le 2$ becomes $x \\le e^2$.",
"Example: $2 \\le log x$ becomes $10^2 \\le x$.",
"Example: $log x \\le 2$ becomes $x \\le 10^2$.",
"You will specify the number ? to use as the base of exponents."
},
{ /* log_ineq3 */
"$u > v$ iff $ln u > ln v$, provided $u > 0$.",
"$u > v$ iff $log u > log v$, provided $u > 0$.",
"Example: $ln x > 2$ becomes $x > e^2$.",
"Example: $2 > ln x$ becomes $e^2 > x$.",
"Example: $log x > 2$ becomes $x > 10^2$.",
"Example: $2 > log x$ becomes $10^2 > x$.",
"You will specify the number ? to use as the base of exponents."
},
{ /* log_ineq4 */
"$u \\ge v$ iff $ln u \\ge ln v$, provided u > 0",
"$u \\ge v$ iff $log u \\ge log v$, provided u >0",
"Example: $ln x \\ge 2$ becomes $x \\ge e^2$.",
"Example: $2 \\ge ln x$ becomes $e^2 \\ge x$.",
"Example: $log x \\ge 2$ becomes $x \\ge 10^2$.",
"Example: $2 \\ge log x$ becomes $10^2 \\ge x$.",
"You will specify the number ? to use as the base of exponents.",
"Example: $n < 2^n$ for $n > M$, for a specific but unspecified number $M$",
"Example: $ln n < \\sqrt n$ for $n > M$, for a specific but unspecified number $M$"
},
{ /* logarithms_base10 */
"Example: $10^(\\log 3x)$ becomes $3x$.",
"Example: log 100 becomes 2",
"The log of 1 is zero since $10^0 = 1$.",
"The log of 10 is 1, since $10^1 = 1$.",
"Convert logarithms base 10 to natural logarithms.",
"Express a power using base 10 and a log in the exponent.",
"Factor an integer (less than 4 billion). Example: $360 = 2^3\\times 3^2\\times 5$.",
"Example: $400 = 10^2\\times 4$. Does not factor completely, only takes out tens.",
"Example: $10^(2 \\log x)$ becomes $x^2$.",
"Example: $log (4/5) = - log (5/4)$",
"Example: $log(3,4/5) = - log(3, 5/4)$"
},
{ /* logarithms */
"Example: $log x^3 = 3 log x$",
"Example: $log 3x = log 3 + log x$",
"Example: $log 1/2 = -log 2$",
"Example: $log x/2 = log x - log 2$",
"Example: $log 2 + log x = log 2x$",
"Example: $log x - log 2 = log a/2$",
"Example: $log x + log 2 - log 3 =log 2x/3$",
"Example: $2 log x = log x^2$",
"Example: $log \\sqrt 3 = \\onehalf log 3$",
"Example: $log ^3\\sqrt x = (1/3) log x$",
"The log of 1 is 0 since $10^0 = 1$.",
"Factor an integer (less than 4 billion). Example: $360 = 2^3\\times 3^2\\times 5$.",
"Example: $400 = 10^2\\times 4$. Does not factor completely, only takes out tens.",
"You will be asked to enter a. Example: $log x = \\onehalf log x^2$",
"Evaluate logs using decimal approximations.",
"Convert base 10 logarithms to natural logarithms."
},
{ /* logarithms_base_e */
"This fundamental law connects natural logs and the exponential function.",
"In words: $e$ is the base of natural logarithms.",
"The natural log of 1 is 0, since $e^0 = 1$.",
"Example: $ln e^2 = 2$",
"Express an arbitrary power using a power of $e$ and a natural logarithm.",
"Eliminate a natural log in an exponent of $e$."
},
{ /* natural_logarithms */ /* menu 70 */
"Example: $ln x^2 = 2 ln x$",
"Example: $ln 2x = ln 2 + ln x$",
"Example: $ln 1/2 = -ln 24$",
"Example: $4ln x/2 = ln x - ln 24$",
"The natural log of 1 is 0, since $e^0 = 1$.",
"Factor an integer (less than 4 billion). Example: $360 = 2^3\\times 3^2\\times 5$.",
"Example: $ln (x-1) + ln (x+1) = ln (x-1)(x+1)$",
"Example: $ln x - ln 2 = ln x/2$",
"Example: $ln x + ln 2 - ln 3 = ln (2x/3)$",
"Example: $2 ln x = ln x^2$",
"Example: $ln \\sqrt 3 = \\onehalf ln 3$",
"Example: $ln ^3\\sqrt x = (1/3) ln x$",
"You will be asked to enter $a$. Example: $ln (1 + 1/n) = 1/n ln(1+1/n)^n$",
"Evaluate natural logarithms using decimal approximations.",
"Example: $ln (4/5) = - ln (5/4)$"
},
{ /* reverse_trig */
"Example: $sin x cos(\\pi /2) + cos x sin(\\pi /2) = sin(x+\\pi /2)$",
"Example: $sin x cos(\\pi /2) - cos x sin(\\pi /2) = sin(x-\\pi /2)$",
"Example: $cos x cos(\\pi /2) - sin x sin(\\pi /2) = cos(x+\\pi /2)$",
"Example: $cos x cos(\\pi /2) + sin x sin(\\pi /2) = cos(x-\\pi /2)$",
"Example: $(sin 4u)/(1+cos 4u) = tan 2u$",
"Example: $(1-cos 4u)/sin 4u = tan 2u$",
"Example: $(1+cos 4u)/sin 4u = cot 2u$",
"Example: $(sin 4u)/(1-cos 4u) = cot 2u$",
"Example: $(tan x + tan \\pi /2)/(1-tan x tan \\pi /2) = tan(x+\\pi /2)$",
"Example: $(tan x - tan \\pi /2)/(1+tan x tan \\pi /2) = tan(x-\\pi /2)$",
"Example: $(cot x cot(\\pi /4) - 1)/(cot x + cot \\pi /4) = cot(x+\\pi /4)$",
"Example: $(1 + cot x cot \\pi /4)/(cot \\pi /4 - cot x) = cot(x-\\pi /4)$",
"Example: $1-cos(\\pi /3)$ becomes $2sin^2 \\pi /6$"
},
{ /* complex_polar_form */
"Convert $x + iy$ to polar form $r e^(i\\theta )$.",
"Express a complex exponential in terms of cosine and sine.",
"Since $e^(i\\theta )$ lies on the unit circle, its absolute value is 1.",
"Since $Re^(i\\theta )$ lies on the circle of radius $R$, its absolute value is $R$.",
"If the sign of $R$ is unknown, you need the absolute value on the right.",
"Example: $-2 = 2e^(i\\pi )$",
"Example: $$root(3,-2) = e^(pi i/3) root(3,2)$$",
"Example: $2/(3e^t) = 2e^(-t)/3$",
"Example: $x^3 = 1$ becomes $$x = e^(2k pi i/3)$$",
"Example: $$x = e^(2k pi i/3)$$ becomes $$[x=1, x=e^(2 pi i/3), x=e^(4 pi i/3)]$$"
},
{ /* logs_to_any_base */
"Example: $$2^(log(2,3)) = 3$$",
"Example: $$5^(2 log(5,x))=x^2$$",
"The log to the base $b$ of $b$ is 1.",
"Example: $$log(2,2^5) = 5$$",
"Example: $log 2x = log 2 + log x$",
"Example: $log (\\onehalf ) = -log 2$",
"Example: $log x/2 = log x - log 2$",
"The log to any base of 1 is zero, since $b^0 = 1$.",
"Example: $$log(6,x)=log(2*3,x)$$",
"Example: $log(3^2,x) = \\onehalf log (3,x)$",
"Example: $log x^2 = 2 log x$",
"Example: $$log(2, 84) = log(2,2^2 21)$$",
"Example: $log 2 + log x = log 2x$",
"Example: $log x - log 2 = log x/2$",
"Example: $log x + log 2 - log 3 = log 2x/3$",
"Example: $2 log x = log x^2$"
},
{ /* change_base */
"Convert logarithms base $b$ to natural logarithms",
"Convert logarithms base $b$ to logarithms base 10",
"Convert logarithms base $b$ to logarithms base a",
"Example: $log(3^2,x) = (1/2) log (3,x)$",
"Definition of log",
"In words: $e$ is the base of natural logarithms.",
"Convert logarithms base 10 to natural logarithms.",
"Convert natural logarithms to logarithms base 10.",
"Example: $x^5 becomes 3^5 log(3,x)$"
},
{ /* evaluate_trig_functions */
"sin 0 = 0",
"cos 0 = 1",
"tan 0 = 0",
"Sine is zero at multiples of $\\pi $.",
"Cosine is 1 at even multiples of $2\\pi $.",
"Tangent is zero at multiples of $\\pi $.",
"Example: $sin 370\\deg = sin 10\\deg $",
"Example: $sin 9\\pi /4 = sin \\pi /4$",
"Examples: $sin 3\\pi /2 = -1; cos 180\\deg = -1; cot 90\\deg = 0$.",
"Examples: $sin 30\\deg = 1/2; cos \\pi /3 = 1/2; tan 2\\pi /3 = -\\sqrt 3$.",
"Examples: $sin 45\\deg = 1/\\sqrt 2; tan 3\\pi /4 = -1$.",
"$\\pi $ radians = 180 degrees = half a circle of arc",
"180 degrees = $\\pi $ radians = half a circle of arc",
"Example: $15\\deg = 45\\deg - 30\\deg $. Use this to evaluate $sin 15\\deg $ exactly.",
"Evaluate trig functions using decimal approximations."
},
{ /* basic_trig */
"Express tan in terms of sin and cos",
"Express cot in terms of tan",
"Express cot in terms of sin and cos",
"Definition of sec",
"Definition of csc",
"Definition of tan",
"Definition of cot"
},
{ /* trig_reciprocals */
"The reciprocal of sine is the cosecant.",
"The reciprocal of cosine is the secant",
"The reciprocal of the tangent is the cotangent",
"The reciprocal of the tangent can be expressed in terms of sin and cos.",
"The reciprocal of the cotangent is the tangent",
"The reciprocal of the cotangent can be expressed in terms of sin and cos.",
"The reciprocal of the secant is the cosine",
"The reciprocal of the cosecant is the sine.",
"The reciprocal of the sine is the cosecant",
"Definition of sec",
"Express tan in terms of cot"
},
{ /* trig_squares */
"This fundamental identity is the Pythagorean theorem in disguise.",
"Use this form of $sin^2 u + cos^2 u = 1$ to simplify $1 - sin^2 u$.",
"Use this form of $sin^2 u + cos^2 u = 1$ to simplify $1 - cos^2 u$.",
"Express $sin^2$ in terms of $cos^2$.",
"Express $cos^2$ in terms of $sin^2$.",
"To remember this identity, divide $sin^2 + cos^2 = 1$ by $cos^2$.",
"Use this to simplify $tan^2 u + 1$.",
"Use this to simplify $sec^2 u - 1$.",
"Express $sec^2$ in terms of $tan^2$.",
"Express $tan^2$ in terms of $sec^2$.",
"Example: $sin^5 t = sin t (1-cos^2 t)^2$",
"Example: $cos^5 t = cos t (1-sin^2 t)^2$",
"Example: $tan^5 t = tan (sec^2 t-1)^2$",
"Example: $sec^5 t = sec t (tan^2 t+1)^2$",
"Example: $(1-cos t)^2(1+cos t)^2 = sin^4 t$",
"Example: $(1-sin t)^2(1+sin t)^2 = cos^4 t$"
},
{ /* csc_and_cot_identities */
"To remember this identity, divide $sin^2 + cos^2 = 1 by sin^2$.",
"Use this to simplify $cot^2 u + 1$.",
"Use this to simplify $csc^2 u - 1$.",
"Express $csc^2$ in terms of $cot^2$.",
"Express $cot^2$ in terms of $csc^2$.",
"Example: $csc \\pi /6 = sec \\pi /3$",
"Example: $cot \\pi /6 = tan \\pi /3$",
"Example: $cot^5 t = cot (csc^2 t-1)^2$",
"Example: $csc^5 t = csc t (cot^2 t+1)^2$"
},
{ /* trig_sum */
"Example: $sin(x+\\pi /4)= sin x cos \\pi /4 + cos x sin \\pi /4$",
"Example: $sin(x-\\pi /4)= sin x cos \\pi /4 - cos x sin \\pi /4$",
"Example: $cos(x+\\pi /4)= cos x cos \\pi /4 - sin x sin \\pi /4$",
"Example: $cos(x-\\pi /4)= cos x cos \\pi /4 + sin x sin \\pi /4$",
"Example: $tan(x+\\pi /4)=(tan x+tan \\pi /4)/(1-tan x tan \\pi /4)$",
"Example: $tan(x-\\pi /4)=(tan x-tan \\pi /4)/(1+tan x tan \\pi /4)$",
"Example: $cot(x+\\pi /4)=(cot x cot \\pi /4-1)/(cot x+cot \\pi /4)$",
"Example: $cot(x-\\pi /4)=(1+cot x cot \\pi /4)/(cot \\pi /4-cot x)$"
},
{ /* double_angle */
"Examples: $sin 4x = 2 sin 2x cos 2x$; $sin 40\\deg = 2 sin 20\\deg sin 20\\deg $",
"Examples: $cos 4x = cos^2 x - sin^2 x$; $cos 40\\deg = cos^2 20\\deg - sin^2 20\\deg $",
"Express $cos 2\\theta $ in terms of $sin^2 \\theta $.",
"Express $cos 2\\theta $ in terms of $cos^2 \\theta $.",
"Express $cos 2\\theta $ in terms of $cos^2 \\theta $.",
"Express $cos 2\\theta $ in terms of $sin^2 \\theta $.",
"Express $tan 2\\theta $ in terms of $tan \\theta $.",
"Express $cot 2\\theta $ in terms of $cot \\theta $.",
"Express $sin \\theta cos \\theta $ in terms of $sin 2\\theta $",
"Express $2 sin \\theta cos \\theta $ in terms of $sin 2\\theta $",
"Express $cos^2 \\theta - sin^2 \\theta $ as a single trig function, $cos(2\\theta )$",
"Use this to get rid of $sin^2$ in favor of a single trig function.",
"Use this to get rid of $cos^2$ in favor of a single trig function."
},
{ /* multiple_angles */
"Example: $3\\theta = 2\\theta + \\theta $",
"Example: $7\\theta = 3\\theta + 4\\theta $; you enter the 3 when you are asked for it.",
"This triple-angle formula can save you several steps.",
"This triple-angle formula can save you several steps.",
"Example: $sin 7\\theta = -sin^7 \\theta + 21 cos^2 \\theta sin^5 \\theta + ...$",
"Example: $cos 7\\theta = cos^7 \\theta - 21 cos^5 \\theta sin^2 \\theta + ...$"
},
{ /* verify_identities */
"Example: $x/3 = 3/4$ becomes $4x = 9$",
"Example: $3 = x$ becomes $x = 3$",
"The specified term will be moved from the left to the right side.",
"The specified term will be moved from the right to the left side.",
"Add a specified term to both sides",
"Subtract a specified term from both sides",
"Multiply both sides by a specified term.",
"Example: $1 - sin^2 x + tan x = tan x + cos^2 x$ becomes $1-sin^2 x = cos^2 x$.",
"Example: $\\sqrt (1-sin^2 x) = cos x$ becomes $1-sin^2 x = cos^2 x$.",
"Example: $tan^2 x = sin^2 x / cos^2 x$ becomes $tan x = sin x / cos x$",
"Example: $tan^3 x = sin^3 x / cos^3 x$ becomes $tan x = sin x / cos x$",
"You will be asked what function to apply.",
arithhelp,
"Use this to disprove a false identity or to test one you can't verify.",
"Introduce a new letter by a definition, to simplify the expression."
},
{ /* solve_by_30_60_90 */
"These angles are $30\\deg $ above the plus and minus x-axes.",
"These angles are $30\\deg $ below the plus and minus x-axes.",
"These angles are the multiples of $60\\deg $ above the x-axis.",
"These angles are the multiples of $60\\deg $ below the x-axis.",
"That is, plus or minus $30\\deg $.",
"That is, plus or minus $30\\deg $ from the negative x-axis.",
"That is, plus or minus $60\\deg $.",
"That is, plus or minus $120\\deg $.",
"That is, $30\\deg $ plus multiples of $\\pi $ (not $2\\pi $, note $210\\deg $ is included).",
"That is, $-30\\deg $ plus multiples of $\\pi $ (not $2\\pi $, note $150\\deg $ is included).",
"That is, $60\\deg $ plus multiples of $\\pi $ (not $2\\pi $, note $240\\deg $ is included).",
"That is, $-60\\deg $ plus multiples of $\\pi $ (not $2\\pi $, note $120\\deg $ is included)."
},
{ /* solve_by_45_45_90 */
"These angles are $45\\deg $ up from the plus and minus x-axes.",
"These angles are $45\\deg $ down from the plus and minus x-axes.",
"These angles are $45\\deg $ right from the plus and minus y-axes.",
"These angles are $45\\deg $ left from the plus and minus y-axes.",
"That is, $45\\deg $ plus multiples of $\\pi $ (not $2\\pi $, note $225\\deg $ is included).",
"That is, $-45\\deg $ plus multiples of $\\pi $ (not $2\\pi $, note $135\\deg $ is included).",
},
{ /* zeroes_of_trig_functions */
"$sin u$ is zero at multiples of $\\pi $.",
"$sin u$ is 1 when $u$ is $\\pi /2$ plus a multiple of $2\\pi $.",
"$sin u$ is -1 when $u$ is $3\\pi /2$ plus a multiple of $2\\pi $.",
"$cos u$ is 0 when $u$ is an odd multiple of $\\pi /2$.",
"$cos u$ = 1 when $u$ is a multiple of $2\\pi $.",
"$cos u$ = -1 when $u$ is an odd multiple of $\\pi $.",
"Example: $tan x^2 = 0$ becomes $sin x^2 = 0$.",
"Example: $cot x^2 = 0$ becomes $cos x^2 = 0$."
},
{ /* inverse_trig_functions */
"Example: $sin x = 3/4$ becomes $x = (-1)^n arcsin 3/4 + n\\pi $",
"Exmaple: $sin x = 3/4$ becomes $[x = arcsin 3/4 + 2n\\pi , x = -arcsin 3/4 + (2n+1)\\pi ]$",
"Example: $cos x = 3/4$ becomes $[x = arccos 3/4+2n\\pi , x = -arccos 3/4 + 2n\\pi ]$",
"Example: $tan x = 3$ becomes $x = arctan 3 + n\\pi $",
"Example: $arcsin(\\onehalf ) = \\pi /6$. Only a few values will evaluate exactly.",
"Example: $arccos(\\onehalf ) = \\pi /3$. Only a few values will evaluate exactly.",
"Example: $arctan 1 = \\pi /4$. Only a few values will evaluate exactly.",
"If $cot z = x$ then $tan z = 1/x$.",
"If $sec z = x$ then $cos z = 1/x$.",
"If $csc z = x$ then $sin z = 1/x$.",
"arcsin is an odd function",
"arccos is not quite odd but obeys this similar identity.",
"arctan is an odd function.",
"Put the solutions in the form $c + 2n\\pi $, if $2\\pi $ is the period.",
"Example: $sin u = 2$ has no solution.",
"Example: $cos u = 2$ has no solution."
},
{ /* invsimp */
"If $sin \\theta = x$ then $tan \\theta = x/\\sqrt (1-x^2)$.",
"If $cos \\theta = x$ then $tan \\theta = \\sqrt (1-x^2)/x$.",
"The defining property of arctan.",
"The defining property of arcsin.",
"If $cos \\theta = x$ then $sin \\theta = \\sqrt (1-x^2)$.",
"If $tan \\theta = x$ then $sin \\theta = x/\\sqrt (x^2+1)$.",
"If $sin \\theta = x$ then $cos \\theta = \\sqrt (1-x^2)$",
"The defining property of arccos",
"If $tan \\theta = x$ then $cos \\theta = 1/\\sqrt (x^2+1)$",
"If $sin \\theta = x$ then $sec \\theta = 1/\\sqrt (1-x^2)$",
"If $cos \\theta = x$ then $sec \\theta = 1/x$",
"If $tan \\theta = x$ then $sec \\theta = \\sqrt (x^2+1)$",
"Example: $arctan (tan \\pi /3) = \\pi /3$",
"Example: $arcsin(sin \\pi /3) = \\pi /3$",
"Example: $arccos(cos \\pi /5) = \\pi /5$",
"$c1$ is constant on intervals where $tan x$ is defined, a constant of integration."
},
{ /* adding_arctrig_functions */
"The angle whose sin is $x$ and the angle whose cosine is $x$ are complementary.",
"That is, the sum is $\\pm \\pi /2$, depending on the sign of x.",
#if 0 /* Perhaps add these later */
"$arcsin x \\pm arcsin y = arcsin[x\\sqrt (1-y^2)\\pm y\\sqrt (1-x^2)]$",
"$arccos x + arccos y = arccos[xy-\\sqrt ((1-x^2)(1-y^2))]$",
"$arccos x - arccos y = arccos[xy+\\sqrt ((1-x^2)(1-y^2))]$",
"$arctan x + arctan y = arctan[(x+y)/(1-xy)]$",
"$arctan x - arctan y = arctan[(x-y)/(1+xy)]$",
#endif
},
{ /* complementary_trig */
"Cosine is the sine of the complement.",
"Sine is the cosine of the complement.",
"Cotangent is the tangent of the complement.",
"Tangent is the cotangent of the complement.",
"Cosecant is the secant of the complement.",
"Secand is the cosecant of the complement.",
"Example: $sin (\\pi /3) = cos (\\pi /6)$",
"Example: $cos (\\pi /3) = sin (\\pi /6)$",
"Example: $tan (\\pi /3) = sin (\\pi /6)$",
"Example: $cot (\\pi /3) = tan (\\pi /6)$",
"Example: $sec (\\pi /3) = csc (\\pi /6)$",
"Example: $csc (\\pi /3) = sec (\\pi /6)$"
},
{ /* complementary_degrees */
"Cosine is the sine of the complement.",
"Sine is the cosine of the complement.",
"Cotangent is the tangent of the complement.",
"Tangent is the cotangent of the complement.",
"Cosecant is the secant of the complement.",
"Secand is the cosecant of the complement.",
"Example: $sin (30\\deg ) = cos (60\\deg )$",
"Example: $cos (30\\deg ) = sin (60\\deg )$",
"Example: $tan (30\\deg ) = sin (60\\deg )$",
"Example: $cot (30\\deg ) = tan (60\\deg )$",
"Example: $sec (30\\deg ) = csc (60\\deg )$",
"Example: $csc (30\\deg ) = sec (60\\deg )$",
"Example: $15\\deg +10\\deg = (15+10)\\deg = 25\\deg $. Only numbers can be directly added.",
"Example: $2\\times 30\\deg = (2\\times 30)\\deg = 60\\deg $",
"Example: $60\\deg /2 = (30)\\deg $"
},
{ /* trig_odd_and_even */
"sin is an odd function.",
"cos is an even function.",
"tan is an odd function.",
"cot is an odd function.",
"sec is an even function.",
"csc is an odd function.",
"$sin^2$ is an even function.",
"$cos^2$ is an even function.",
"$tan^2$ is an even function.",
"$cot^2$ is an even function.",
"$sec^2$ is an even function.",
"$csc^2$ is an even function."
},
{ /* trig_periodic */
"sin is periodic with period $2\\pi $. Example: $sin (9\\pi /4) = sin (\\pi /4)$",
"cos is periodic with period $2\\pi $. Example: $cos (9\\pi /4) = cos (\\pi /4)$",
"tan is periodic with period $\\pi $. Example: $tan (3\\pi /4) = tan (\\pi /4)$",
"sec is periodic with period $2\\pi $. Example: $sec (9\\pi /4) = sec (\\pi /4)$",
"csc is periodic with period $2\\pi $. Example: $csc (9\\pi /4) = csc (\\pi /4)$",
"cot is periodic with period $\\pi $. Example: $cot (3\\pi /4) = cot (\\pi /4)$",
"sin^2 is periodic with period $\\pi $. Example: $sin^2 (3\\pi /4) = sin^2 (\\pi /4)$",
"cos^2 is periodic with period $\\pi $. Example: $cos^2 (3\\pi /4) = cos^2 (\\pi /4)$",
"sec^2 is periodic with period $\\pi $. Example: $sec^2 (3\\pi /4) = sec^2 (\\pi /4)$",
"csc^2 is periodic with period $\\pi $. Example: $csc^2 (3\\pi /4) = csc^2 (\\pi /4)$",
"Example: $sin 200\\deg = -sin 20\\deg $",
"Example: $sin 160\\deg = sin 20\\deg $",
"Example: $cos 200\\deg = -cos 20\\deg $",
"Example: $cos 160\\deg = -cos 20\\deg $"
},
{ /* half_angle_identities */
"Express $sin^2$ in terms of a single trig function instead of a power.",
"Express $cos^2$ in terms of a single trig function instead of a power.",
"Express $sin^2$ in terms of a single trig function instead of a power.",
"Express $cos^2$ in terms of a single trig function instead of a power.",
"Change a product of trig functions into a single trig function.",
"There are two formulas for $tan (\\theta /2)$. Choose the best one by context.",
"There are two formulas for $tan (\\theta /2)$. Choose the best one by context.",
"There are two formulas for $cot (\\theta /2)$. Choose the best one by context.",
"There are two formulas for $cot (\\theta /2)$. Choose the best one by context.",
"Express $sin(\\theta /2)$ in terms of $cos \\theta $",
"Express $sin(\\theta /2)$ in terms of $cos \\theta $",
"Express $cos(\\theta /2)$ in terms of $cos \\theta $",
"Express $cos(\\theta /2)$ in terms of $cos \\theta $",
"Example: $60\\deg = 2\\times 30\\deg $."
},
{ /* product_and_factor_identities */
"The reverse of the double angle formula.",
"Example: $sin (x+\\pi /4) cos (x-\\pi /4) = \\onehalf [sin(2x)+sin(\\pi /2)]$",
"Example: $cos (x+\\pi /4) sin (x-\\pi /4) = \\onehalf [sin(2x)-sin(\\pi /2)]$",
"Example: $sin (x+\\pi /4) sin (x-\\pi /4) = \\onehalf [cos(\\pi /2)-cos(2x)]$",
"Example: $cos (x+\\pi /4) cos (x-\\pi /4) = \\onehalf [cos(2x)+cos(\\pi /2)]$",
"Write a sum of sines can be as a product of a sine and a cosine.",
"Write a difference of sines as a product of a sine and a cosine.",
"Write a sum of cosines as a product of a sine and a cosine.",
"Write a difference of cosines as a product of a sine and a cosine.",
"Substitute two new variables for the two different expressions inside the trig functions."
},
{ /* limits */
"Calculate the function near the limit point, at values you will specify.",
"The limit of a sum is the sum of the limits (if defined).",
"The limit of a difference is the difference of the limits (if defined).",
"Example: $lim(t->3,\\pi ) = \\pi $",
"Example: lim(t->3,t) = 3",
"Pull out a constant through the limit sign.",
"Pull out a minus sign through a limit.",
"The limit of a product is the product of the limits (if defined).",
"The limit of a (constant) power is the power of the limit.",
"Example: lim(x->3,2^x) = 2^lim(x->3,x)",
"The limit of a power is the power of the limits (if defined).",
"Watch out for the case when the limit is zero. It still works if $u\\ge 0$.",
"The limit of an odd root is the root of the limit.",
"Watch out for the case when the limit is zero. It still works if $u\\ge 0$.",
"Calculate the limit of a polynomial in the limit variable in one step.",
"Example: lim(x->0,|x^3|) = |lim(x->0,x^3|"
},
{ /* limits_of_quotients */
"Pull constants out of the numerator and denominator through the limit sign.",
"Applies only if the numerator is constant.",
"Does not work if lim u and lim v are both zero or infinity.",
"Factor powers of (x-a) out of both numerator and denominator, if possible.",
"Calculate the limit of a quotient of polynomials in one step.",
"Use this law to prepare to push the limit through the power.",
"Example: This will multiply num and denom of $(x-1)/(\\sqrt x-1)$ by $\\sqrt x+1$.",
"Example: in the limit of (x-1)^2 sin x/ tan x as x->0, pull out lim (x-1)^2.",
"$ab + ac = a(b+c)$, where $a$ does not depend on the limit variable.",
"You will be asked what to multiply num and denom by.",
"You will get a limit of a compound fraction, not a quotient of limits.",
"You will get a quotient of limits, not a limit of a compound fraction.",
"Example: use this on $(sin x cos h + cos x sin h - sin x)/h$"
},
{ /* quotients_of_roots */
"Example: $\\sqrt x/2 = \\sqrt (x/4)$",
"Example: $\\sqrt x/(-2) = -\\sqrt (x/4)$",
"Example: $^3\\sqrt a/2 = ^3\\sqrt (a/8)$",
"Example: $^4\\sqrt x/(-2) = -^4\\sqrt (x/16) (b<0, n even)$",
"Example: $2/\\sqrt x = \\sqrt (4/x)$",
"Example: $(x-1)/\\sqrt x = -\\sqrt ((x-1)^2/x)$ when $x\\le 1$",
"Example: $2/+^3\\sqrt x = ^3\\sqrt (8/x)$",
"Example: $(x-1)/^3\\sqrt x = -^3\\sqrt (x-1)^n/x)$ when $x\\le 1$"
},
{ /* lhopitalmenu */
"Replace an indeterminate limit of a quotient by the limit of the derivatives.",
"Uses all the rules about derivatives to get the answer in one step.",
"Example: lim x ln x = lim (ln x)/(1/x). Then use L'Hospital's rule.",
"Example: $lim x (ln x)^2 = lim (ln x)^2/(1/x)$. Then use L'Hospital's rule.",
"Example: lim x^(-3) e^x = lim e^x/x^3.",
"Example: lim x^3 e^x = lim x^3/e^(-x). Then use L'Hospital's rule.",
"Examples: $lim f(x) tan x = lim f(x)/cot x$; $lim f(x) sin x = lim f(x)/csc x$.",
"You will be asked which factor to move to the denominator.",
"Put fractions over a common denominator and simplify."
},
{ /* special_limits */
"For small t, sin t is approximately t.",
"For small t, tan t is approximately t.",
"cos t goes to 1 quite rapidly, faster than t goes to zero.",
"cos t goes to 1 like t^2, as t goes to 0. The coefficient is $\\onehalf $.",
"For example (1+ .001)^1000 is pretty close to e.",
"For small t, ln(1+t) is approximately t.",
"For small t, e^t-1 is approximately t.",
"For small t, e^t-1 is approximately t.",
"Any power of t, even a fractional power, will kill the singularity in ln.",
"cos (1/t) oscillates between -1 and 1 infinitely many times as t->""0.",
"sin (1/t) oscillates between -1 and 1 infinitely many times as t->""0.",
"tan (1/t) has large oscillations and is not even everywhere defined near t=0.",
"cos t oscillates between -1 and 1 infinitely many times as t->$\\infty $.",
"sin t oscillates between -1 and 1 infinitely many times as t->$\\infty $.",
"tan t has large oscillations and is not even everywhere defined as t->$\\infty $."
},
{ /* hyper_limits */
"For small t, sinh t is approximately t.",
"For small t, tanh t is approximately t.",
"cosh t goes to 1 quite rapidly, faster than t goes to zero.",
"cosh t goes to 1 like t^2, as t goes to zero. The coefficient is $\\onehalf $.",
},
{ /* advanced_limits */
"Push limit through ln.",
"Example: lim sin x^2 = sin lim x^2",
"lim(t->a,f(g(t)))=lim(u->g(a),f(u))",
"Evaluate a limit in one step, if within MathXpert's capabilities.",
"Example: $$lim(x->0, x^x) = lim(x->0,e^(x ln x))$$",
"You will be asked for the factor to move to the denominator.",
"Example, limit of $\\sqrt x$ as x->0 is undefined since $\\sqrt x$ is not defined for x < 0.",
"Example: $$lim(x->0, x^x) = e^(lim(x->0, ln x^x))$$",
"Example: lim x sin(1/x) as x->0 = 0 since $|sin(1/x)| \\le 1$.",
"Rationalize the numerator, except that no fraction is originally present.",
"Drop terms in numerator and denominator which are dominated by other terms.",
"Example: lim (x + x^2 sin x) = lim x as x->0 since (x^2 sin x)/x ->0",
"Replaces u+v by u if v/u->0.",
"Example: $sin(undefined) = undefined$",
"Example: $lim e^(1/x) = e^(\\lim 1/x)$",
"Push limit through ln"
},
{ /* logarithmic_limits */
"Any power of t, even a fractional power, will kill the singularity in ln.",
"Any power of t, even a fractional power, will kill the singularity in ln.",
"Any power of t, even a fractional power, will kill the singularity in ln.",
"Any power of t, even a fractional power, will kill the singularity in ln.",
"Any power of t, even a fractional power, will kill the singularity in ln.",
"Any power of t, even a fractional power, will kill the singularity in ln.",
"Any power of t, even a fractional power, will kill the singularity in ln.",
"Any power of t, even a fractional power, will kill the singularity in ln.",
"Any power of t, even a fractional power, will kill the singularity in ln.",
"Any power of t, even a fractional power, will kill the singularity in ln.",
"Any power of t, even a fractional power, will kill the singularity in ln.",
"Any power of t, even a fractional power, will kill the singularity in ln."
},
{ /* limits_at_infinity */
"For large t, 1/t^n is small.",
"For large t, t^n is large",
"For large t, e^t is large",
"For large negative t, e^t is small.",
"For large t, ln t is large.",
"For large t, $\\sqrt t$ is large.",
"For large t, $^n\\sqrt t$ is large.",
"The arctan of a large positive (or negative) number is almost $\\pi /2$ (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 $.",
"tanh of a large positive (or negative) number is almost 1 (or -1).",
"Rationalize the numerator, except that no fraction is originally present.",
"Push limit through sin",
"Push limit through cos",
"$lim(t->\\infty,f(t))=lim(t->0+,f(1/t))$",
"Drop terms in numerator and denominator which are dominated by other terms."
},
{ /* infinite_limits */
"Example: $lim 1/t^4 ->\\infty $ as t->0",
"Example: the two-sided limit, lim 1/t^3 as t->0, is undefined.",
"Example: the right-hand limit, lim 1/t^3 as t->0+, is $\\infty $.",
"Example: the left-hand limit, lim 1/t^3 as t->0-, is $-\\infty $.",
"Example: lim 1/t as t->0 is undefined.",
"This one-sided limit is $-\\infty $, but the two-sided limit is undefined.",
"The given one-sided limits are $\\pm \\infty $, but the two-sided limits are undefined.",
"The given one-sided limits are $\\pm \\infty $, but the two-sided limits are undefined.",
"The given one-sided limits are $\\pm \\infty $, but the two-sided limits are undefined.",
"The given one-sided limits are $\\pm \\infty $, but the two-sided limits are undefined.",
"Example: $lim(t->0, ln(1+t) e^t)$ becomes $lim(t->0, ln(1+t)/t) lim(t->0,te^t)$.",
"Example: $lim(t->0,t ln(1+t))$ becomes $lim(t->0, t^2) lim(t->0,ln(1+t)/t)$.",
},
{ /* infinities */
"Example: $\\infty /2 = \\infty $",
"Example: $1/\\infty = 0$",
"Example: $2\\times \\infty = \\infty $",
"This rule is shorthand for $lim uv = \\infty $ if $lim u = \\infty $ and $lim v = \\infty $.",
"Example: $\\infty + 2 = \\infty $",
"This rule is shorthand for $lim u+v = \\infty $ if $lim u = \\infty $ and $lim v = \\infty $.",
"Example: $e^\\infty = \\infty $",
"Example: $(\\onehalf)^\\infty = 0$",
"Example: $e^(-\\infty) = 0$",
"Example: $(\\onehalf)^(-\\infty) = \\infty $",
"Example: $\\infty ^3 = \\infty $",
"You cannot cancel $\\infty -\\infty $. This expression is undefined."
},
{ /* zero_denom */
"0+ means that the 0 came from a term that is positive near the limit point.",
"0- means that the 0 came from a term that is negative near the limit point.",
"If the sign of the denom near the limit point alternates or is not known.",
"0+ means that the 0 came from a term that is positive near the limit point.",
"0- means that the 0 came from a term that is negative near the limit point.",
"If the sign of the denom near the limit point alternates or is not known.",
"This is shorthand for $lim u/v^2 = \\infty $ if $lim u = \\infty $ and lim v = 0.",
"This is shorthand for $lim u/v^2^n = \\infty $ if $lim u = \\infty $ and lim v = 0.",
"This is shorthand for $lim a/u^2 = \\infty $ if a>0 and lim u = 0.",
"This is shorthand for $lim a/u^2 = -\\infty $ if a<0 and lim u = 0.",
"This is shorthand for $lim a/u^2^n = \\infty $ if a>0 and lim u = 0.",
"This is shorthand for $lim a/u^2^n = -\\infty $ if a<0 and lim u = 0."
},
{ /* more_infinities */
"This is shorthand for $lim ln u = \\infty $ if $lim u = \\infty $.",
"This is shorthand for $lim \\sqrt u = \\infty $ if $lim u = \\infty $.",
"This is shorthand for $lim ^n\\sqrt u = \\infty $ if $lim u = \\infty $.",
"The arctan of a large positive (or negative) number is near $\\pi /2$ (or $-\\pi /2$).",
"The arccot of a large positive number is near 0.",
"The arccot of a large negative number is near $\\pi $.",
"The arcsec of a large number is near $\\pi /2$.",
"The arccsc of a large number is near 0.",
"None of sin, cos, tan, sec, csc, tan have limits at $\\infty $.",
"cosh of a large number x is approximately e^x/2, which is large.",
"sinh of a large number x is approximately e^x/2, which is large.",
"tanh of a large number x is approximately 1, since cosh and sinh are both approximately e^x",
"This is shorthand for $lim ln u = -\\infty $ if $lim u = 0$ and $0<u$."
},
{ /* polynomial_derivs */
"The derivative of a constant is zero.",
"The derivative of x with respect to x is 1",
"The derivative of a sum is the sum of the derivatives.",
"Pull a minus sign out of a derivative.",
"Pull a constant out of a derivative.",
"This is called the power rule.",
"Differentiate a polynomial at once, in one step.",
"Express f'(x) using the d/dx notation for the derivative."
},
{ /* derivatives */
"This is the definition of the derivative as a limit.",
"Differentiate a polynomial at once, in one step.",
"The derivative of a sum is the sum of the derivatives.",
"Pull a minus sign out of a derivative.",
"Pull a constant out of a derivative.",
"Pull a constant out of the denominator.",
"This is called the power rule.",
"This is called the product rule.",
"Although this is a special case of the quotient rule, memorize it separately.",
"This is called the quotient rule.",
"Use this rule on $\\sqrt $, rather than always converting to fractional exponents.",
"Convert roots to fractional exponents in order to differentiate.",
"Use this rule, rather than convert to negative exponents and back again.",
"Use this rule rather than expand |x| by cases.",
"Express f'(x) using the d/dx notation for derivatives."
},
{ /* dif_trig */
"The derivative of sine is cosine.",
"The derivative of cosine is minus the sine",
"The derivative of tangent is secant squared.",
"The derivative of secant is secant tangent.",
"The derivative of cotangent is cosecant squared.",
"The derivative of cosecant is cosecant cotangent."
},
{ /* dif_explog */
"The exponential function is its own derivative.",
"Every exponential function is its own derivative except for a constant $ln c$.",
"Use this rule to differentiate a power with nonconstant base and exponent.",
"The derivative of $ln x$ is $1/x$.",
"$ln |x|$ has the same derivative as $ln x$ but is defined for negative $x$ too.",
"Using this formula is called logarithmic differentiation.",
"Example: $d/dx e^(\\sin x) = e^(\\sin x) d/dx sin x$",
"Example: $d/dx 2^(\\sin x)=(ln 2)2^(\\sin x) d/dx sin x$",
"Example: $d/dx ln sin x = (1/sin x)(d/dx sin x)$",
"Example: $d/dx ln |x^3| = (1/x^3) d/dx x^3$",
"When $d/dx ln(cos x)$ occurs, this rule does it in one step.",
"When $d/dx ln(sin x)$ occurs, this rule does it in one step."
},
{ /* dif_inverse_trig */
"If you forget this, differentiate x = tan y and solve for dy/dx.",
"If you forget this, differentiate x = sin y and solve for dy/dx.",
"If you forget this, differentiate x = cos y and solve for dy/dx.",
"If you forget this, differentiate x = cot y and solve for dy/dx.",
"If you forget this, differentiate x = sec y and solve for dy/dx.",
"If you forget this, differentiate x = csc y and solve for dy/dx.",
"Example: d/dx arctan x^2 = d/dx(x^2)/(1+x^4)",
"Example: $d/dx arcsin x^2 = d/dx(x^2)/\\sqrt (1-x^4)$",
"Example: $d/dx arccos x^2 = -d/dx(x^2)/\\sqrt (1-x^4)$",
"Example: $d/dx arccot x^2 = -d/dx(x^2)/(1+x^4)$",
"Example: $d/dx arcsec x^2 = d/dx(x^2)/(|x^2|\\sqrt (x^4-1))$",
"Example: $d/dx arccsc x^2 = -d/dx(x^2)/(|x^2|\\sqrt (x^4-1))$"
},
{ /* chain_rule (113) */
"Example: d/dx (1+x^2)^100 = 100(1+x^2)^99 d/dx x^2",
"Example: $d/dx \\sqrt (1+x^2) = (d/dx x^2)/(2\\sqrt (1+x^2))$",
"Example d/dx sin x^2 = (cos x^2) d/dx x^2",
"Example: d/dx cos x^2 = -(sin x^2) d/dx x^2",
"Example: d/dx tan x^2 = (sec^2 x^2) d/dx x^2",
"Example: d/dx sec x^2 = (sec x^2 tan x^2) d/dx x^2",
"Example: cot x^2 = -(csc^2 x^2) d/dx x^2",
"Example: csc x^2 = -(csc x^2 cot x^2) d/dx x^2",
"Example: d/dx |sin x| = (sin x d/dx sin x)/|sin x|",
"The chain rule applied to any function f, with or without a definition.",
"Introduce a new letter to stand for the selected term.",
"Replace a defined variable by its definition throughout the line."
},
{ /* maxima_and_minima */
"experiment numerically",
"Add the points where $f'(x)=0$ to the list of points considered.",
"Add the endpoints of the interval to the list of points considered.",
"Add the points where $f'(x)$ undefined to the list of points considered.",
"consider limits at open ends",
"reject point outside interval",
"make table of decimal $y$-values for the listed $x$-values.",
"make table of exact $y$-values for the listed $x$-values.",
"choose maximum value(s) from the table.",
"choose minimum value(s) from the table.",
"evaluate derivative in one step",
"solve simple equation",
"evaluate limit in one step",
"eliminate integer parameter",
"For a constant function, the max and min are equal."
},
{ /* implicit_diff */
"Evaluate a derivative at once, in one step.",
"Perform algebraic simplification.",
"Solve an equation in one step. Will fail on complicated equations."
},
{ /* related_rates */
"differentiate both sides of an equation valid for all $t$ in some interval.",
"MathXpert will evaluate the derivative",
"Eliminate a derivative by substituting an expression known to be equal to it.",
"solve simple equation"
},
{ /* simplify */
"Perform algebraic simplification, collecting, cancelling, ordering, etc.",
"Use various laws to eliminate compound fractions in one step.",
"Put a sum containing fractions over a common denominator and simplify.",
"$ab+ac = a(b+c)$; factors out the greatest explicit common factor",
"Use simple factoring identities to factor as much as possible in one step.",
"Multiply out a product of sums and then collect and/or cancel the terms.",
"Factor out the greatest common divisor of numerator and denominator.",
"Solve an equation in one step. Will fail on complicated equations.",
"Example: write $(x+1)^2 -2x$ as polynomial in x+1, get $(x+1)^2-2(x+1) + 2$.",
"Express in standard polynomial form in the main variable.",
"Example: 3x^2 - 2x + 1 becomes 3(x^2 - 2/3 x + 1/3)",
"Change $x^\\onehalf $ to $\\sqrt x$ throughout the selected expression.",
"Change fractional exponents to roots throughout the selected expression.",
"Change roots to fractional exponents throughoug the selected expression."
},
{ /* higher_derivatives */
"Differentiate an identity.",
"The second derivative is the derivative of the derivative.",
"Example: d^3u/dx^3= d/dx d^2u/dx^2",
"The derivative of the derivative is the second derivative.",
"The derivitive of the n-th derivative is the n+1-st derivative.",
"Calculate a derivative at once, in one step.",
"Compute the value of the current line at a specified point."
},
{ /* basic_integration */
"The integral of 1 with respect to t is just t.",
"The integral of a constant c is ct.",
"Special case of the power rule if we consider t as t to the first power.",
"Pull a constant out of an integral.",
"Pull a minus sign out of an integral.",
"This is called the additivity of the integral.",
"The integral of a difference is the difference of the integrals.",
"This is called the linearity of the integral.",
"This is the power rule for integration.",
"Use this rule instead of always converting to negative exponents.",
"Integrate a polynomial at once, in one step.",
"Don't forget the absolute value; ln |t| is a more natural function than ln t.",
"Don't forget the absolute value; ln |t| is a more natural function than ln t.",
"Multiply out products of sums in the integrand.",
"Example: $\\int (t+1)^2 dt = \\int t^2+2t+1 dt$",
"Use this formula rather than expanding |t| by cases."
},
{ /* trig_integration */
"The integral of sine is minus cosine.",
"The integral of cosine is sine.",
"The integral of tangent is -ln cosine, but don't forget the absolute value.",
"The integral of cotangent is ln sine, but don't forget the absolute value.",
"This amazing formula is due to Euler.",
"This formula is almost like the integral of secant, but one sign is different.",
"The derivative of tangent is secant squared.",
"The derivative of cotangent is minus cosecant squared.",
"If you forget this, remember to write $tan^2$ as $sec^2 - 1$.",
"If you forget this, remember to write $cot^2$ as $csc^2 - 1$.",
"The derivative of secant is secant tangent.",
"The derivative of cosecant is minus cosecant cotangent."
},
{ /* trig_integration2 */
"Example: $\\int sin 2t dt = -(1/2) cos 2t$",
"Example: $\\int cos 2t dt = (1/2) sin 2t$",
"Example: $\\int tan 2t dt = -(1/2) ln |cos 2t|$",
"Example: $\\int cot 2t dt = (1/2) ln |sin 2t|$",
"Example: $\\int sec 2t dt = (1/2) ln |sec 2t + tan 2t|$",
"Example: $\\int csc 2t dt = (1/2) ln |csc 2t - cot 2t|$",
"Example: $\\int sec^2 2t dt = (1/2) tan 2t$",
"Example: $\\int csc^2 2t dt = -(1/2) cot 2t$",
"Example: $\\int tan^2 2t dt = (1/2) tan 2t - t$",
"Example: $\\int cot^2 2t dt = -(1/2) cot 2t - t$",
"Example: $\\int sec 2t tan 2t dt = (1/2) sec 2t$",
"Example: $\\int csc 2t cot 2t dt = -(1/2) csc 2t$"
},
{ /* integrate_exp */
"The exponential function is its own integral, as well as derivative.",
"Example: $\\int e^(2t) dt =(1/2) e^(2t)$",
"The function e^(-t) is minus its own integral.",
"Example: $\\int e^(-2t)dt = -(1/2) e^(-2t)$",
"Example: $$integral(e^(t/2),t) = 2e^(t/2)$$",
"Example: $\\int 3^t dt = (1/ln 3) 3^t$",
"Example: $$integral(t^t,t) = integral(e^t ln t,t)$$",
"If you forget this, integrate by parts, taking parts $ln t$ and 1.",
"This is the definition of Erf; the integral has no simpler form.",
},
{ /* integrate_by_substitution */
"Introduce a new letter for the specified expression.",
"MathXpert will try to find an applicable substitution.",
"Apply this to the equation defining your new variable.",
"Calculate a derivative at once, in one step.",
"Use this when you have calculated du/dx to get the original integral back.",
"Separate out du/dx from the integrand and write the rest as a function of u.",
"This is the substitution rule proper, for which you have been preparing.",
"Replace a defined variable by its definition throughout the current line.",
"Integrate by substitution in one step using the specified expression.",
"Integrate by substitution in one step; let MathXpert choose the substitution.",
},
{ /* integrate_by_parts */
"Integrate by parts, using the selected term as the part u to differentiate.",
"Integrate by parts, letting MathXpert choose the parts.",
"This creates an equation which can sometimes be solved for the integral.",
"Transfer the integral to the left side in order to solve for it.",
"Calculate a derivative at once, in one step",
"Integrate by substitution in one step, using the selected term to define u.",
"integrate by substitution in one step, letting MathXpert choose a substitution.",
"Evaluate an integral in one step, if it is not too complicated."
},
{ /* fundamental_theorem */
"This is the derivative form of the Fundamental Theorem of Calculus.",
"This is the integral form of the Fundamental Theorem of Calculus."
},
{ /* definite_integration */
"This is the definition of the symbols on the left side.",
"This is often simpler than ln f(b) - ln f(a)",
"An integral changes sign if its upper and lower limit are exchanged.",
"This is called the additivity of the integral.",
"You will be asked for the point at which to break the integral",
"Example: a definite integral $\\int |(t-1)(t+1)| dt$ should be broken at -1 and 1.",
"Specify parameter value, then use approximate numerical integration.",
"Use approximate numerical integration to get a decimal answer.",
"When the upper and lower limits are the same, a definite integral is zero."
},
{ /* improper_integrals */
"Converts an improper integral to a limit of proper integrals.",
"Converts an improper integral to a limit of proper integrals.",
"Converts an improper integral to a limit of proper integrals.",
"Converts an improper integral to a limit of proper integrals.",
"If $u$ does not tend to 0 as $t->\\infty $, then $\\int u dt$ from c to $\\infty $ diverges.",
"If $u$ does not tend to 0 as $t->-\\infty $, then $\\int u dt$ from $-\\infty $ to c diverges."
},
{ /* oddandeven */
"An odd function, integrated over a symmetric interval, yields zero.",
"An even function contributes equally to the integral for plus and minus x."
},
{ /* trig_substitutions */
"Example: substitute $x = sin \\theta $ to integrate $\\sqrt (1-x^2)$",
"Example: substitute $x = tan \\theta $ to integrate $\\sqrt (1+x^2)$",
"Example: substitute $x = sec \\theta $ to integrate $\\sqrt (x^2-1)$",
"Example: substitute $x = sinh \\theta $ to integrate $\\sqrt (1+x^2)$",
"Example: substitute $x = a cosh \\theta $ to integrate $\\sqrt (x^2-1)$",
"Example: substitute $x = a tanh \\theta $ to integrate $\\sqrt (1-x^2)$",
"You will be asked to enter the definition of x in terms of a new variable",
"Calculate a derivative at once, in one step.",
"Evaluate an integral at once, in one step, if it isn't too complicated."
},
{ /* trigonometric_integrals */
"Use this to get rid of $sin^2 t$ in an integral.",
"Use this to get rid of $cos^2 t$ in an integral",
"Use this to integrate an odd power of sin x (also with powers of cos).",
"Use this to integrate an odd power of cos x (also with powers of sin).",
"Use this to integrate an even power of sec x (also with powers of tan).",
"Use this to integrate an even power csc x (also with powers of cot).",
"Use this to integrate an odd power of tan x with power of sec present too.",
"Use this to integrate an odd power of cot x with powers of csc present too.",
"Express $tan^2 x$ in terms of $sec^2 x$ to prepare for u = sec x",
"Express $cot^2 x$ in terms of $csc^2 x$ to prepare for u = csc x",
"$\\int sec^n x dx = -1/(n-1) sec^n x tan x + (n-2)/(n-1)\\int sec^(n-2) x dx$",
"$\\int csc^n x dx = -1/(n-1) csc^n x cot x + (n-2)/(n-1)\\int csc^(n-2) x dx$",
"This works on any trigonometric integral, but other methods may be simpler.",
},
{ /* trigrationalize */
"Use this to get rid of 1-cos x in the denominator.",
"Use this to get rid of 1+cos x in the denominator.",
"Use this to get rid of 1-sin x in the denominator.",
"Use this to get rid of 1+sin x in the denominator.",
"Use this to get rid of sin x - cos x in the denominator.",
"Use this to get rid of cos x + sin x in the denominator."
},
{ /* integrate_rational*/
"Example: (x^2 + 2x + 2)/(x+1) = x + 1 + 1/(x+1)",
"Use all applicable factoring rules to factor the denominator.",
"Factor out the greatest common divisor of numerator and denominator",
"Factor out all repeated factors (greatest common divisor of u and u')",
"Example: x^3-x+1 = (x+1.32472)(x^2 - 1.32472 x + 0.754878)",
"Example: 2x/(x^2-1) = 1/(x-1) + 1/(x+1)",
"Example: x^2 + 4x = (x+2)^2 - 4",
"Example: $\\int 1/(3t-1) dt = (1/3) ln |3t-1|$",
"Example: $\\int 1/(3t+1)^3 dt = -1/6 (3t+1)^2$",
"Example: $\\int 1/(t^2+4)dt=(1/2)arctan(t/2)$",
"Example: $\\int 1/(t^2-4)dt=(1/2)arccoth(t/2)$",
"Example: $\\int 1/(t^2-4)dt=(1/4)ln|(t-2)/(t+2)|$",
"Example: $\\int 1/(4-t^2)dt=(1/2)arctanh(t/2)$",
"Example: $\\int 1/(4-t^2)dt=(1/4)ln|(t+2)/(2-t)|$"
},
{ /* integrate_sqrtdenom */
"Example: $x^2 + 4x = (x+2)^2 - 4$",
"Example: $\\int 1/\\sqrt (4-t^2)dt = arcsin(t/2)$",
"Example: $\\int 1/\\sqrt (t^2-3)dt)=ln|t+\\sqrt (t^2-3)|$",
"Example: $\\int 1/(t\\sqrt (t^2-4))dt=(1/2)arccos(t/2)$",
"That is, integrate by substitution. You specify the substitution."
},
{ /* integrate_arctrig */
"If you forget this, derive it using integration by parts.",
"If you forget this, derive it using integration by parts.",
"If you forget this, derive it using integration by parts.",
"If you forget this, derive it using integration by parts.",
"If you forget this, derive it using integration by parts.",
"If you forget this, derive it using integration by parts.",
"If you forget this, derive it using integration by parts.",
"If you forget this, derive it using integration by parts."
},
{ /* simplify_calculus */
"Perform algebraic simplification.",
"Use various laws of fractions to eliminate compound fractions in one step.",
"Put sums containing fractions over a common denominator and simplify.",
"ab+ac = a(b+c). Factors out the greatest explicit common factor.",
"Example: x^3 + 2x^2 + x becomes x(x+1)^2",
"Multiply out products of sums and collect and/or cancel the resulting terms.",
"Factor out the greatest common divisor of numerator and denominator.",
"Solve an equation in one step, if it is not too complicated.",
"Calculate a derivative at once, in one step.",
"Evaluate a limit at once, if MathXpert can do it at all.",
"Integrate by substitution. You will be asked for a substitution.",
"Evaluate an integral in one step, if it isn't too complicated.",
"Example: 3 + c_1 becomes c_2"
},
{ /* 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, but it's ln tanh(u/2), not ln tanh u.",
"The integral of sech is arctan of sinh."
},
{ /* series_geom1 */
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1."
},
{ /* series_geom2 */
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1."
},
{ /* series_geom3 */
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1."
},
{ /* series_geom4 */
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1."
},
{ /* series_geom5 */
"Expand $x^k/(1-x)$ in a geometric series.",
"Expand $x^k/(1-x)$ in a geometric series.",
"Expand $x^k/(1-x)$ in a geometric series.",
"Expand $x^k/(1+x)$ in a geometric series.",
"Expand $x^k/(1+x)$ in a geometric series.",
"Expand $x^k/(1+x)$ in a geometric series.",
"Formula for the sum of a geometric series starting from an arbitrary term.",
"Formula for the sum of a geometric series starting from an arbitrary term.",
"Formula for the sum of a geometric series starting from an arbitrary term.",
"Formula for the sum of a geometric series starting from an arbitrary term.",
"Formula for the sum of a geometric series starting from an arbitrary term.",
"Formula for the sum of a geometric series starting from an arbitrary term.",
},
{ /* series_ln */
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1."
},
{ /* series_trig */
"This converges for all x",
"This converges for all x",
"This converges for all x",
"This converges for all x",
"This converges for all x",
"This converges for all x",
"This converges for all x",
"This converges for all x",
"This converges for all x",
"This converges for all x",
"This converges for all x",
"This converges for all x",
},
{ /* series_exp */
"This converges for all x",
"This converges for all x",
"This converges for all x",
"This converges for all x",
"This converges for all x",
"This converges for all x",
"This converges for all x",
"This converges for all x",
"This converges for all x",
"This converges for all x",
"This converges for all x",
"This converges for all x",
},
{ /* series_atan */
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This is called the binomial series. It converges for |x|<1.",
"This is called the binomial series. It converges for |x|<1.",
"This is called the binomial series. It converges for |x|<1.",
"This is called the binomial series. It converges for |x|<1.",
"This is called the binomial series. It converges for |x|<1.",
"This is called the binomial series. It converges for |x|<1."
},
{ /* series_bernoulli */
"This converges for |x|<\\pi/2.",
"This converges for |x|<\\pi/2.",
"This converges for |x|<\\pi/2.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|<1.",
"This converges for |x|< \\pi/2.",
"This converges for |x|< \\pi/2.",
"This converges for |x|< \\pi/2.",
"This converges for |s|>1.",
"This converges for |s|>1.",
"This converges for |s|>1.",
"This is called the alternating harmonic series"
},
{ /* series_appearance */
"Express an infinite series using the first two terms and ... ",
"Express an infinite series using the first three terms and ... ",
"Example: $1 + x + ... + x^n + ...$",
"Replace the ... notation with sigma notation",
"One more term of the series will be visible.",
"You will enter how many more terms you want to see.",
"Show the visible part of the series with factorial evaluated.",
"Show the visible part of the series with factorial not evaluated.",
"Show the visible part of the series using decimal coefficients.",
"Do not evaluate the coefficients to decimal form."
},
{ /* series_algebra */
"(a_1-a_0) + (a_2-a_1) + ...= - a_0.",
"The result is a double sum: $(\\sum a_n)(\\sum b_m) = \\sum \\sum a_nb_m$",
"The result is a power series whose coefficients are given by finite sums.",
"The division will be carried out in one step.",
"The division will be carried out in one step.",
"The division will be carried out in one step.",
"The result is a double sum: $(\\sum a_n)^2 = \\sum \\sum a_na_m$",
"The result is a power series whose coefficients are given by finite sums.",
"The result is a series whose coefficients are defined by a recurrence relation.",
"$\\sum u + \\sum v = \\sum (u + v)$ if the limits of summation are the same.",
"$\\sum u - \\sum v = \\sum (u - v)$ if the limits of summation are the same."
},
{ /* series_manipulations */
"The series will be broken into a finite sum plus a new series.",
"Example: change the lower limit from 1 to 0 and subtract the extra term.",
"Example: in a sum involving $x^(n-1)$, add 1 to the index variable.",
"Example: in a sum involving $x^(n+1)$, subtract 1 from the index variable.",
"The index variable can be renamed without changing the value of the series.",
"This law is only valid if the resulting series all converge.",
"Power series and some other series can be differentiated term by term.",
"Power series and some other series can be differentiated term by term.",
"Power series and some other series can be integrated term by term.",
"Power series and some other series can be integrated term by term.",
"Use decimal arithmetic to calculate the sum of a specified number of terms.",
"This is useful if you can expand the derivative in a series.",
"Using a definite integral saves solving for a constant of integration.",
"This is useful if you can expand the integral in a series.",
"Substitute zero (or another value) and solve for the constant.",
"Separate terms with even and odd indices into two different series."
},
{ /* series_convergence_tests */
"Example: $\\sum (n-1)/n$ diverges because $lim(n->\\infty ,(n-1)/n) = 1$",
"If $u$ is positive and decreasing, $\\sum u$ converges if and only if $\\int u dx$ converges.",
"The limit of the ratio of successive terms, if not 1, determines convergence.",
"Limit of the $n$-th root of the $n$-th term, if not 1, determines convergence.",
"Example: $\\sum |sin n|/2^n$ converges since $\\sum 1/2^n$ converges and $|sin n|< 1$.",
"Example: $\\sum ln(n)/n$ diverges since $\\sum 1/n$ diverges and $ln(n)/n < 1/n $.",
"If $lim a_n/b_n > 0$ and $a_n>0$ and $b_n>0$ then $\\sum a$ converges iff $\\sum b$ converges.",
"Replace the $n$-th term of a decreasing series by $2^n$ times the $2^n$-th term.",
"State the result of the test about convergence or divergence.",
"State the result of the test about convergence or divergence.",
"State the result of the test about convergence or divergence.",
"State the result of the test about convergence or divergence.",
"Make the comparison series the current expression so it can be manipulated.",
"Make the comparison series the current expression so it can be manipulated.",
"State the result of the test about convergence or divergence.",
"State the result of the test about convergence or divergence."
},
{ /* series_convergence2 */
"State the result of the comparison test as a bound on the original series",
"State the result of the comparison test: the original series is divergent.",
"The harmonic series diverges to infinity.",
"The sum of the reciprocals of the squares is $\\pi^2/6$.",
"This infinite series defines the $\\zeta$ function",
"The values of $\\zeta$ at even integers are given by this formula."
},
{ /* complex_functions */
"To take the ln of a complex number, first convert to polar form.",
"The ln of a complex number is the ln of the modulus + i times the argument.",
"Since the argument of i (the angle in its polar form) is $\\pi /2$",
"Since the argument of -1 (the angle in its polar form) is $\\pi $",
"Since the argument of a negative number is $\\pi $",
"This famous formula links the trig and complex exponential functions.",
"This famous formula links the trig and complex exponential functions.",
"Halve the argument and take the square root of the modulus.",
"Divide the argument by n and take the n-th root of the modulus.",
"This famous formula links the trig and complex exponential functions.",
"This famous formula links the trig and complex exponential functions.",
"This formula, due to Euler, links several fundamental constants.",
"This formula, due to Euler, links several fundamental constants.",
"This formula, due to Euler, links several fundamental constants.",
"The complex exponential function is periodic, with period $2\\pi i$.",
"To compute a complex power, express it using the exponential function.",
},
{ /* complex_hyperbolic */
"Express complex sin in terms of sinh",
"Express complex cos in terms of cosh",
"Express complex cosh in terms of cos",
"Express complex sinh in terms of sin",
"Express complex tan in terms of tanh",
"Express complex cot in terms of coth",
"Express complex tanh in terms of tan",
"Express complex coth in terms of cot",
"Fundamental relation between complex exponential and trig functions",
"Fundamental relation between complex exponential and trig functions",
"Definition of complex cos, used in reverse",
"Definition of complex sin, used in reverse",
"Definition of complex cos, used in reverse",
"Definition of complex sin, used in reverse"
},
{ /* hyperbolic_functions */
"This formula defines the hyperbolic cosine function.",
"Definition of cosh, used in reverse.",
"This formula defines the hyperbolic sine function.",
"Definition of sinh, used in reverse.",
"Definition of cosh, used in reverse.",
"Definition of sinh, used in reverse.",
"cosh is an even function.",
"sinh is an odd function.",
"The sum of cosh and sinh simplifies to an exponential.",
"The difference of cosh and sinh simplifies to an exponential.",
"This is also the minimum value of cosh.",
"The graph of sinh passes through the origin, since it is an odd function.",
"Express e^x in terms of hyperbolic functions,",
"Express e^(-x) in terms of hyperbolic functions."
},
{ /* hyperbolic2 */
"This identity is analogous to $sin^2 + cos^2 = 1$, but notice the different sign.",
"This identity is analogous to $sin^2 + cos^2 = 1$, but notice the different sign.",
"This identity is analogous to $sin^2 + cos^2 = 1$, but notice the minus sign.",
"This identity is analogous to $cos^2 = 1 - sin^2$, but notice the different sign.",
"This identity is analogous to $sin^2 = 1 - cos^2$, but notice the different sign.",
"This identity is analogous to $1 + tan^2 = sec^2$, but notice the different sign.",
"This identity is analogous to $sec^2 - 1 = tan^2$, but notice the different sign."
},
{ /* more_hyperbolic */
"Definition of the hyperbolic tangent.",
"Definition of tanh in reverse",
"Definition of the hyperbolic cotangent.",
"Definition of coth in reverse",
"Definition of the hyperbolic secant.",
"Definition of sech in reverse.",
"Definition of the hyperbolic cosecant.",
"Definition of csch in reverse.",
"Analogous to $sec^2-tan^2 = 1$, but notice the different sign.",
"Analogous to $tan^2 = sec^2-1$, but notice the different signs.",
"Analogous to $sec^2 = 1 + tan^2$, but notice the different sign.",
"Analogous to the formula for sin(u+v), but the sign is different.",
"Analogous to the formula for cos(u+v), but the sign is different.",
"Analogous to the formula for sin 2u.",
"Analogous to the formula for $cos 2u$, but the sign is different.",
"Surprise: tanh(ln u) is not as complicated as it looks."
},
{ /* inverse_hyperbolic */
"arcsinh is a logarithm of an algebraic function.",
"arccosh is a logarithm of an algebraic function.",
"arctanh is a logarithm of a rational function.",
"The defining property of arcsinh.",
"The defining property of arccosh.",
"The defining property of arctanh.",
"The defining property of arccoth.",
"The defining property of arcsech.",
"The defining property of arccsch."
},
{ /* 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 u 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 */
"Similar to the formula for the derivative of arcsin, but with a sign change.",
"Similar to the formula for the derivative of arccos, but with a sign change.",
"Similar to the formula for the derivative of arctan, but with a sign change.",
"Similar to the formula for the derivative of arccot, but with a sign change.",
"Similar to the formula for the derivative of arcsec, but with a sign change.",
"Similar to the formula for the derivative of arccsc, but with a sign change."
},
{ /* sg_function1 */
"sgn(x) is the sign of x, 1 if x is positive, -1 if x is negative.",
"sgn(x) is the sign of x, 1 if x is positive, -1 if x is negative.",
"sgn(x) is the sign of x, 1 if x is positive, -1 if x is negative.",
"sgn is an odd function.",
"sgn is an odd function.",
"sgn can be expressed in terms of absolute value.",
"sgn can be expressed in terms of absolute value.",
"Use this inside an integral if the integrand is nonzero.",
"Also works on fractional exponents even/odd.",
"Also works on fractional exponents odd/odd.",
"Use this to get sgn in the numerator.",
"sgn is not differentiable at zero, but it's constant elsewhere.",
"sgn can be integrated directly using this formula.",
"This law is valid only if the integrand is nonzero.",
"If necessary, handle the cases of positive and negative sign separately.",
"If necessary, handle the cases of positive and negative sign separately."
},
{ /* sg_function2 */
"Example: sgn(3x) = sgn(x)",
"Example: sgn(ax) = sgn(x) if a<0 has been assumed.",
"Example: sgn(2x/3) = sgn(x)",
"Example: sgn(x/a) = sgn(x) if a<0 has been assumed.",
"Example: sgn(x^3) = sgn(x)",
"Example: sgn(1/c) = sgn(c)",
"Example: sgn(3/c) = sgn(c)",
"Example: a sgn(a) = |a|",
"Example: |a| sgn(a) = a"
},
{ /* bessel_functions */
"The derivative of J_0 is minus J_1.",
"The derivative of J_1 is given in terms of J_0 and J_1.",
"The derivative of J_n is given in terms of J_(n-1) and J_n.",
"The derivative of Y_0 is minus Y_1.",
"The derivative of Y_1 is given in term of Y_0 and Y_1.",
"The derivative of Y_n is given in terms of Y_(n-1) and Y_n."
},
{ /* modified_bessel_functions */
"The derivative of I_0 is minus J_1.",
"The derivative of I_1 is given in terms of I_0 and I_1.",
"The derivative of I_n is given in terms of I_(n-1) and I_n.",
"The derivative of K_0 is minus K_1.",
"The derivative of K_1 is given in term of K_0 and K_1.",
"The derivative of K_n is given in terms of K_(n-1) and K_n."
},
{ /* functions_menu */
"Apply user-defined function."
},
{"" /* automode_only, this menu never appears! */
},
{"" /* automode_only2, also never appears */
},
{"" /* automode_only3, also never appears */
}
};
/*_____________________________________________________________*/
const char ** English_ophelp(int n)
/* returns an array of strings of tooltips for the n-th menu */
{ int nitems; /* number of menus represented in ophelp1 */
nitems = sizeof(ophelp1) / (MAXLENGTH * sizeof(char *));
if(n < nitems)
return (const char **) ophelp1[n];
if(n >= MAXMENUS || n >= nitems)
assert(0);
return (const char **) ophelp1[0]; // avoid a warning message
}
Sindbad File Manager Version 1.0, Coded By Sindbad EG ~ The Terrorists