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yyy/english hints2.c � 9� Change $1 / sin$ to csc Change $1 / cos$ to sec Change $1 / tan$ to cot Change $1 / tan$ to $cos / sin$ Change $1 / cot$ to tan Change $1 / cot$ to $sin / cos$ Change $1 / sec$ to cos Change $1 / csc$ to sin Express sin in terms of csc Express cos in terms of sec Express tan in terms of cot Use the law $sin^2 u + cos^2 u = 1$. Notice an expression matching the pattern $1 - sin^2 u$. Notice an expression matching the pattern $1 - cos^2 u$ Try rewriting $sin^2$ as $1 - cos^2$ Try rewriting $cos^2$ as $1 - sin^2$ Use the law $sec^2 u - tan^2 u = 1$. Notice an expression matching the pattern $tan^2 u + 1$. Notice an expression matching the pattern $sec^2 u - 1$. Try rewriting $sec^2$ as $tan^2 + 1$ Try rewriting $tan^2$ as $sec^2 u - 1$ Get rid of all powers of $sin$ using $sin^(2n+1) u = sin u (1-cos^2 u)^n$ Get rid of all powers of $cos$ using $cos^(2n+1) u = cos u (1-sin^2 u)^n$ Get rid of all powers of $tan$ using $tan^(2n+1) u = tan u (sec^2 u-1)^n$ Get rid of all powers of $sec$ using $sec^(2n+1) u = sec u (tan^2 u+1)^n$ Combine powers of $(1-cos t)$ and powers of $(1+cos t)$ to a power of $sin^2 t$ Combine powers of $(1-sin t)$ and powers of $(1+sin t)$ to a power of $cos^2 t$ Notice an expression matching the pattern $csc^2 u - cot^2 u$ Notice an expression matching the pattern $cot^2 u + 1$ Notice an expression matching the pattern $csc^2 u - 1$ Try rewriting $csc^2$ as $cot^2 + 1$ Try rewriting $cot^2$ as $csc^2 - 1$ Express $csc(\pi /2-\theta )$ in terms of $sec \theta $ Express $cot(\pi /2-\theta )$ in terms of $tan \theta $ Get rid of all powers of $cot$ using $cot^(2n+1) u = cot u (csc^2 u-1)^n$ Get rid of all powers of $csc$ using $csc^(2n+1) u = csc u (cot^2 u+1)^n$ Use the formula for $sin(u+v)$ Use the formula for $sin(u-v)$ Use the formula for $cos(u+v)$ Use the formula for $cos(u-v)$ Use the formula for $tan(u+v)$ Use the formula for $tan(u-v)$ Use the formula for $cot(u+v)$ Use the formula for $cot(u-v)$ Use the double-angle formula for sin You have a formula of the form $cos(2\theta )$. There are three different double-angle formulas beginning with $cos(2\theta )$. Choose carefully, thinking about what will come next. Select the sum containing $cos(2\theta )+1$. Select the sum containing $cos(2\theta )-1$. Use the double-angle formula for tan Use the double-angle formula for cot A product of sin times cos can be simplified to a single trig function using the law: $sin \theta cos \theta = \onehalf sin 2\theta $ A product of sin times cos can be simplified to a single trig function using the law: $2 sin \theta cos \theta = sin 2\theta $ Combine some terms to get the cosine of a double angle. Expand a trig function by writing $n\theta $ as $(n-1)\theta + \theta $ and using a sum formula. There is a formula for expanding $sin(3\theta )$. There is a formula for expanding $cos(3\theta )$. You can expand $sin n\theta $ as a polynomial in $sin \theta $ and $cos \theta $. You can expand $cos n\theta $ as a polynomial in $sin \theta $ and $cos \theta $. You could cross multiply. You might switch the sides. Transfer a suitable term from left to right. Transfer a suitable term from right to left. Add something to both sides. Subtract something from both sides. Multiply both sides by something. Cancel a term from both sides. Raise both sides to the same power. Take the square root of both sides. Take the $n$-th root of both sides. Apply a function to both sides. Perhaps it isn't even a true identity. Check it numerically. If it's not an identity, you should soon find a number that makes the sides unequal. Make a substitution. When does $sin(u) = 1/2$ ? When does $sin(u) = -1/2$ ? When does $sin(u) = \sqrt 3/2$ ? When does $sin(u) = -\sqrt 3/2$ ? When does $cos(u) = \sqrt 3/2$ ? When does $cos(u) = -\sqrt 3/2$ ? When does $cos(u) = 1/2$ ? When does $cos(u) = -1/2$ ? When does $tan(u) = 1/\sqrt 3$ ? When does $tan(u) = -1/\sqrt 3$ ? When does $tan(u) = \sqrt 3$ ? When does $tan(u) = -\sqrt 3$ ? When does $sin(u) = 1/\sqrt 2$ ? When does $sin(u) = -1/\sqrt 2$ ? When does $cos(u) = 1/\sqrt 2$ ? When does $cos(u) = -1/\sqrt 2$ ? When does $tan(u) = 1$ ? When does $tan(u) = -1$ ? When does $sin u = 0$ ? When does $sin u = 1$ ? When does $sin u = -1$ ? When does $cos u = 0$ ? When does $cos u = 1$ ? When does $cos u = -1$ ? When does $tan u = 0$ ? When does $cot u = 0$ ? You can get rid of the sin by taking the arcsin, but there will be multiple solutions. You can get rid of the cos by taking the arccos, but there will be multiple soltuions. Try taking the arctan to get rid of the tangent. Evaluate the arcsin exactly. Evaluate the arccos exactly. Evaluate the arctan exactly. Get rid of the arccot, using the law $arccot x = arctan (1/x)$ Get rid of the arcsec, using the law $arcsec x = arccos (1/x)$ Get rid of the arccsc, using the law $arccsc x = arcsin (1/x)$ arcsin is an odd function. Although arccos is neither an odd nor an even function, it does satisfy the law $arccos(-x) = \pi -arccos x$ arctan is an odd function Your solutions involve an integer parameter, so there are infinitely many of them. If the original equation is periodic with period $2\pi $, you should rewrite your solutions so the solutions have the form $c + 2n\pi $. Then you will only need to check the solutions in one period. Remember the values of sin are all between $-1$ and 1. Remember the values of cos are all between $-1$ and 1. $tan(arcsin x)$ is actually an algebraic function of $x$. $tan(arccos x)$ is actually an algebraic function of $x$. $tan(arctan x)$ is just $x$. $sin(arcsin x)$ is just $x$. $sin(arccos x)$ is actually an algebraic function of $x$. $sin(arctan x)$ is actually an algebraic function of $x$. $cos(arcsin x)$ is actually an algebraic function of $x$. $cos(arccos x)$ is just $x$. $cos(arctan x)$ is actually an algebraic function of $x$. $sec(arcsin x)$ is actually an algebraic function of $x$. $sec(arccos x)$ is just $1/x$. $sec(arctan x)$ is actually an algebraic function of $x$. $arctan(tan \theta )$ is just $\theta $, if $-\pi /2\le \theta \le \pi /2$ $arcsin(sin \theta )$ is just $\theta $, if $-\pi /2\le \theta \le \pi /2$ $arccos(cos \theta )$ is just $\theta $, if $0\le \theta \le \pi $ $arctan(tan x)$ in general is not equal to $x$, but it is $x$ minus a certain multiple of $pi$, so it can be espressed as $x + c1$ where $c1$ is constant on intervals where $tan x$ is defined. $arcsin x$ and $arccos x$ are complementary angles. $arctan x$ and $arctan 1/x$ are complementary angles, but watch out for the signs if $x$ is negative. Remember cos means sin of the complement. So the cosine of the complement is the sin. That is, $cos(\pi /2-\theta ) = sin \theta $. Remember cos means sin of the complement. That is, $sin(\pi /2-\theta ) = cos \theta $. Remember cot means tan of the complement. So the cot of the complement is the tan. That is, $cot(\pi /2-\theta ) = tan \theta $. Remember cot means tan of the complement. That is, $tan(\pi /2-\theta ) = cot \theta $. Remember csc means sec of the complement. So the csc of the complement is the sec. That is, $csc(\pi /2-\theta ) = sec \theta $. Remember csc means sec of the complement. That is, $sec(\pi /2-\theta ) = csc \theta $. Rewrite the sine as cosine of the complement. Rewrite the cosine as sine of the complement. Rewrite the tangent as cot of the complement. Rewrite the cotangent as tan of the complement. Rewrite the secant as csc of the complement. Rewrite the cosecant as sec of the complement. Remember cos means sin of the complement. That is, $sin(90\deg -\theta ) = cos \theta $. Remember cot means tan of the complement. That is, $tan(90\deg -\theta ) = cot \theta $. Remember csc means sec of the complement. That is, $sec(90\deg -\theta ) = csc \theta $. Combine the degrees into a single expression. sin is an odd function. cos is an even function. tan is an odd function. cot is an odd function. sec is an even function. csc is an odd function. sin squared is an even function. cos squared is an even function. tan squared is an even function. cot squared is an even function. sec squared is an even function. csc squared is an even function. sin is periodic; use the formula expressing this fact. cos is periodic; use the formula expressing this fact. tan is periodic; use the formula expressing this fact. sec is periodic; use the formula expressing this fact. csc is periodic; use the formula expressing this fact. cot is periodic; use the formula expressing this fact. $sin^2$ is periodic withx period $\pi $, even though the period of sin is $2\pi .$ $cos^2$ is periodic with period $\pi $, even though the period of cos is $2\pi .$ $sec^2$ is periodic with period $\pi $, even though the period of sec is $2\pi .$ $csc^2$ is periodic with period $\pi $, even though the period of csc is $2\pi .$ Reduce the angle using $sin u = -sin(u-\pi )$ Reduce the angle using $sin u = sin(\pi -u)$ Reduce the angle using $cos u = -cos(u-\pi )$ Reduce the angle using $cos u = -cos(\pi -u)$ Get rid of $sin^2$ using a half-angle identity. Get rid of $cos^2$ using a half-angle identity. A product of sin and cos can be simplified using the law: $sin \theta cos \theta = \onehalf sin 2\theta $ Use a half-angle identity Write $\theta $ as $2(\theta /2)$; you can find this operation with the half-angle identities. You can express $sin x cos x$ as $\onehalf sin 2x$ You can write $sin x cos y$ as a sum of sines whose frequencies are the sum and difference of $x$ and $y$ You can write $cos x sin y$ as a difference of sines whose frequencies are the sum and difference of $x$ and $y$ You can write $sin x sin y$ as a difference of cosines whose frequencies are the sum and difference of $x$ and $y$ You can write $cos x cos y$ as a sum of cosines whose frequencies are the sum and difference of $x$ and $y$ You can write $sin x + sin y$ as a product of sines and cosines whose frequencies are the sum and difference of $x$ and $y$ You can write $sin x - sin y$ as a product of sines and cosines whose frequencies are the sum and difference of $x$ and $y$ You can write $cos x + cos y$ as a product of cosines whose frequencies are the sum and difference of $x$ and $y$ You can write $cos x - cos y$ as a product of sines whose frequencies are the sum and difference of $x$ and $y$ Substitute u,v for the expressions in the trig functions. Experiment numerically. The limit of a sum is the sum of the limits, at least if the limits exist. The limit of a difference is the difference of the limits, at least if the limits exist. The limit of a constant is that constant. The limit of $x$ as $x$ goes to $c$ is just $c$ itself. You can pull a constant through the limit. You can pull a minus sign through the limit. The limit of a product is the product of the limits, at least if the limits exist. The limit of a (constant) power is the power of the limit. The limit of $c^v$ is $c$ raised to the power $lim v$, when $c$ is constant. $lim u^v=(lim u)^(lim v)$ The limit of a square root is the square root of the limit, provided it is positive. The limit of an odd root is the root of the limit. The limit of a root is the root of the limit, provided it is positive. You can use MathXpert to calculate limits of polynomials in one step. Push the limit inside the absolute value sign. You can pull a constant out from the numerator using $lim cu/v = c lim u/v$ The limit of a reciprocal is the reciprocal of the limit; more generally for $c$ constant we have $lim c/v = c/lim v$ The limit of a quotient is the quotient of the limits, at least if the limit in the denominator is nonzero. Factor out powers of $(x-a)$ in a limit as $x$ approaches $a$. You can use MathXpert to calculate the limit of a rational function in one step. Sometimes it helps to write $a^n/b^n as (a/b)^n$. Rationalize the fraction. Look for that operation with the limit of quotients operations. Simplify your limit by pulling out a simple part of it which has a nonzero finite limits. This means to express $lim uv$ as $lim u lim v$, where $lim u$ is finite and nonzero. For example, you might pull out $sin(x)/x$ from the limit of $sin^2(x) /x$ as $x$ approaches 0. Factor out a constant. Multiply both numerator and denominator by something. The aim is to make the limit in the denominator nonzero. Divide both numerator and denominator by something. The aim is to make the limit in the denominator nonzero. Divide both numerator and denominator by something and then push the limit into numerator and denominator. Choose the quantity to divide by so that the denominator will have a nonzero limit. With the limits of quotients operations you will find an algebraic formula which may be helpful: $$(ab+ac+d)/q = a(b+c)/q + d/q$$ You can bring the denominator inside the square root (squaring it). You can bring the denominator under the square root (squaring it), but watch out for the sign. You can bring the denominator under the radical. You can bring the denominator under the radical, but watch out for the sign. You can bring the numerator inside the square root (squaring it). You can bring the numerator under the square root (squaring it), but watch out for the sign. You can bring the numerator under the radical. You can bring the numerator under the radical, but watch out for the sign. Use L'H�pital's rule. You can ask MathXpert to evaluate the derivative in one step Put everything but the logarithm in the denominator, and then use L'H�pital's rule. Select the whole limit term to find the right operation. Put the negative exponent in the denominator as a positive exponent, and then use L'H�pital's rule. Move the exponential function to the denominator, and then use L'H�pital's rule. Move a trig function to the denominator (using a trig identity), and then use L'H�pital's rule. Convert the product to a fraction by moving one or more factors to the denominator, creating a compound fraction. Put the fractions over a common denominator and simplify. There is a special limit formula involving $(sin t)/t$ There is a special limit formula involving $(tan t)/t$ There is a special limit formula involving $(1-cos t)/t$ There is a special limit formula involving $(1-cos t)/t^2$ There is a special limit formula involving $(1+t)^(1/t)$ There is a special limit formula involving $(ln(1+t))/t$ There is a special limit formula involving $(e^t-1)/t$ There is a special limit formula involving $(e^(-t)-1)/t$ The singularity of $ln x$ at the origin is so weak that any positive power of $t$ will kill it. MathXpert has an operation for handling such a limit in one step, or you can put the power in the denominator and use L'H�pital's rule. The function $cos(1/t)$ makes infinitely many oscillations between -1 and 1 as $t$ approaches 0. The function $sin(1/t)$ makes infinitely many oscillations between -1 and 1 as $t$ approaches 0. The function $tan(1/t)$ behaves quite wildly as $t$ approaches 0. The function $cos t$ makes infinitely many oscillations between -1 and 1 as $t$ approaches infinity. The function $sin t$ makes infinitely many oscillations between -1 and 1 as $t$ approaches infinity. The function $tan t$ takes on all real values for arbitrarily large $t$, so it can't approach any limit as $t$ approaches infinity. There is a special limit formula involving $(sinh t)/t$ There is a special limit formula involving $(tanh t)/t$ There is a special limit formula involving $(cosh t -1)/t$ There is a special limit formula involving $(cosh t - 1)/t^2$ The limit of a ln is the ln of the limit, at least if it's positive. Limits of continuous functions are calculated by $lim f(u)=f(lim u)$. In fact, this is the \it definition \rm of continuity. You can change the limit variable using the formula for composition of functions. Namely, $$lim(t->a,f(g(t)))=lim(u->g(a),f(u))$$ You can ask MathXpert to evaluate a simple limit in one step. To calculate the limit of a non-constant power, first make the base be constant, using the law $lim u^v = lim e^(v ln u)$. If the limit of a product seems to be indeterminate, you can try the law: $lim uv = lim v/(1/u)$. Sometimes the resulting limit of a quotient can be evaluated. A limit is undefined if the function whose limit is being taken limit fails to be defined in a suitable neighborhood of the limit point. Try the law: $$lim(t->a, u) = e^(lim(t->a, ln u))$$ Maybe you can remove a troublesome term, perhaps an oscillatory factor, using the squeeze theorem. You can try something similar to rationalizing the numerator, even though there is no numerator: $$lim(t->a, sqrt(u)-v)=lim(t->a, (sqrt(u)-v)(sqrt(u)+v)/(sqrt(u)+v))$$ You can neglect all but the leading terms in numerator and denominator. A complicated limit can be replaced by the limit of the leading term. You can replace a sum by its leading term in a limit under certain conditions, but not always. You must take care that the leading terms don't cancel out to zero, causing you to lose the real answer among the terms you neglected. An expression with undefined parts is itself undefined $lim(e^u) = e^(lim u)$ $lim(ln u) = ln(lim u)$ An algebraic function always dominates a logarithm. For $t$ large, $t^n$ is large too, so $1/t^n$ is small. For $t$ large, $t^n$ is large too. For $t$ large, $e^t$ is large too. For $t$ large and negative, $e^t$ is very small. For $t$ large, $ln t$ is large too. For $t$ large, $\sqrt t$ is large, too. For $t$ large, $^n\sqrt t$ is large, too. For $abs(t)$ large, $arctan t$ is close to $pi/2$ or $-pi/2$ The arccot of a large positive number is close to zero. The arccot of a large negative number is close to $pi$ For $abs(t)$ large, $tanh t$ is close to 1 or -1. $lim \sqrt u-v=lim (\sqrt u-v)(\sqrt u+v)/\sqrt u+v)$ $lim(sin u) = sin(lim u)$ if the limit is finite. $lim(cos u) = cos(lim u)$ if the limit is finite Limits at infinity can be transformed to limits at zero if $f(t)$ is replaced by $f(1/t)$. For $u$ small, $1/u^2^n$ is large. For $u$ small, $1/u^n$ is large, but if $n$ is odd, it has opposite signs for $u$ positive and $u$ negative, which makes trouble for the two-sided limit as $u$ approaches zero. For $u$ small and positive, $1/u^n$ is large. For $u$ small and negative, $1/u^n$ is large and (if $n$ is odd) negative. If the denominator goes to zero and the numerator does not, then the limit is undefined. For $t$ small and positive, $ln t$ is large and negative. tan $t$ has singularities at odd multiples of $\pi /2$. But it approaches the singularities with different signs from the left and right. cot $t$ has singularities at multiples of $\pi $. But it approaches the singularities with different signs from the left and right. sec $t$ has singularities at odd multiples of $\pi /2$. But it approaches the singularities with different signs from the left and right. csc $t$ has singularities at multiples of $\pi $. But it approaches the singularities with different signs from the left and right. Multiply one factor and divide the other by something chosen to make it possible to calculate the limits. $\pm \infty /$positive = $\pm \infty $ nonzero$/\pm \infty = 0$ positive$\times \pm \infty = \pm \infty $ $\pm \infty \times \infty = \pm \infty $ $\pm \infty +$ finite$ = \pm \infty $ $\infty + \infty = \infty $ $u^\infty = \infty $ if $u > 1$ $u^\infty = 0$ if $0 < u < 1$ $u^(-\infty ) = 0$ if $u > 1$ $u^(-\infty ) = \infty $ if $0 < u < 1$ $\infty ^n = \infty $ if $n > 0$ You have a sum containing infinities of different signs; such a sum is undefined. $a/0+ = \infty $ if $a>0$ $a/0- = -\infty $ if $a>0$ $a/0 =$ undefined $\infty /0+ = \infty $ $\infty /0- = -\infty $ $\infty /0 = $ undefined $\infty /0^2 = \infty $ $\infty /0^2^n = \infty $ $a/0^2 = \infty if a > 0$ $a/0^2 = -\infty if a < 0$ $a/0^2^n = \infty if a > 0$ $a/0^2^n = -\infty if a < 0$ $ln \infty = log \infty = \infty $ $\sqrt \infty = \infty $ $^n\sqrt \infty = \infty $ $arctan \pm \infty = \pm \pi /2$ $arccot \infty = 0$ $arccot -\infty = \pi $ $arcsec \pm \infty = \pi /2$ $arccsc \pm \infty = 0$ trig limits at $\infty $ are undefined, because the trig function oscillate (or worse) $cosh \pm \infty = \infty $ $sinh \pm \infty = \pm \infty $ $tanh \pm \infty = \pm 1$ $ln 0 = -\infty $ The derivative of a constant is zero. Here a 'constant' means anything that doesn't depend on the variable with respect to which you are differentiating. You have an expression $dx/dx$. This should evaluate to 1. The derivative of a sum is the sum of the derivatives. You can pull a minus sign out through the derivative sign You can pull a constant out through the derivative sign Use the 'power law' for differentiating a power. You can use MathXpert to differentiate a polynomial in one step. By definition, $f'(x) = d/dx f(x)$. Use the formula that defines a derivative as a certain limit. It's with the other operations for derivatives. You can ask MathXpert to differentiate a polynomial in one step. The derivative of a sum (or difference) is the sum (or difference) of the derivatives. You have a constant in the denominator. Pull it out using: $$diff(u/c,x)=(1/c)diff(u,x)$$. Any constants in the numerator will also come out. Use the 'product rule' for derivatives There is a simple formula for the derivative of a reciprocal: $$diff(1/v,x) = -diff(v,x)/v^2$$ It's well worth memorizing this special case of the quotient rule. Use the 'quotient rule' for derivatives There is a formula for the derivative of a square root. Often it is much simpler to differentiate a square root directly, rather than convert it to a fractional exponent and use the power law. To differentiate a root, first convert it to fractional exponent form. To differentiate a power in the denominator, you don't have to first convert it to a negative exponent as so many students do. You can use the power law directly in the form: $$diff(c/x^n,x) = -nc/x^(n+1)$$ There is a simple formula for differentiating absolute values: $d/dx |x| = x/|x|$. If your textbook omits this formula, check it yourself by considering separately the cases when $x$ is positive and negative. Of course, both sides of the formula are undefined when $x=0$. By definition, $f'(x) = d/dx f(x)$ The derivative of sin is cos The derivative of cos is $-sin$ The derivative of tan is $sec^2$ The derivative of sec is sec tan The derivative of cot is $-csc^2$ The derivative of csc is - csc cot $e^x$ is its own derivative Exponential functions are their own derivatives, except for a constant:$ d/dx c^x = (ln c) c^x$ To differentiate a power with a non-constant exponent, make the base constant by using the law: $$ diff(u^v,x) = diff(e^(v ln u),x)$$ The derivative of $ln x is 1/x$ The derivative of $ln |x| = 1/x$ Try rewriting $dy/dx$ as $y (d/dx) ln y$ Use the formula: $d/dx e^u = e^u du/dx$ To differentiate a power with constant base, use the formula: $$diff(c^u,x)=(ln c)c^u diff(u,x)$$ To differentiate a logarithm, use the formula: $$diff(ln u,x) = (1/u)(diff(u,x))$$ Use the formula: $$diff(ln abs(u),x) = (1/u) diff(u,x)$$ There is a formula for differentiating $ln(cos x)$ in one step. There is a formula for differentiating $ln(sin x)$ in one step. $d/dx arctan x = 1/(1+x^2)$ $d/dx arcsin x = 1/\sqrt (1-x^2)$ $d/dx arccos x = -1/\sqrt (1-x^2)$ $d/dx arccot x = -1/(1+x^2)$ $d/dx arcsec x = 1/(|x|\sqrt (x^2-1))$ $d/dx arccsc x = -1/(|x|\sqrt (x^2-1))$ $d/dx arctan u = (du/dx)/(1+u^2)$ $d/dx arcsin u = (du/dx)/\sqrt (1-x^2)$ $d/dx arccos u = -(du/dx)/\sqrt (1-x^2)$ $d/dx arccot u = -(du/dx)/(1+u^2)$ $d/dx arcsec u=(du/dx)/(|u|\sqrt (u^2-1))$ $d/dx arccsc u=-(du/dx)/(|u|\sqrt (u^2-1))$ Use the chain rule form of the power rule: $$diff(u^n,x) = nu^(n-1) diff(u,x)$$ Use the chain rule with the rule for differentiating square roots: $$diff(sqrt(u),x) = diff(u,x)/(2 sqrt(u))$$ Use the chain rule with the formula for the derivative of sin Use the chain rule with the formula for the derivative of cos Use the chain rule with the formula for the derivative of tan Use the chain rule with the formula for the derivative of sec Use the chain rule with the formula for the derivative of cot Use the chain rule with the formula for the derivative of csc Use the chain rule with the formula for the derivative of absolute value Use the chain rule in the form $$diff(f(u),x) = f'(u) diff(u,x)$$ Now eliminate your defined variable. Consider points where $f'(x)=0$ Consider endpoints of interval Are there points where $f'(x)$ is undefined? Consider the limits at open ends of the interval. Reject a point outside interval Make a table of decimal $y$-values Make a table of exact $y$-values Choose the maximum value(s) from your table. Choose the minimum value(s) from your table. You can ask MathXpert to evaluate a derivative in one step. Now solve the equation. Get rid of the integer parameter. This function is constant, so the max equals the min equals the value. Evaluate the derivative. Simplify the expression. Solve the equation. Differentiate the equation. Eliminate the derivative of a variable by substituting for it. Eliminate the compound fractions. Factor out a common term. Try to factor. Multiply out and simplify. Is there a common factor in numerator and denominator? Write it as a polynomial in some variable or expression. Express some expression in polynomial form. Make the leading coefficient of some polynomial 1. Convert fractional exponents of 1/2 to square roots. Convert fractional exponents to roots. Eliminate roots and square roots in favor of fractional exponents. Differentiate the identity using the law: $u=v => du/dx = dv/dx$. Express the second derivative using $$diff(u,x,2) = (diff(diff(u,x),x)$$ $$diff(u,x,n) = diff(diff(u,x,n-1),x)$$ The derivative of the derivative is the second derivative. Differentiating an $n$-th derivative produces an $n+1$-st derivative. Evaluate numerically at a point. $\int 1 dt = t$ There is a constant integrand, so use the law $$integral(c,t) = ct$$ $\int t dt = t^2/2$ $\int cu dt = c\int u dt (c constant)$ Bring the minus sign out of the integral using $$integral(-u,t) = -integral(u,t)$$ The integrand is a sum, so you can use the property known as linearity of the integral: $$integral(u+v,t) = integral(u,t) + integral(v,t) $$ The integrand is a difference, so you can use the property known as linearity of the integral: $$integral(u-v,t) = integral(u,t) - integral(v,t) $$ The integrand is a sum or difference, so you can use the property known as linearity of the integral: $$integral(au+bv,t) = a integral(u,t) + b integral(v,t) $$ This property also works with a minus sign, or with a mixture of plus and minus signs. $\int t^n dt=t^(n+1)/(n+1) (n # -1)$ $\int 1/t^(n+1) dt= -1/(nt^n) (n # 0)$ The integrand is a polynomial. You can ask MathXpert to integrate it in one step. $\int (1/t) dt = ln |t|$ $\int 1/(t\pm a) dt = ln |t\pm a|$ Multiply out the integrand, obtaining a sum of simpler terms. expand $(a+b)^n$ in integrand $\int |t| dt = t|t|/2$ Integrate the sine. Integrate the cosine. Integrate the tangent. Integrate the cotangent. Integrate the secant. Integrate the cosecant. Integrate the square of the secant. Integrate the square of the cosecant. There is a formula for the integral of $tan^2 t$, or you can do it by parts. There is a formula for the integral of $cot^2 t$, or you can do it by parts. $sec t tan t$ can be directly integrated, since it is the derivative of $sec t$. $csc t cot t$ can be directly integrated, since it is the derivative of $csc t$. The exponential function is its own integral: $$integral(e^t,t) = e^t$$ An exponential function is its own integral, but if the exponent contains a constant the integral has a corresponding factor: $\int e^at dt =(1/a) e^at$ $\int e^(-t)dt = -e^(-t)$ $\int e^(-at)dt = -(1/a) e^(-at)$ $\int e^(t/a)dt = a e^(t/a)$ An exponential function is its own integral, except that if the base is not $e$, then a constant factor must be thrown in. $\int u^v dt = \int (e^(v ln u) dt$ $\int ln t = t ln t - t$ $\int e^(-t^2) dt = \sqrt \pi /2 Erf(t)$ Try integration by substitution compute $du/dx$ Evaluate the derivative Get back your original integral with 'show integral again' Express the integrand as a function of the new variable, by choosing: integrand = $f(u) \times du/dx$ Eliminate the original variable of integration entirely now. Integrate by substitution. Try integration by parts. Set the current line equal to the original problem, getting an equation. Isolate the original integral on the left side of the equation. Integrate by substitution You can ask MathXpert to evaluate a simple integral in one step. Use the fundamental theorem of calculus Get rid of the bar for function evaluation. Switch the limits of integration, introducing a minus sign. Combine two definite integrals with the same integrand into one integral, if they represent integration over different parts of the same interval. It may help to break a definite integral up into two (or more) integrals, introducing an intermediate point (or points) as a new limit of integration. Break up the integral into two or more integrals whose endpoints are at the zeroes of the integrand. Then you will be able to get rid of absolute value. You can ask MathXpert to calculate the numerical value of an integral, if the integral has a numerical value. Notice that the upper and lower limits of integration are the same. Express an improper integral as a limit of proper integrals. If the integrand does not tend to zero at $\infty $, an improper integral diverges. If the integrand does not tend to zero at $-\infty $, an improper integral diverges. The integral of an odd function over an interval whose midpoint is the origin has to be zero. The integral of an even function over an interval whose midpoint is the origin is twice the integral over the positive half of the interval. Use a trig substitution Use an inverse substitution Get rid of the $sin^2$ term in the integrand using: $sin^2 t = (1-cos 2t)/2$ in integral. You can find this formula with the trigonometric integral formulas as well as with the trig formulas. Get rid of the $cos^2$ term in the integrand using: $cos^2 t = (1+cos 2t)/2$ in integral. You can find this formula with the trigonometric integral formulas as well as with the trig formulas. Make a substitution $u=cos x$ after using $sin^2=1-cos^2$. Select the whole integral to see this choice. Make a substitution $u=sin x$ after using $cos^2=1-sin^2$. Select the whole integral to see this choice. Make a substitution $u=tan x$ after using $sec^2=1+tan^2$. Select the whole integral to see this choice. Make a substitution $u=cot x$ after using $csc^2=1+cot^2$. Select the whole integral to see this choice. Make a substitution $u=sec x$ after using $tan^2=sec^2-1$. Select the whole integral to see this choice. Make a substitution $u=csc x$ after using $cot^2=csc^2-1$. Select the whole integral to see this choice. Use the identity $tan^2 x = sec^2 x - 1$ in the integrand. Select the whole integral to see this choice. Use the identity $cot^2 x = csc^2 x - 1$ in the integrand. Select the whole integral to see this choice. Use a reduction formula to reduce this to another similar integral, but with a lower power of sec. Use a reduction formula to reduce this to another similar integral, but with a lower power of csc. Use the Weierstrass substitution: $u = tan(x/2)$. Select the whole integral to see this choice. Multiply both numerator and denominator by $1+cos x$. Multiply both numerator and denominator by $1-cos x$. Multiply both numerator and denominator by $1+sin x$. Multiply both numerator and denominator by $1-sin x$. Multiply both numerator and denominator by $sin x + cos x$. Multiply both numerator and denominator by $cos x - sin x$. Use polynomial division to reduce to the case in which the numerator is of lower degree than the denominator Factor the denominator if you can. Is there any common factor in the numerator and denominator? You can ask MathXpert to perform 'square-free factorization', which will find any repeated factors. This operation uses an algorithm not usually taught in textbooks. You can use MathXpert to factor a polynomial numerically. Close decimal approximations to the roots will be used. Expand the integrand in partial fractions. Complete the square in the denominator. A reciprocal of a linear function integrates to a logarithm. A reciprocal of a power of a linear function integrates to another such function. You could reduce the integral by substitution to a power of the variable, but you might as well do it in one step. A reciprocal of a sum of squares integrates to an arctan. A reciprocal of a difference of squares integrates to an arccoth, an arctanh, or a logarithm. Complete the square in the denominator A reciprocal of a square root of a difference of squares integrates to an arcsin. A reciprocal of a square root of a sum of squares integrates to a logarithm. Look on the menu for integrating square roots in the denominator. Make a rationalizing substitution. There is an integration formula for arcsin There is an integration formula for arccos There is an integration formula for arctan There is an integration formula for arccot There are two integration formulas for arccsc--be careful. There are two integration formulas for arcsec--be careful. Eliminate compound fractions. Put fractions over a common denominator and simplify. Try to factor Evaluate the limit Change the integral by substitution Absorb numbers into the constant of integration. The integral of sinh is cosh. The integral of cosh is sinh. The integral of tanh is ln cosh. The integral of coth is ln sinh. The integral of csch is $ln tanh(u/2)$. The integral of $sech u$ is $arctan (sinh u)$. Expand $1/(1-x)$ in a power series. Expand $1/(1+x)$ in a power series. Sum the series for $1/(1-x)$. Sum the series for $1/(1+x)$. Expand $x/(1-x)$ in a power series. Expand $x/(1+x)$ in a power series. Sum the series for $x/(1-x)$. Sum the series for $x/(1+x)$. Expand $1/(1-x^k)$ in a power series. Expand $x^m/(1-x^k)$ in a power series. Sum the series for $1/(1-x^k)$. Sum the series for $x^m/(1-x^k)$. Expand $1/(1+x^k)$ in a power series. Expand $x^m/(1+x^k)$ in a power series. Sum the series for $1/(1+x^k)$. Sum the series for $x^m/(1+x^k)$. You can expand $x^k/(1-x)$ as a geometric series You can expand $x^k/(1+x)$ as a geometric series Sum the geometric series. Expand $ln(1-x)$ in a power series. Expand $ln(1+x)$ in a power series. Sum the power series for $ln(1-x)$. Sum the power series for $ln(1+x)$. Expand $sin x$ in a power series. Expand $cos x$ in a power series. Sum the series for $sin x$. Sum the series for $cos x$. Expand $e^x$ in a power series. Sum the series for $e^x$. Expand $e^-x$ in a power series. Sum the series for $e^-x$. Expand $arctan x$ in a power series. Sum the series for arctan. Use the binomial series to expand a power of a sum. Sum the binomial series You may want to express the series in the form $a_0 + a_1 + ... $ You may want to express the series in the form $a_0 + a_1 + a_2 + ... $ You may want to express the series using ... instead of sigma notation. Express the series using sigma notation. Show another term before ... Show more terms before ... You have a telescoping series. Multiply series Two power series can be multiplied to produce a new power series. A power series can be divided by a polyomial, using a process like long division. A polynomial can be divided by a power series , using a process like long division. Two power series can be divided, using a process like long division. The square of a series can be written as a double series. The square of a power series can be written as another power series. A power of a power series can be expressed as another power series. Combine the sum of two series into a single series. Combine the difference of two series into a single series. Split off the first few terms of an infinite series. Perhaps by decreasing the lower limit of a series (subtracting the new terms) you can bring your series into a standard form. Add something to the index variable to bring the series into a more manageable form. Subtract something from the index variable to bring the series into a more manageable form. Rename the index variable Break a series $\sum (a+b)$ into a sum of series $\sum a + \sum b$. Differentiate term by term. Pull a derivative out of the series. Integrate term by term. Pull an integral out of the series. Calculate the first few terms. Write the function as the integral of its derivative. Then expand the derivative in a series and integrate term-by-term. Write the function as a definite integral of its derivative. Then expand the derivative in a series and integrate term-by-term. Write the function as the derivative of its integral. Then expand the integral in a series and differentiate term-by-term. Solve for the constant of integration in order to eliminate it. Separate the terms with even and odd indices, getting two new series. You can show that a series is divergent by showing its general term does not tend to zero. Use the integral test. Use the ratio test. Use the root test. Use the comparison test to prove convergence. Find a convergent series with larger general term. Use the comparison test to prove divergence. Find a divergent series with smaller general term. Use the limit comparison test. Use the condensation test. Finish the integral test. Finish the root test. Finish the ratio test. Finish the divergence test. Finish the comparison test. Finish the limit comparison test. Finish the condensation test. You have finished showing the convergence of the comparison series. Now state the positive result about the convergence of the original series. To see this choice, select the entire current line. You have finished showing the divergence of the comparison series. Now state the negative result about the convergence of the original series. To see this choice, select the entire current line. The harmonic series $$sum(1/k,k,1,infinity)$$ is divergent, since its partial sum up to $n$ terms is approximately $ln n$. There is a formula for $$sum(1/k^2,k,1,infinity$$ Express a complex number in polar form to calculate its logarithm, using the law $$ln(u+iv) = ln(r e^(i theta))$$ Use the formula for complex logarithms: $$ln(re^(i theta))=ln r + i theta$$ There is a subtlety here: in applying this law, if $\theta $ is not between $-\pi $ and $\pi $, it will be reduced to that range. The natural logarithm of i is $i\pi /2$, since $\pi /2$ is the argument of i The natural logarithm of -1 is $i\pi $, since $-1 = e^(i\pi )$ The natural logarithm of -a is $ln a + i\pi $, since $-1 = e^(i\pi )$. This formula assumes $a$ is positive. Expand cos in terms of complex exponentials. Expand sin in terms of complex exponentials. To take a complex square root, you take the square root of the radius and half of the argument. To take a complex $n$-th root, you take the $n$-th root of the radius, and divide the argument by $n$. Expand the complex exponential using cos and sin Use Euler's famous identity: $$e^(i pi) = -1 $$ Use Euler's famous identity: $$e^(-i pi) = -1 $$ $e^(2n\pi i) = 1$, because as $\theta $ varies, $e^i\theta $ traces the unit circle. As $\theta $ varies, $e^i\theta $ traces the unit circle. Therefore you can get rid of multiples of $2 pi i$ in the exponent. Rewrite the complex exponential so it has base $e$, using the law $$u^v = e^(v ln u)$$ $sin(it)$ can be expressed using the hyperbolic sine, instead of expanding in complex exponentials. $cos(it)$ can be expressed using the hyperbolic cosine, instead of expanding in complex exponentials. $sinh(it)$ can be expressed as $i sin t$, instead of expanding in exponentials. $cosh(it)$ can be expressed as $cos t$, instead of expanding in exponentials. $tan(it)$ can be expressed using the hyperbolic tangent,instead of expanding in complex exponentials. $cot(it)$ can be expressed using the hyperbolic cotangent, instead of expanding in complex exponentials. $tanh(it)$ can be expressed as $i tan t$, instead of expanding in exponentials. $coth(it)$ can be expressed as $-i cot t$, instead of expanding in exponentials. Use a complex exponential to express $cos t + i sin t$ Use a complex exponential to express $cos t - i sin t$ Simplify an expression in complex exponentials to a cosine. Simplify an expression in complex exponentials to a sine. Use the definition of cosh Combine exponentials into a cosh term Use the definition of sinh Combine exponentials into a sinh term cosh is an even function sinh is an odd function Combine the cosh and sinh terms using: $cosh u + sinh u = e^u$ Combine the cosh and sinh terms using: $cosh u - sinh u = e^(-u)$ Remember $cosh 0 = 1$ Remember $sinh 0 = 0$ Express $e^x$ in terms of hyperbolic functions Express $e^(-x)$ in terms of hyperbolic functions Use the identity $sinh^2u + 1 = cosh^2 u$ Use the identity $cosh^2 u - 1 = sinh^2u $ Use the identity $cosh^2 u - sinh^2u = 1$ Use the identity $cosh^2 u = sinh^2u + 1$ Use the identity $sinh^2u = cosh^2 u - 1$ Use the identity $1 - tan^2u = sech^2u$ Use the identity $1 - sech^2u = tan^2u$ Express tanh in terms of sinh and cosh. Combine sinh and cosh into tanh. Express coth in terms of cosh and sinh Combine cosh and sinh into coth Express sech as the reciprocal of cosh The reciprocal of cosh is sech Express csch as the reciprocal of sinh The reciprocal of sinh is csch Use the formula $tanh^2 u + sech^2 u = 1$. Use the formula $tanh^2 u = 1 - sech^2 u$. Use the formula $sech^2 u = 1 - tanh^2 u$. Use the formula for sinh of a sum or difference Use the formula for cosh of a sum or difference Use the double-angle formula: $sinh 2u = 2 sinh u cosh u$ Use the double-angle formula: $cosh 2u = cosh^2 u + sinh^2 u$ There is a formula to simplify $tanh(ln u)$. There is a formula to express arcsinh in terms of logarithms. There is a formula to express arccosh in terms of logarithms. There is a formula to express arctanh in terms of logarithms. The derivative of sinh is cosh The derivative of cosh is sinh The derivative of tanh is $sech^2$ The derivative of coth is $-csch^2$ The derivative of sech is $- sech tanh$ The derivative of csch is $- csch coth$ The derivative of ln sinh is coth The derivative of ln cosh is tanh The derivative of arcsinh is actually an algebraic function The derivative of arccosh is actually an algebraic function The derivative of arctanh is actually an algebraic function The derivative of arccoth is actually an algebraic function The derivative of arcsech is actually an algebraic function The derivative of arccsch is actually an algebraic function Eliminate the sg function, since its argument is positive. Eliminate the sg function, since its argument is negative. Eliminate the sg function, since its argument is zero. sg is an odd function Express sg in terms of absolute value Express $|x|$ as $x sg(x)$ An even power is always positive An odd power has the same sign as its base, so $sg(x)$ to an odd power is $sg(x)$ Bring sg to the numerator using $1/sg(x) = sg(x)$ sg(x) is constant when x is nonzero, in which case its derivative is zero. sg(x) can be integrated directly. sg(x) can be pulled through the integral sign if the integrand is nonzero. sg(x) is used to combine the cases of $x$ positive and $x$ negative, but sometimes they have to be treated separately. Drop positive factors inside the sg function. Drop negative factors inside the sg function, adding a minus sign in front. The sign of an odd power of $x$ is the same as the sign of $x$. $1/x$ has the same sign as $x$. $c/x$ has the same sign as $x$, if $x$ is positive. Express $x sg(x)$ as $|x|$. Express $|x| sg(x)$ as $x$. The derivative of $J0$ is $-J1$ $d/dx J1(x) = J0(x) - J1(x)/x$ $d/dx J(n,x)=J(n-1,x)-(n/x)J(n,x)$ The derivative of $Y0$ is $-Y1$ $d/dx Y1(x) = Y0(x) - Y1(x)/x$ $d/dx Y(n,x)=Y(n-1,x)-(n/x)Y(n,x)$ The derivative of $I0$ is $-I1$ $d/dx I1(x) = I0(x) - I1(x)/x$ $d/dx I(n,x)=I(n-1,x)-(n/x)I(n,x)$ The derivative of $K0$ is $-K1$ $d/dx K1(x) = -K0(x) - K1(x)/x$ $d/dx K(n,x)= -K(n-1,x)-(n/x)K(n,x)$ Use a defined function Multiply out products of sums and collect the resulting terms. Multiply out using $a(b+c) = ab+ac$, and then make a cancellation. Put factors in order. The fractions need to be put over a common denominator before calculating the limit. Begin by factoring the denominators if necessary. The fractions need to be put over a common denominator before calculating the limit. The fractions need to be put over a common denominator before calculating the limit. Begin by eliminating negative exponents. Express the square root using a fractional exponent. Expand the cosine of a double angle. Eliminate $sin^2 t$ by expressing it in terms of $cos^2 t$. Eliminate $cos^2 t$ by expressing it in terms of $sin^2 t$. Eliminate $tan^2 t$ by expressing it in terms of $sec^2 t$. Eliminate $sec^2 t$ by expressing it in terms of $tan^2 t$. Multiply coefficients Evaluate a simple square root. Add or subtract something to both sides. Factor one of the summands to make a common factor explicit. After that you can factor out the common factor. Make a substitution Rewrite trig functions in terms of sin and cos so that common denominators can be found. Use $ab+ac = a(b+c)$ to create the middle term of a quadratic expression. Factor one or both sides of an identity if the result will permit a cancellation. One side is a perfect square (or other power). Factor it. Get all the logarithms to have the same argument by using the law for logarithms of a power. 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Sindbad File Manager Version 1.0, Coded By Sindbad EG ~ The Terrorists