Chapter 4: Transcendental FunctionsContent check — Calculus (Guichard), Chapter 4: Transcendental Functions

Find all values of $\theta$ such that $\sin (\theta )=-1$⁠; give your answer in radians.

Find all values of $\theta$ such that $\cos (2\theta )=1/2$⁠; give your answer in radians.

Use an angle sum identity to compute $\cos (\pi /12)$⁠.

Use an angle sum identity to compute $\tan (5\pi /12)$⁠.

Find all of the solutions of ${\displaystyle 2\sin (t)-1-{\sin }^2(t)=0}$ in the interval $[\mathrm{0,2}\pi ]$⁠.

Compute ${\displaystyle \underset{x\to 0}{\lim }\frac{\sin (5x)}{x}}$

Compute ${\displaystyle \underset{x\to 0}{\lim }\frac{\sin (7x)}{\sin (2x)}}$

Compute ${\displaystyle \underset{x\to 0}{\lim }\frac{\cot (4x)}{\csc (3x)}}$

Compute ${\displaystyle \underset{x\to 0}{\lim }\frac{\tan x}{x}}$

Compute ${\displaystyle \underset{x\to \pi /4}{\lim }\frac{\sin x-\cos x}{\cos (2x)}}$

For all $x\ge 0$⁠, $4x-9\le f(x)\le x^2-4x+7$⁠. Find ${\displaystyle \underset{x\to 4}{\lim }f(x)}$⁠.

For all $x$⁠, $2x\le g(x)\le x^4-x^2+2$⁠. Find ${\displaystyle \underset{x\to 1}{\lim }g(x)}$⁠.

Find the derivatives of the following functions.

${\displaystyle {\sin }^2(\sqrt{x})}$

Find the derivatives of the following functions.

${\displaystyle \sqrt{x}\sin x}$

Find the derivatives of the following functions.

${\displaystyle \frac{1}{\sin x}}$

Find the derivatives of the following functions.

${\displaystyle \frac{x^2+x}{\sin x}}$

Find the derivatives of the following functions.

${\displaystyle \sqrt{1-{\sin }^{2}x}}$

Find the derivatives of the following functions.

${\displaystyle \sin x\cos x}$

Find the derivatives of the following functions.

${\displaystyle \sin (\cos x)}$

Find the derivatives of the following functions.

${\displaystyle \sqrt{x\tan x}}$

Find the derivatives of the following functions.

${\displaystyle \tan x/(1+\sin x)}$

Find the derivatives of the following functions.

${\displaystyle \cot x}$

Find the derivatives of the following functions.

${\displaystyle \csc x}$

Find the derivatives of the following functions.

${\displaystyle x^3\sin (23x^2)}$

Find the derivatives of the following functions.

${\displaystyle {\sin }^{2}x+{\cos }^{2}x}$

Find the derivatives of the following functions.

${\displaystyle \sin (\cos (6x))}$

Find the derivatives of the following functions.

Compute ${\displaystyle \frac{d}{d\theta }\frac{\sec \theta }{1+\sec \theta }}$⁠.

Find the derivatives of the following functions.

Compute ${\displaystyle \frac{d}{dt}{t}^5\cos (6t)}$⁠.

Find the derivatives of the following functions.

Compute ${\displaystyle \frac{d}{dt}\frac{t^3\sin (3t)}{\cos (2t)}}$⁠.

Find the derivatives of the following functions.

Find all points on the graph of ${\displaystyle f(x)={\sin }^2(x)}$ at which the tangent line is horizontal.

Find the derivatives of the following functions.

Find all points on the graph of ${\displaystyle f(x)=2\sin (x)-{\sin }^2(x)}$ at which the tangent line is horizontal.

Find the derivatives of the following functions.

Find an equation for the tangent line to ${\displaystyle {\sin }^2(x)}$ at $x=\pi /3$⁠.

Find the derivatives of the following functions.

Find an equation for the tangent line to ${\displaystyle {\sec }^{2}x}$ at $x=\pi /3$⁠.

Find the derivatives of the following functions.

Find an equation for the tangent line to ${\displaystyle {\cos }^{2}x-{\sin }^2(4x)}$ at $x=\pi /6$⁠.

Find the derivatives of the following functions.

Find the points on the curve ${\displaystyle y=x+2\cos x}$ that have a horizontal tangent line.

Expand ${\displaystyle {\log }_{10}({(x+45)}^7(x-2))}$⁠.

Expand ${\displaystyle {\log }_2\frac{x^3}{3x-5+(7/x)}}$⁠.

Write ${\displaystyle {\log }_{2}3x+}$ ${\displaystyle 17{\log }_2(x-2)-}$ ${\displaystyle 2{\log }_2(x^2+4x+1)}$ as a single logarithm.

Solve ${\displaystyle {\log }_2(1+\sqrt{x})=6}$ for $x$⁠.

Solve ${\displaystyle 2^{x^2}=8}$ for $x$⁠.

Solve ${\displaystyle {\log }_2({\log }_3(x))=1}$ for $x$⁠.

Find the derivatives of the functions.

${\displaystyle 3^{x^2}}$

Find the derivatives of the functions.

${\displaystyle \frac{\sin x}{e^x}}$

Find the derivatives of the functions.

${\displaystyle {(e^x)}^2}$

Find the derivatives of the functions.

${\displaystyle \sin (e^x)}$

Find the derivatives of the functions.

${\displaystyle e^{\sin x}}$

Find the derivatives of the functions.

${\displaystyle x^{\sin x}}$

Find the derivatives of the functions.

${\displaystyle x^3{e}^x}$

Find the derivatives of the functions.

${\displaystyle x+2^x}$

Find the derivatives of the functions.

${\displaystyle {(1/3)}^{x^2}}$

Find the derivatives of the functions.

${\displaystyle e^{4x}/x}$

Find the derivatives of the functions.

${\displaystyle \ln (x^3+3x)}$

Find the derivatives of the functions.

${\displaystyle \ln (\cos (x))}$

Find the derivatives of the functions.

${\displaystyle \sqrt{\ln (x^2)}/x}$

Find the derivatives of the functions.

${\displaystyle \ln (\sec (x)+\tan (x))}$

Find the derivatives of the functions.

${\displaystyle x^{\cos (x)}}$

Find the derivatives of the functions.

${\displaystyle x\ln x}$

Find the derivatives of the functions.

$\ln (\ln (3x))$

Find the derivatives of the functions.

${\displaystyle \frac{1+\ln (3x^2)}{1+\ln (4x)}}$

Find the derivatives of the functions.

${\displaystyle \frac{x^8{(x-23)}^{1/2}}{27x^6{(4x-6)}^8}}$

Find the value of $a$ so that the tangent line to $y=\ln (x)$ at $x=a$ is a line through the origin. Sketch the resulting situation.

If ${\displaystyle f(x)=\ln (x^3+2)}$ compute ${\displaystyle f^{\prime }(e^{1/3})}$⁠.

Find a formula for the derivative $y^{\prime }$ at the point $(x,y)$⁠:

${\displaystyle y^2=1+x^2}$

Find a formula for the derivative $y^{\prime }$ at the point $(x,y)$⁠:

${\displaystyle x^2+xy+y^2=7}$

Find a formula for the derivative $y^{\prime }$ at the point $(x,y)$⁠:

${\displaystyle x^3+x{y}^2=y^3+y{x}^2}$

Find a formula for the derivative $y^{\prime }$ at the point $(x,y)$⁠:

${\displaystyle 4\cos x\sin y=1}$

Find a formula for the derivative $y^{\prime }$ at the point $(x,y)$⁠:

${\displaystyle \sqrt{x}+\sqrt{y}=9}$

Find a formula for the derivative $y^{\prime }$ at the point $(x,y)$⁠:

${\displaystyle \tan (x/y)=x+y}$

Find a formula for the derivative $y^{\prime }$ at the point $(x,y)$⁠:

${\displaystyle \sin (x+y)=xy}$

Find a formula for the derivative $y^{\prime }$ at the point $(x,y)$⁠:

${\displaystyle \frac{1}{x}+\frac{1}{y}=7}$

A hyperbola passing through $(\mathrm{8,6})$ consists of all points whose distance from the origin is a constant more than its distance from the point (5,2). Find the slope of the tangent line to the hyperbola at $(\mathrm{8,6})$⁠.

The graph of the equation ${\displaystyle x^2-xy+y^2=9}$ is an ellipse. Find the lines tangent to this curve at the two points where it intersects the $x$⁠-axis. Show that these lines are parallel.

Repeat the previous problem for the points at which the ellipse intersects the $y$⁠-axis.

Find the points on the ellipse from the previous two problems where the slope is horizontal and where it is vertical.

Find an equation for the tangent line to ${\displaystyle x^4=y^2+x^2}$ at ${\displaystyle (2,\sqrt{12})}$⁠. (This curve is the kampyle of Eudoxus.)

Find an equation for the tangent line to ${\displaystyle x^{2/3}+y^{2/3}=a^{2/3}}$ at a point ${\displaystyle (x_1,y_1)}$ on the curve, with ${\displaystyle x_1\ne 0}$ and ${\displaystyle y_1\ne 0}$⁠. (This curve is an astroid.)

Find an equation for the tangent line to ${\displaystyle {(x^2+y^2)}^2=x^2-y^2}$ at a point ${\displaystyle (x_1,y_1)}$ on the curve, when ${\displaystyle y_1\ne 0}$⁠. (This curve is a lemniscate.)

The inverse of $\cot$ is usually defined so that the range of arccot is $(0,\pi )$⁠. Sketch the graph of $y=\mathrm{arccot}x$⁠. In the process you will make it clear what the domain of arccot is. Find the derivative of the arccotangent.

Find the derivative of ${\displaystyle \arcsin (x^2)}$⁠.

Find the derivative of ${\displaystyle \arctan (e^x)}$⁠.

Find the derivative of ${\displaystyle \arccos (\sin x^3)}$

Find the derivative of ${\displaystyle \ln ({(\arcsin x)}^2)}$

Find the derivative of ${\displaystyle \arccos e^x}$

Find the derivative of $\arcsin x+\arccos x$

Find the derivative of ${\displaystyle {\log }_5(\arctan (x^x))}$

Compute the limits.

${\displaystyle \underset{x\to 0}{\lim }\frac{\cos x-1}{\sin x}}$

Compute the limits.

${\displaystyle \underset{x\to \infty }{\lim }\frac{e^x}{x^3}}$

Compute the limits.

${\displaystyle \underset{x\to \infty }{\lim }\sqrt{x^2+x}-\sqrt{x^2-x}}$

Compute the limits.

${\displaystyle \underset{x\to \infty }{\lim }\frac{\ln x}{x}}$

Compute the limits.

${\displaystyle \underset{x\to \infty }{\lim }\frac{\ln x}{\sqrt{x}}}$

Compute the limits.

${\displaystyle \underset{x\to \infty }{\lim }\frac{e^x+e^{-x}}{e^x-e^{-x}}}$

Compute the limits.

${\displaystyle \underset{x\to 0}{\lim }\frac{\sqrt{9+x}-3}{x}}$

Compute the limits.

${\displaystyle \underset{t\to 1^{+}}{\lim }\frac{(1/t)-1}{t^2-2t+1}}$

Compute the limits.

${\displaystyle \underset{x\to 2}{\lim }\frac{2-\sqrt{x+2}}{4-x^2}}$

Compute the limits.

${\displaystyle \underset{t\to \infty }{\lim }\frac{t+5-2/t-1/t^3}{3t+12-1/t^2}}$

Compute the limits.

${\displaystyle \underset{y\to \infty }{\lim }\frac{\sqrt{y+1}+\sqrt{y-1}}{y}}$

Compute the limits.

${\displaystyle \underset{x\to 1}{\lim }\frac{\sqrt{x}-1}{\sqrt[3]{x}-1}}$

Compute the limits.

${\displaystyle \underset{x\to 0}{\lim }\frac{{(1-x)}^{1/4}-1}{x}}$

Compute the limits.

${\displaystyle \underset{t\to 0}{\lim }(t+}$ ${\displaystyle \frac{1}{t})({(4-t)}^{3/2}-}$ ${\displaystyle 8)}$

Compute the limits.

${\displaystyle \underset{t\to 0^{+}}{\lim }(\frac{1}{t}+\frac{1}{\sqrt{t}})(\sqrt{t+1}-1)}$

Compute the limits.

${\displaystyle \underset{x\to 0}{\lim }\frac{x^2}{\sqrt{2x+1}-1}}$

Compute the limits.

${\displaystyle \underset{u\to 1}{\lim }\frac{{(u-1)}^3}{(1/u)-u^2+3u-3}}$

Compute the limits.

${\displaystyle \underset{x\to 0}{\lim }\frac{2+(1/x)}{3-(2/x)}}$

Compute the limits.

${\displaystyle \underset{x\to 0^{+}}{\lim }\frac{1+5/\sqrt{x}}{2+1/\sqrt{x}}}$

Compute the limits.

${\displaystyle \underset{x\to 0^{+}}{\lim }\frac{3+x^{-1/2}+x^{-1}}{2+4x^{-1/2}}}$

Compute the limits.

${\displaystyle \underset{x\to \infty }{\lim }\frac{x+x^{1/2}+x^{1/3}}{x^{2/3}+x^{1/4}}}$

Compute the limits.

${\displaystyle \underset{t\to \infty }{\lim }\frac{1-\sqrt{\frac{t}{t+1}}}{2-\sqrt{\frac{4t+1}{t+2}}}}$

Compute the limits.

${\displaystyle \underset{t\to \infty }{\lim }\frac{1-\frac{t}{t-1}}{1-\sqrt{\frac{t}{t-1}}}}$

Compute the limits.

${\displaystyle \underset{x\to -\infty }{\lim }\frac{x+x^{-1}}{1+\sqrt{1-x}}}$

Compute the limits.

${\displaystyle \underset{x\to \pi /2}{\lim }\frac{\cos x}{(\pi /2)-x}}$

Compute the limits.

${\displaystyle \underset{x\to 0}{\lim }\frac{e^x-1}{x}}$

Compute the limits.

${\displaystyle \underset{x\to 0}{\lim }\frac{x^2}{e^x-x-1}}$

Compute the limits.

${\displaystyle \underset{x\to 1}{\lim }\frac{\ln x}{x-1}}$

Compute the limits.

${\displaystyle \underset{x\to 0}{\lim }\frac{\ln (x^2+1)}{x}}$

Compute the limits.

${\displaystyle \underset{x\to 1}{\lim }\frac{x\ln x}{x^2-1}}$

Compute the limits.

${\displaystyle \underset{x\to 0}{\lim }\frac{\sin (2x)}{\ln (x+1)}}$

Compute the limits.

${\displaystyle \underset{x\to 1}{\lim }\frac{x^{1/4}-1}{x}}$

Compute the limits.

${\displaystyle \underset{x\to 1^{+}}{\lim }\frac{\sqrt{x}}{x-1}}$

Compute the limits.

${\displaystyle \underset{x\to 1}{\lim }\frac{\sqrt{x}-1}{x-1}}$

Compute the limits.

${\displaystyle \underset{x\to \infty }{\lim }\frac{x^{-1}+x^{-1/2}}{x+x^{-1/2}}}$

Compute the limits.

${\displaystyle \underset{x\to \infty }{\lim }\frac{x+x^{-2}}{2x+x^{-2}}}$

Compute the limits.

${\displaystyle \underset{x\to \infty }{\lim }\frac{5+x^{-1}}{1+2x^{-1}}}$

Compute the limits.

${\displaystyle \underset{x\to \infty }{\lim }\frac{4x}{\sqrt{2x^2+1}}}$

Compute the limits.

${\displaystyle \underset{x\to 0}{\lim }\frac{3x^2+x+2}{x-4}}$

Compute the limits.

${\displaystyle \underset{x\to 0}{\lim }\frac{\sqrt{x+1}-1}{\sqrt{x+4}-2}}$

Compute the limits.

${\displaystyle \underset{x\to 0}{\lim }\frac{\sqrt{x+1}-1}{\sqrt{x+2}-2}}$

Compute the limits.

${\displaystyle \underset{x\to 0^{+}}{\lim }\frac{\sqrt{x+1}+1}{\sqrt{x+1}-1}}$

Compute the limits.

${\displaystyle \underset{x\to 0}{\lim }\frac{\sqrt{x^2+1}-1}{\sqrt{x+1}-1}}$

Compute the limits.

${\displaystyle \underset{x\to \infty }{\lim }(x+5)(\frac{1}{2x}+\frac{1}{x+2})}$

Compute the limits.

${\displaystyle \underset{x\to 0^{+}}{\lim }(x+5)(\frac{1}{2x}+\frac{1}{x+2})}$

Compute the limits.

${\displaystyle \underset{x\to 1}{\lim }(x+5)(\frac{1}{2x}+\frac{1}{x+2})}$

Compute the limits.

${\displaystyle \underset{x\to 2}{\lim }\frac{x^3-6x-2}{x^3+4}}$

Compute the limits.

${\displaystyle \underset{x\to 2}{\lim }\frac{x^3-6x-2}{x^3-4x}}$

Compute the limits.

${\displaystyle \underset{x\to 1+}{\lim }\frac{x^3+4x+8}{2x^3-2}}$

Compute the limits.

The function ${\displaystyle f(x)=\frac{x}{\sqrt{x^2+1}}}$ has two horizontal asymptotes. Find them and give a rough sketch of $f$ with its horizontal asymptotes.

Compute the following limits:

a. ${\displaystyle \underset{x\to \infty }{\lim }\cosh x}$

b. ${\displaystyle \underset{x\to \infty }{\lim }\sinh x}$

c. ${\displaystyle \underset{x\to \infty }{\lim }\tanh x}$

d. ${\displaystyle \underset{x\to \infty }{\lim }(\cosh x-\sinh x)}$