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# 4.E: Transcendental Functions (Exercises)

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These are homework exercises to accompany David Guichard's "General Calculus" Textmap. Complementary General calculus exercises can be found for other Textmaps and can be accessed here.

## 4.1: Trigonometric Functions

Some useful trigonometric identities are in chapter 18.

Ex 4.1.1 Find all values of $$\theta$$ such that $$\sin(\theta) = -1$$; give your answer in radians. (answer)

Ex 4.1.2 Find all values of $$\theta$$ such that $$\cos(2\theta) = 1/2$$; give your answer in radians. (answer)

Ex 4.1.3 Use an angle sum identity to compute $$\cos(\pi/12)$$. (answer)

Ex 4.1.4 Use an angle sum identity to compute $$\tan(5\pi/12)$$. (answer)

Ex 4.1.5 Verify the identity $$\cos^2(t)/(1-\sin(t)) = 1+\sin(t)$$.

Ex 4.1.6 Verify the identity $$2\csc(2\theta)=\sec(\theta)\csc(\theta)$$.

Ex 4.1.7 Verify the identity $$\sin(3\theta) - \sin(\theta) = 2\cos(2\theta) \sin(\theta)$$.

Ex 4.1.8 Sketch $$y=2\sin(x)$$.

Ex 4.1.9 Sketch $$y=\sin(3x)$$.

Ex 4.1.10 Sketch $$y=\sin(-x)$$.

Ex 4.1.11 Find all of the solutions of $$2\sin(t) -1 -\sin^2(t) =0$$ in the interval $$[0,2\pi]$$. (answer)

## 4.3: A hard Limit

Ex 4.3.1 Compute $$\lim_{x\to 0} {\sin (5x)\over x}$$ (answer)

Ex 4.3.2 Compute $$\lim_{x\to 0 } {\sin(7x)\over\sin (2x)}$$ (answer)

Ex 4.3.3 Compute $$\lim_{x\to 0 } {\cot (4x) \over\csc (3x)}$$ (answer)

Ex 4.3.4 Compute $$\lim_{x\to 0 } {\tan x\over x}$$ (answer)

Ex 4.3.5 Compute $$\lim_{x\to \pi/4} {\sin x -\cos x \over\cos (2x)}$$ (answer)

Ex 4.3.6 For all $$x\geq 0$$, $$4x-9 \leq f(x) \leq x^2 - 4x +7$$. Find $$\lim_{x\to4}f(x)$$. (answer)

Ex 4.3.7 For all $$x$$, $$2x \leq g(x) \leq x^4 - x^2 +2$$. Find $$\lim_{x\to1}g(x)$$. (answer)

Ex 4.3.8 Use the Squeeze Theorem to show that $$\lim_{x\to0} x^4 \cos(2/x)=0$$.

## 4.4: The Derivative of Sin x Part II

Find the derivatives of the following functions.

Ex 4.4.1$$\sin^2(\sqrt{x})$$ (answer)

Ex 4.4.2$$\sqrt{x}\sin x$$ (answer)

Ex 4.4.3 $${1\over \sin x}$$ (answer)

Ex 4.4.4 $${x^2+x\over \sin x}$$ (answer)

Ex 4.4.5 $$\sqrt{1-\sin^2x }$$ (answer)

## 4.5: Derivatives of the Trigonometric Functions

Find the derivatives of the following functions.

Ex 4.5.1 $$\sin x\cos x$$ (answer)

Ex 4.5.2 $$\sin(\cos x)$$ (answer)

Ex 4.5.3 $$\sqrt{x\tan x }$$ (answer)

Ex 4.5.4 $$\tan x/(1+\sin x)$$ (answer)

Ex 4.5.5 $$\cot x$$ (answer)

Ex 4.5.6 $$\csc x$$ (answer)

Ex 4.5.7 $$x^3 \sin (23x^2 )$$ (answer)

Ex 4.5.8 $$\sin ^2 x + \cos ^2 x$$ (answer)

Ex 4.5.9 $$\sin (\cos (6x) )$$ (answer)

Ex 4.5.10 Compute $${d\over d\theta}{\sec \theta\over 1+\sec \theta}$$. (answer)

Ex 4.5.11 Compute $${d\over dt}t^5 \cos (6t)$$. (answer)

Ex 4.5.12 Compute $${d\over dt}{t^3 \sin (3t)\over\cos (2t)}$$. (answer)

Ex 4.5.13 Find all points on the graph of $$f(x)=\sin^2(x)$$ at which the tangent line is horizontal. (answer)

Ex 4.5.14 Find all points on the graph of $$f(x) = 2\sin(x) - \sin^2(x)$$ at which the tangent line is horizontal. (answer)

Ex 4.5.15 Find an equation for the tangent line to $$\sin^2(x)$$ at $$x=\pi/3$$. (answer)

Ex 4.5.16 Find an equation for the tangent line to $$\sec ^2 x$$ at $$x=\pi/3$$. (answer)

Ex 4.5.17 Find an equation for the tangent line to $$\cos ^2 x - \sin ^2 (4x)$$ at $$x=\pi/6$$. (answer)

Ex 4.5.18 Find the points on the curve $$y= x+ 2\cos x$$ that have a horizontal tangent line. (answer)

Ex 4.5.19 Let $C$ be a circle of radius $$r$$. Let $$A$$ be an arc on $$C$$ subtending a central angle $$\theta$$. Let $$B$$ be the chord of $$C$$ whose endpoints are the endpoints of $$A$$. (Hence, $$B$$ also subtends $$\theta$$.) Let $$s$$ be the length of $$A$$ and let $$d$$ be the length of $$B$$. Sketch a diagram of the situation and compute $$\lim_{\theta \to 0^+ } s/d$$.

## 4.6: Exponential and Logarithmic Functions

Ex 4.6.1 Expand $$\log_{10} ((x+45)^7 (x-2))$$.

Ex 4.6.2 Expand $$\log_2 {x^3\over 3x-5 +(7/x)}$$.

Ex 4.6.3 Write $$\log_2 3x + 17 \log_2 (x-2) - 2\log_2 (x^2 + 4x + 1)$$ as a single logarithm.

Ex 4.6.4 Solve $$\log_2 (1+ \sqrt{x} ) = 6$$ for $$x$$.

Ex 4.6.5 Solve $$2^{x^2} = 8$$ for $$x$$.

Ex 4.6.6 Solve $$\log_2 (\log_3 (x) ) = 1$$ for $$x$$.

## 4.7: Derivatives of the exponential and logarithmic Functions

In 1--19, find the derivatives of the functions.

Ex 4.7.1$3^{x^2}\) (answer) Ex 4.7.2$ {\sin x \over e^x}\) (answer)

Ex 4.7.3$(e^x)^2\) (answer) Ex 4.7.4$ \sin(e^x)\) (answer)

Ex 4.7.5$e^{\sin x}\) (answer) Ex 4.7.6$ x^{\sin x}\) (answer)

Ex 4.7.7$x^3e^x\) (answer) Ex 4.7.8$ x+2^x\) (answer)

Ex 4.7.9$(1/3)^{x^2}\) (answer) Ex 4.7.10$ e^{4x}/x\) (answer)

Ex 4.7.11$\ln(x^3+3x)\) (answer) Ex 4.7.12$ \ln(\cos(x))\) (answer)

Ex 4.7.13$\ds\sqrt{\ln(x^2)}/x\) (answer) Ex 4.7.14$ \ln(\sec(x) + \tan(x))\) (answer)

## 4.11: Hyperbolic Functions

Ex 4.11.1 Show that the range of $$\sinh x$$ is all real numbers. (Hint: show that if $$y=\sinh x$$ then $$x =\ln (y+\sqrt{y^2+1})$$.)

Ex 4.11.2 Compute the following limits:

1. $$\lim_{x\to \infty } \cosh x$$
2. $$\lim_{x\to \infty } \sinh x$$
3. $$\lim_{x\to \infty } \tanh x$$
4. $$\lim_{x\to \infty } (\cosh x -\sinh x)$$

Ex 4.11.3 Show that the range of $$\tanh x$$ is $$(-1,1)$$. What are the ranges of $$\coth$$, $$\sech$$, and $$\csch$$? (Use the fact that they are reciprocal functions.)

Ex 4.11.4 Prove that for every $$x,y\in\R$$, $$\sinh (x+y) =\sinh x \cosh y + \cosh x \sinh y$$. Obtain a similar identity for $$\sinh(x-y)$$.

Ex 4.11.5 Prove that for every $$x,y\in\R$$, $$\cosh (x+y) =\cosh x \cosh y + \sinh x \sinh y$$. Obtain a similar identity for $$\cosh(x-y)$$.

Ex 4.11.6 Use exercises 4 and 5 to show that $$\sinh(2x)=2\sinh x \cosh x$$ and $$\cosh(2x)=\cosh^2 x +\sinh^2 x$$ for every $$x$$. Conclude also that $$(\cosh (2x) -1)/2 = \sinh ^2 x$$.

Ex 4.11.7 Show that $${d\over dx} (\tanh x) =\sech^2 x$$. Compute the derivatives of the remaining hyperbolic functions as well.

Ex 4.11.8 What are the domains of the six inverse hyperbolic functions?

Ex 4.11.9 Sketch the graphs of all six inverse hyperbolic functions.