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)

4.2: The Derivative of Sin x Part I

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)

Ex 4.7.15$ x^{\cos(x)}\) (answer)

Ex 4.7.16$ x\ln x$

Ex 4.7.17$\ln (\ln (3x) )$

Ex 4.7.18$ {1+\ln (3x^2 )\over 1+ \ln(4x)}$

Ex 4.7.19$ {x^8 (x-23)^{1/2}\over 27 x^6(4x-6)^8 }$

Ex 4.7.20Find 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. (answer)

Ex 4.7.21If $ f(x) = \ln(x^3 + 2)$ compute $ f'(e^{1/3})$.

Ex 4.7.22If $ y=\log_a x$ then $ a^y=x$. Use implicit differentiation to find $ y'$.

4.8: Implicit Differentiation

In exercises 1--8, find a formula for the derivative $y'$ at the point $(x,y)$:

Ex 4.8.1 $ y^2=1+x^2$ (answer)

Ex 4.8.2 $ x^2+xy+y^2=7$ (answer)

Ex 4.8.3 $ x^3+xy^2=y^3+yx^2$ (answer)

Ex 4.8.4 $ 4\cos x \sin y = 1$ (answer)

Ex 4.8.5 $ \sqrt{x} + \sqrt{y} = 9$ (answer)

Ex 4.8.6 $ \tan(x/y) = x+ y$ (answer)

Ex 4.8.7 $ \sin (x+y ) =xy$ (answer)

Ex 4.8.8 ${1\over x} + {1\over y} = 7$ (answer)

Ex 4.8.9 A hyperbola passing through $(8,6)$ 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 $(8,6)$. (answer)

Ex 4.8.10 Compute $y'$ for the ellipse of example 4.8.3.

Ex 4.8.11 The graph of the equation $ 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. (answer)

Ex 4.8.12 Repeat the previous problem for the points at which the ellipse intersects the $y$-axis. (answer)

Ex 4.8.13 Find the points on the ellipse from the previous two problems where the slope is horizontal and where it is vertical. (answer)

Ex 4.8.14 Find an equation for the tangent line to $ x^4 = y^2 + x^2$ at $ (2, \sqrt{12})$. (This curve is the kampyle of Eudoxus.) (answer)

Ex 4.8.15 Find an equation for the tangent line to $ x^{2/3} + y^{2/3} = a^{2/3}$ at a point $ (x_1 ,y_1)$ on the curve, with $ x_1 \neq 0$ and $ y_1 \neq 0$. (This curve is an astroid.) (answer)

Ex 4.8.16 Find an equation for the tangent line to $ (x^2 +y^2 )^2 =x^2 -y^2$ at a point $ (x_1 , y_1)$ on the curve, with $ x_1 \neq 0, -1, 1$. (This curve is a lemniscate.) (answer)

Remark 4.8.5 {Definition} Two curves are orthogonal if at each point of intersection, the angle between their tangent lines is $\pi/2$. Two families of curves, $\cal{A}$ and $\cal{B}$, are orthogonal trajectories of each other if given any curve $C$ in $\cal{A}$ and any curve $D$ in $\cal{B}$ the curves $C$ and $D$ are orthogonal. For example, the family of horizontal lines in the plane is orthogonal to the family of vertical lines in the plane.

Ex 4.8.17 Show that $ x^2 -y^2 =5$ is orthogonal to $ 4x^2 +9y^2 =72$. (Hint: You need to find the intersection points of the two curves and then show that the product of the derivatives at each intersection point is $-1$.)

Ex 4.8.18 Show that $ x^2 +y^2 = r^2$ is orthogonal to $y=mx$. Conclude that the family of circles centered at the origin is an orthogonal trajectory of the family of lines that pass through the origin.

Note that there is a technical issue when $m=0$. The circles fail to be differentiable when they cross the $x$-axis. However, the circles are orthogonal to the $x$-axis. Explain why. Likewise, the vertical line through the origin requires a separate argument.

Ex 4.8.19 For $k\not= 0$ and $c \neq 0$ show that $ y^2 -x^2 =k$ is orthogonal to $yx =c$. In the case where $k$ and $c$ are both zero, the curves intersect at the origin. Are the curves $ y^2 -x^2 =0$ and $yx=0$ orthogonal to each other?

Ex 4.8.20 Suppose that $m\neq 0$. Show that the family of curves $ \{y=mx+b \mid b\in \R \}$ is orthogonal to the family of curves $ \{y=-(x/m)+c \mid c \in \R\}$.

4.9: Inverse Trigonometric Functions

Ex 4.9.1 Show that the derivative of $\arccos x$ is $ -{1\over \sqrt{1-x^2}}$.

Ex 4.9.2 Show that the derivative of $\arctan x$ is $ {1\over 1+x^2}$.

Ex 4.9.3 The inverse of $\cot$ is usually defined so that the range of arccot is $(0, \pi )$. Sketch the graph of $y=\arccot x$. In the process you will make it clear what the domain of arccot is. Find the derivative of the arccotangent. (answer)

Ex 4.9.4 Show that $\arccot x + \arctan x =\pi/2$.

Ex 4.9.5 Find the derivative of $ \arcsin(x^2)$. (answer)

Ex 4.9.6 Find the derivative of $ \arctan(e^x)$. (answer)

Ex 4.9.7 Find the derivative of $ \arccos (\sin x^3 )$ (answer)

Ex 4.9.8 Find the derivative of $ \ln( (\arcsin x )^2)$ (answer)

Ex 4.9.9 Find the derivative of $ \arccos e^x$ (answer)

Ex 4.9.10 Find the derivative of $\arcsin x + \arccos x$ (answer)

Ex 4.9.11 Find the derivative of $ \log _5 (\arctan (x^x ) )$ (answer)

4.10: Limits Revisited

Compute the limits.

Ex 4.10.1$\ds\lim_{x\to 0} {\cos x -1\over \sin x}$ (answer)

Ex 4.10.2$\ds\lim_{x\to \infty} {e^x\over x^3}$ (answer)

Ex 4.10.3$\ds\lim_{x\to \infty} \sqrt{x^2+x}-\sqrt{x^2-x}$ (answer)

Ex 4.10.4$\ds\lim_{x\to \infty} {\ln x\over x}$ (answer)

Ex 4.10.5$\ds\lim_{x\to \infty} {\ln x\over \sqrt{x}}$ (answer)

Ex 4.10.6$\ds\lim_{x\to\infty} {e^x + e^{-x}\over e^x -e^{-x}}$ (answer)

Ex 4.10.7$\ds\lim_{x\to0}{\sqrt{9+x}-3\over x}$ (answer)

Ex 4.10.8$\ds\lim_{t\to1^+}{(1/t)-1\over t^2-2t+1}$ (answer)

Ex 4.10.9$\ds\lim_{x\to2}{2-\sqrt{x+2}\over 4-x^2}$ (answer)

Ex 4.10.10$\ds\lim_{t\to\infty}{t+5-2/t-1/t^3\over 3t+12-1/t^2}$ (answer)

Ex 4.10.11$\ds\lim_{y\to\infty}{\sqrt{y+1}+\sqrt{y-1}\over y}$ (answer)

Ex 4.10.12$\ds\lim_{x\to1}{\sqrt{x}-1\over \root 1/3\of{x}-1}$ (answer)

Ex 4.10.13$\ds\lim_{x\to0}{(1-x)^{1/4}-1\over x}$ (answer)

Ex 4.10.14$\ds\lim_{t\to 0}{\left(t+{1\over t}\right)((4-t)^{3/2}-8)}$ (answer)

Ex 4.10.15$\ds\lim_{t\to 0^+}\left({1\over t}+{1\over\sqrt{t}}\right) (\sqrt{t+1}-1)$ (answer)

Ex 4.10.16$\ds\lim_{x\to 0}{x^2\over\sqrt{2x+1}-1}$ (answer)

Ex 4.10.17$\ds\lim_{u\to 1}{(u-1)^3\over (1/u)-u^2+3u-3}$ (answer)

Ex 4.10.18$\ds\lim_{x\to 0}{2+(1/x)\over 3-(2/x)}$ (answer)

Ex 4.10.19$\ds\lim_{x\to 0^+}{1+5/\sqrt{x}\over 2+1/\sqrt{x}}$ (answer)

Ex 4.10.20$\ds\lim_{x\to 0^+}{3+x^{-1/2}+x^{-1}\over 2+4x^{-1/2}}$ (answer)

Ex 4.10.21$\ds\lim_{x\to\infty}{x+x^{1/2}+x^{1/3}\over x^{2/3}+x^{1/4}}$ (answer)

Ex 4.10.22$\ds\lim_{t\to\infty} {1-\sqrt{t\over t+1}\over 2-\sqrt{4t+1\over t+2}}$ (answer)

Ex 4.10.23$\ds\lim_{t\to\infty}{1-{t\over t-1}\over 1-\sqrt{t\over t-1}}$ (answer)

Ex 4.10.24$\ds\lim_{x\to-\infty}{x+x^{-1}\over 1+\sqrt{1-x}}$ (answer)

Ex 4.10.25$\ds\lim_{x\to\pi/2}{\cos x\over (\pi/2)-x}$ (answer)

Ex 4.10.26$\ds\lim_{x\to0}{e^x-1\over x}$ (answer)

Ex 4.10.27$\ds\lim_{x\to0}{x^2\over e^x-x-1}$ (answer)

Ex 4.10.28$\ds\lim_{x\to1}{\ln x\over x-1}$ (answer)

Ex 4.10.29$\ds\lim_{x\to0}{\ln(x^2+1)\over x}$ (answer)

Ex 4.10.30$\ds\lim_{x\to1}{x\ln x\over x^2-1}$ (answer)

Ex 4.10.31$\ds\lim_{x\to0}{\sin(2x)\over\ln(x+1)}$ (answer)

Ex 4.10.32$\ds\lim_{x\to1}{x^{1/4}-1\over x}$ (answer)

Ex 4.10.33$\ds\lim_{x\to1^+}{\sqrt{x}\over x-1}$ (answer)

Ex 4.10.34$\ds\lim_{x\to1}{\sqrt{x}-1\over x-1}$ (answer)

Ex 4.10.35$\ds\lim_{x\to\infty}{x^{-1}+x^{-1/2}\over x+x^{-1/2}}$ (answer)

Ex 4.10.36$\ds\lim_{x\to\infty}{x+x^{-2}\over 2x+x^{-2}}$ (answer)

Ex 4.10.37$\ds\lim_{x\to\infty}{5+x^{-1}\over 1+2x^{-1}}$ (answer)

Ex 4.10.38$\ds\lim_{x\to\infty}{4x\over\sqrt{2x^2+1}}$ (answer)

Ex 4.10.39$\ds\lim_{x\to0}{3x^2+x+2\over x-4}$ (answer)

Ex 4.10.40$\ds\lim_{x\to0}{\sqrt{x+1}-1\over \sqrt{x+4}-2}$ (answer)

Ex 4.10.41$\ds\lim_{x\to0}{\sqrt{x+1}-1\over \sqrt{x+2}-2}$ (answer)

Ex 4.10.42$\ds\lim_{x\to0^+}{\sqrt{x+1}+1\over\sqrt{x+1}-1}$ (answer)

Ex 4.10.43$\ds\lim_{x\to0}{\sqrt{x^2+1}-1\over\sqrt{x+1}-1}$ (answer)

Ex 4.10.44$\ds\lim_{x\to\infty}{(x+5)\left({1\over 2x}+{1\over x+2}\right)}$ (answer)

Ex 4.10.45$\ds\lim_{x\to0^+}{(x+5)\left({1\over 2x}+{1\over x+2}\right)}$ (answer)

Ex 4.10.46$\ds\lim_{x\to1}{(x+5)\left({1\over 2x}+{1\over x+2}\right)}$ (answer)

Ex 4.10.47$\ds\lim_{x\to2}{x^3-6x-2\over x^3+4}$ (answer)

Ex 4.10.48$\ds\lim_{x\to2}{x^3-6x-2\over x^3-4x}$ (answer)

Ex 4.10.49$\ds\lim_{x\to1+}{x^3+4x+8\over 2x^3-2}$ (answer)

Ex 4.10.50The function $\ds f(x) = {x\over\sqrt{x^2+1}}$ has two horizontal asymptotes. Find them and give a rough sketch of $f$ with its horizontal asymptotes. (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:

$ \lim_{x\to \infty } \cosh x$
$ \lim_{x\to \infty } \sinh x$
$ \lim_{x\to \infty } \tanh x$
$ \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.

Contributors

David Guichard (Whitman College)

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