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 θ such that sin(θ)=−1; give your answer in radians. (answer)
Ex 4.1.2 Find all values of θ such that cos(2θ)=1/2; give your answer in radians. (answer)
Ex 4.1.3 Use an angle sum identity to compute cos(π/12). (answer)
Ex 4.1.4 Use an angle sum identity to compute tan(5π/12). (answer)
Ex 4.1.5 Verify the identity cos2(t)/(1−sin(t))=1+sin(t).
Ex 4.1.6 Verify the identity 2csc(2θ)=sec(θ)csc(θ).
Ex 4.1.7 Verify the identity sin(3θ)−sin(θ)=2cos(2θ)sin(θ).
Ex 4.1.8 Sketch y=2sin(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 2sin(t)−1−sin2(t)=0 in the interval [0,2π]. (answer)
4.2: The Derivative of Sin x Part I
4.3: A hard Limit
Ex 4.3.1 Compute limx→0sin(5x)x (answer)
Ex 4.3.2 Compute limx→0sin(7x)sin(2x) (answer)
Ex 4.3.3 Compute limx→0cot(4x)csc(3x) (answer)
Ex 4.3.4 Compute limx→0tanxx (answer)
Ex 4.3.5 Compute limx→π/4sinx−cosxcos(2x) (answer)
Ex 4.3.6 For all x≥0, 4x−9≤f(x)≤x2−4x+7. Find limx→4f(x). (answer)
Ex 4.3.7 For all x, 2x≤g(x)≤x4−x2+2. Find limx→1g(x). (answer)
Ex 4.3.8 Use the Squeeze Theorem to show that limx→0x4cos(2/x)=0.
4.4: The Derivative of Sin x Part II
Find the derivatives of the following functions.
Ex 4.4.1sin2(√x) (answer)
Ex 4.4.2√xsinx (answer)
Ex 4.4.3 1sinx (answer)
Ex 4.4.4 x2+xsinx (answer)
Ex 4.4.5 √1−sin2x (answer)
4.5: Derivatives of the Trigonometric Functions
Find the derivatives of the following functions.
Ex 4.5.1 sinxcosx (answer)
Ex 4.5.2 sin(cosx) (answer)
Ex 4.5.3 √xtanx (answer)
Ex 4.5.4 tanx/(1+sinx) (answer)
Ex 4.5.5 cotx (answer)
Ex 4.5.6 cscx (answer)
Ex 4.5.7 x3sin(23x2) (answer)
Ex 4.5.8 sin2x+cos2x (answer)
Ex 4.5.9 sin(cos(6x)) (answer)
Ex 4.5.10 Compute ddθsecθ1+secθ. (answer)
Ex 4.5.11 Compute ddtt5cos(6t). (answer)
Ex 4.5.12 Compute ddtt3sin(3t)cos(2t). (answer)
Ex 4.5.13 Find all points on the graph of f(x)=sin2(x) at which the tangent line is horizontal. (answer)
Ex 4.5.14 Find all points on the graph of f(x)=2sin(x)−sin2(x) at which the tangent line is horizontal. (answer)
Ex 4.5.15 Find an equation for the tangent line to sin2(x) at x=π/3. (answer)
Ex 4.5.16 Find an equation for the tangent line to sec2x at x=π/3. (answer)
Ex 4.5.17 Find an equation for the tangent line to cos2x−sin2(4x) at x=π/6. (answer)
Ex 4.5.18 Find the points on the curve y=x+2cosx 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 θ. Let B be the chord of C whose endpoints are the endpoints of A. (Hence, B also subtends θ.) Let s be the length of A and let d be the length of B. Sketch a diagram of the situation and compute limθ→0+s/d.
4.6: Exponential and Logarithmic Functions
Ex 4.6.1 Expand log10((x+45)7(x−2)).
Ex 4.6.2 Expand log2x33x−5+(7/x).
Ex 4.6.3 Write log23x+17log2(x−2)−2log2(x2+4x+1) as a single logarithm.
Ex 4.6.4 Solve log2(1+√x)=6 for x.
Ex 4.6.5 Solve 2x2=8 for x.
Ex 4.6.6 Solve log2(log3(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(x3+2) compute f′(e1/3).
Ex 4.7.22If y=logax then ay=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 y2=1+x2 (answer)
Ex 4.8.2 x2+xy+y2=7 (answer)
Ex 4.8.3 x3+xy2=y3+yx2 (answer)
Ex 4.8.4 4cosxsiny=1 (answer)
Ex 4.8.5 √x+√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 1x+1y=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 x2−xy+y2=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 x4=y2+x2 at (2,√12). (This curve is the kampyle of Eudoxus.) (answer)
Ex 4.8.15 Find an equation for the tangent line to x2/3+y2/3=a2/3 at a point (x1,y1) on the curve, with x1≠0 and y1≠0. (This curve is an astroid.) (answer)
Ex 4.8.16 Find an equation for the tangent line to (x2+y2)2=x2−y2 at a point (x1,y1) on the curve, with x1≠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 π/2. Two families of curves, A and B, are orthogonal trajectories of each other if given any curve C in A and any curve D in 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 x2−y2=5 is orthogonal to 4x2+9y2=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 x2+y2=r2 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≠0 and c≠0 show that y2−x2=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 y2−x2=0 and yx=0 orthogonal to each other?
Ex 4.8.20 Suppose that m≠0. Show that the family of curves {y=mx+b∣b∈\R} is orthogonal to the family of curves {y=−(x/m)+c∣c∈\R}.
4.9: Inverse Trigonometric Functions
Ex 4.9.1 Show that the derivative of arccosx is −1√1−x2.
Ex 4.9.2 Show that the derivative of arctanx is 11+x2.
Ex 4.9.3 The inverse of cot is usually defined so that the range of arccot is (0,π). Sketch the graph of y=\arccotx. 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 \arccotx+arctanx=π/2.
Ex 4.9.5 Find the derivative of arcsin(x2). (answer)
Ex 4.9.6 Find the derivative of arctan(ex). (answer)
Ex 4.9.7 Find the derivative of arccos(sinx3) (answer)
Ex 4.9.8 Find the derivative of ln((arcsinx)2) (answer)
Ex 4.9.9 Find the derivative of arccosex (answer)
Ex 4.9.10 Find the derivative of arcsinx+arccosx (answer)
Ex 4.9.11 Find the derivative of log5(arctan(xx)) (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 sinhx is all real numbers. (Hint: show that if y=sinhx then x=ln(y+√y2+1).)
Ex 4.11.2 Compute the following limits:
- limx→∞coshx
- limx→∞sinhx
- limx→∞tanhx
- limx→∞(coshx−sinhx)
Ex 4.11.3 Show that the range of tanhx 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∈\R, sinh(x+y)=sinhxcoshy+coshxsinhy. Obtain a similar identity for sinh(x−y).
Ex 4.11.5 Prove that for every x,y∈\R, cosh(x+y)=coshxcoshy+sinhxsinhy. Obtain a similar identity for cosh(x−y).
Ex 4.11.6 Use exercises 4 and 5 to show that sinh(2x)=2sinhxcoshx and cosh(2x)=cosh2x+sinh2x for every x. Conclude also that (cosh(2x)−1)/2=sinh2x.
Ex 4.11.7 Show that ddx(tanhx)=\sech2x. 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.