Answer Key

Chapter 1

1. 1
2. $2\sqrt{3}$
3. $4$
4. $y=3\sin \left[\frac{1}{2}(x-\pi )\right]+1$
5. Three possible answers:​
   ​
   $y=3\cos \left[\frac{1}{2}(x-2\pi )\right]+1$​
   $y=3\cos \left[\frac{1}{2}(x+2\pi )\right]+1$​
   $y=-3\cos \left(\frac{1}{2}x\right)+1$
6. $g\left(x\right)=(1/2)e^{x-1}+1$
7. Domain [−1,1]; range [−π/2,π/2].
8. $y=\ln ({20}^{1/\ln 4}·{16}^{1/\ln 10})\approx 3.3651$
9. $2+\ln 9-2\ln \pi$
10. Green $e^{-x}$; blue $-e^x$; orange $-e^{-x}$; pink $-\ln (-x)$; cyan $\ln (-x)$; purple $-\ln x$; yellow $\ln x$.

Chapter 2

1. $\epsilon =4\delta$
2. $\frac{4}{5}$
3. $2$
4. $0$
5. $0$
6. No two-sided limit; right limit $+\infty$, left limit $-\infty$.
7. $-1/2$
8. $\frac{3}{2}$
9. $0$
10. Prove it

Chapter 3

1. $-2$
2. $f^{\prime }(x)=4x+4-2x^{-3}$
3. $3e^x-4x^{-2}+(1/2)x^{-1/2}$, for x>0.
4. ${\sec }^{2}x-\sin x$, where cos x≠0.
5. For sin x>0: $-{\csc }^{2}x-\csc x\cot x$. For sin x<0: ${\csc }^{2}x-\csc x\cot x$.
6. $x_1=-1$​
   $x_2=-\frac{2}{3}$​
   $x_3=-\frac{133}{225}$
7. Undefined, bad initial guess
8. $12x+8+2x^{-3}$, for x≠0.
9. $f^{\prime \prime }(x)=-\sin x+\cos x$
10. $3e^x+48/\sqrt{x}$ for x>0.

Chapter 4

1. $y\prime =e^x(1+x)/(1-x{e}^x{)}^2$ where $1-x{e}^x\ne 0$.
2. $\left[\right(x^2-4){\sec }^{2}x-2x\tan x]/(x^2-4{)}^2$.
3. $\left[\right(e^x-4)\sin x-2(e^x-4x\left)\cos x\right]/{\sin }^{3}x$.
4. $3/5$
5. $y\prime =(y+3e^x)/(3y^2-x)$, where the original y≠0 and the displayed denominator is nonzero.
6. $y\prime =(\sin y-3x^{2}y-8x\sin (2x^2\left)\right)/(x^3-x\cos y)$.
7. $\frac{dy}{dx}=\frac{2x\cdot \sin y\cdot e^{\cos x}-x^2\cdot \sin y\cdot e^{\cos x}\cdot \sin x}{1-x^2\cdot \cos y\cdot e^{\cos x}}$
8. $y\prime (\pi /4)=\sqrt{2}/2-2$.
9. $\frac{dy}{dx}=\frac{e^x\cdot \cos (4x^2+1)\cdot \sin y-8x\cdot e^x\cdot \sin (4x^2+1)\cdot \sin y-y-2x}{x-e^x\cdot \cos (4x^2+1)\cdot \cos y}$
10. $\frac{dy}{dx}=\frac{y\left({\cos }^{2}x\cdot \sin (\ln (y))+y\cdot \cos x+x\cdot y\cdot \sin x\right)}{\cos x\left(y\cdot \cos x-x\cdot \cos x\cdot \cos (\ln (y))-x\cdot y\right)}$

Chapter 5

1. $100e^4\approx 5460$ dollars/year under continuous compounding.
2. $0$ from the right.
3. 0
4. 0
5. $2/\left(x\ln 4\right)+3^{x{\log }_{2}x}\ln 3·({\log }_{2}x+1/\ln 2)+4/x$, x>0.
6. $f^{\prime }(x)=\frac{1}{x\cdot \ln 2\cdot \left(1+({\log }_{2}x{)}^2\right)}-\frac{3}{\sqrt{\left(1-{\cos }^{-1}(x{)}^2\right)\cdot \left(1-x^2\right)}}$
7. $1$
8. $5\ln 4/\sqrt{4^{2x}-1}+2/\left[x\sqrt{x^2-1}\right(1+\left(\mathrm{arccsc}x{)}^2\right)]+8^{x^{\ln x}}\ln 8·2\ln x·x^{\ln x-1}$, for x>1.
9. $f^{\prime }(0)=2+\ln 3$
10. No real tangent slope at x=0.5.

Chapter 6

1. Conditional on $A\left(t\right)=10000e^{t(e^{-t}+0.07)}$, $A\prime \left(3\right)\approx -423.62$ dollars/year.
2. $1000000e^{-1.5}\approx 223130.16$ dollars under the source’s continuous model.
3. $P=e^{5/3}\approx 5.2945$ dollars; sales 30000 units.
4. $Q=\sqrt{10000000/3}\approx 1825.74$ laptops in the continuous model; maximum profit ≈1217161.24 dollars; integer optimum 1826 laptops.
5. Answers will vary
6. $25$
7. a. $36$​
   b. $4.75$
8. $3$ cm²/s.
9. $\frac{dV}{dt}=48\frac{{\text{cm}}^3}{s}$
10. $-3/\sqrt{255/16}\approx -0.75147$ rad/s, assuming the rigid ladder geometry in the source image.

Chapter 7

1. $1$
2. $5.15$
3. $4\pi$
4. $32$
5. $0.3$
6. $3.2395055\dots$ square units.
7. $\frac{79}{32}=2.46875$ square units.
8. $75/64=1.171875$ square units.
9. $1$
10. The unit-width right sum is about 1.6449; it does not represent a finite exact area from zero.

Chapter 8

1. $\frac{1}{3}$
2. $-\frac{1}{x}-\frac{3}{2}{x}^{-2}+C$
3. $-1/\left(3x\right)-1/\left(2x^2\right)-1/\left(3x^3\right)+C$, x≠0.
4. $6$
5. $-\pi /4$
6. $(1/3)\ln (3x-2+\sqrt{(3x-2{)}^2+1})+C$
7. The printed integral is not defined as a real integral over [0,6].
8. $\arctan (x^2-4)+C$
9. $\ln (x^2+3/2+\sqrt{x^4+3x^2+2})+C$.
10. $(1/2)\ln \left|\right(\sqrt{x}-1)/(\sqrt{x}+1\left)\right|+C$ on 0<x<1 or x>1.

Chapter 9

1. $-\ln |\csc x+\cot x|+C$
2. $0$
3. $512/5-32(4-\sqrt{2}{)}^{3/2}+(24/5\left)\right(4-\sqrt{2}{)}^{5/2}\approx 20.951317$.
4. $3{\sin }^{-1}\left(\frac{x}{3}\right)+\frac{3}{2}\sin \left(2{\sin }^{-1}\left(\frac{x}{3}\right)\right)+C$
5. $\ln \left|x\right|-2x^2+C$, x≠0.
6. $\pi /4$, treating x=2 as a removable hole.
7. $\pi$
8. $\ln \left(\cos \right(\ln \left(\cos x\right))+3)+C$, on cos x>0.
9. $-\ln \left|\ln \right(\cos \left(\ln x\right))+3|+C$ where x>0, cos(ln x)>0, and ln(cos(ln x))+3≠0.
10. $x+\frac{2}{3}(1-x^2{)}^{\frac{3}{2}}+C$

Chapter 10

1. $V=4\pi a{b}^2/3$, for $a>0$ and $b>0$.
2. $0.5854\phantom{l}{\text{units}}^3$
3. $V\left(r\right)=4\pi r^3/3$, for $r\ge 0$.
4. $0.333$
5. $V=\frac{93\pi }{5}\approx 58.4336$ cubic units.
6. For the clarified cavity below $y=0$, $V=4\alpha (6+{\alpha }^2)\approx 76.2594$ mm³, where $\alpha =\sqrt{(3+\sqrt{21})/2}$.
7. $V=\pi [\frac{1-e^{-2}}{2}-\frac{1}{5}]\approx 0.729894$ cubic units.
8. Prove it is $V=\frac{1}{3}\pi r^{2}h$
9. $V=33{\pi }^2/16+3\pi /2\approx 25.0684$ cubic units.
10. $5.5{\pi }^2$​