Chapter 1
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Chapter 1
Calculus Volume 1Chapter 1
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Chapter 1
Checkpoint
1.1
$f(1) = 3$ and $f(a + h) = a^{2} + 2ah + h^{2} - 3a - 3h + 5$
1.2
Domain = $\left\{ x \middle| x \leq 2 \right\},$ range = $\left\{ y \middle| y \geq 5 \right\}$
1.3
$x = 0,2,3$
1.4
$\left( \frac{f}{g} \right)(x) = \frac{x^{2} + 3}{2x - 5}.$ The domain is $\left\{ {x|x \neq \frac{5}{2}} \right\}.$
1.5
$\left( {f \circ g} \right)(x) = 2 - 5\sqrt{x}.$
1.6
$(g \circ f)(x) = 0.63x$
1.7
$f(x)$ is odd.
1.8
Domain = $(\text{−}\infty,\infty),$ range = $\left\{ y \middle| y \geq -4 \right\}.$
1.9
$m = 1\text{/}2.$ The point-slope form is
$y - 4 = \frac{1}{2}\left( {x - 1} \right).$
The slope-intercept form is
$y = \frac{1}{2}x + \frac{7}{2}.$
1.10
The zeros are $x = 1 \pm \sqrt{3}\text{/}3.$ The parabola opens upward.
1.11
The domain is the set of real numbers $x$ such that $x \neq 1\text{/}2.$ The range is the set $\left\{ y \middle| y \neq 5\text{/}2 \right\}.$
1.12
The domain of $f$ is $\text{(−∞, ∞).}$ The domain of $g$ is $\left\{ x \middle| x \geq 1\text{/}5 \right\}.$
1.13
Algebraic
1.14
1.15
$C(x) = \left\{ \begin{matrix}
{49,0 < x \leq 1} \\
{70,1 < x \leq 2} \\
{91,2 < x \leq 3}
\end{matrix} \right.$
1.16
Shift the graph $y = x^{2}$ to the left 1 unit, reflect about the $x$-axis, then shift down 4 units.
1.17
$7\pi\text{/}6;$ 330°
1.18
$\text{cos}(3\pi\text{/}4) = \text{−}\sqrt{2}\text{/}2;\ \text{sin}(\text{−}\pi\text{/}6) = -1\text{/}2$
1.19
$10$ ft
1.20
$\theta = \frac{3\pi}{2} + 2n\pi,\frac{\pi}{6} + 2n\pi,\frac{5\pi}{6} + 2n\pi$ for $n = 0, \pm 1, \pm 2\text{,…}$
1.21
$\begin{matrix}
{1 + \theta} & = & {1 + \frac{\theta}{\theta}} \\
& = & {\frac{\theta}{\theta} + \frac{\theta}{\theta}} \\
& = & \frac{\theta + \theta}{\theta} \\
& = & \frac{1}{\theta} \\
& = & \theta
\end{matrix}$
1.22
To graph $f(x) = 3\mspace{2mu}\text{sin}\left( {4x} \right) - 5,$ the graph of $y = \text{sin}(x)$ needs to be compressed horizontally by a factor of 4, then stretched vertically by a factor of 3, then shifted down 5 units. The function $f$ will have a period of $\pi\text{/}2$ and an amplitude of 3.
1.23
No.
1.24
$f^{-1}(x) = \frac{2x}{x - 3}.$ The domain of $f^{-1}$ is $\left\{ x \middle| x \neq 3 \right\}.$ The range of $f^{-1}$ is $\left\{ y \middle| y \neq 2 \right\}.$
1.25
1.26
The domain of $f^{-1}$ is $\left( {0,\infty} \right).$ The range of $f^{-1}$ is $(\text{−}\infty,0).$ The inverse function is given by the formula $f^{-1}(x) = -1\text{/}\sqrt{x}.$
1.27
$f(4) = 900;f(10) = 24,300.$
1.28
$x\text{/}\left( {2y^{3}} \right)$
1.29
$A(t) = 750e^{0.04t}.$ After $30$ years, there will be approximately $\text{\$}2,490.09.$
1.30
$x = \frac{\text{ln}\mspace{2mu} 3}{2}$
1.31
$x = \frac{1}{e}$
1.32
$1.29248$
1.33
The magnitude $8.4$ earthquake is roughly $10$ times as severe as the magnitude $7.4$ earthquake.
1.34
$(x^{2} + x^{-2})\text{/}2$
1.35
$\frac{1}{2}\text{ln}(3) \approx 0.5493.$
Section 1.1 Exercises
1.
a\. Domain = $\left\{ {-3,-2,-1,0,1,2,3} \right\},$ range = $\left\{ {0,1,4,9} \right\}$ b. Yes, a function
3.
a\. Domain = $\left\{ {0,1,2,3} \right\},$ range = $\left\{ {-3,-2,-1,0,1,2,3} \right\}$ b. No, not a function
5.
a\. Domain = $\left\{ {3,5,8,10,15,21,33} \right\},$ range = $\left\{ {0,1,2,3} \right\}$ b. Yes, a function
7.
a\. $-2$ b. 3 c. 13 d. $-5x - 2$ e. $5a - 2$ f. $5a + 5h - 2$
9.
a\. Undefined b. 2 c. $\frac{2}{3}$ d. $- \frac{2}{x}$ e $\frac{2}{a}$ f. $\frac{2}{a + h}$
11.
a\. $\sqrt{5}$ b. $\sqrt{11}$ c. $\sqrt{23}$ d. $\sqrt{-6x + 5}$ e. $\sqrt{6a + 5}$ f. $\sqrt{6a + 6h + 5}$
13.
a\. 9 b. 9 c. 9 d. 9 e. 9 f. 9
15.
$x \geq \frac{1}{8};y \geq 0;x = \frac{1}{8};$ no *y*-intercept
17.
$x \geq -2;y \geq -1;x = -1;y = -1 + \sqrt{2}$
19.
$x \neq 4;y \neq 0;$ no *x*-intercept; $y = - \frac{3}{4}$
21.
$x > 5;y > 0;$ no intercepts
23.
25.
27.
29.
Function; a. Domain: all real numbers, range: $y \geq 0$ b. $x = \pm 1$ c. $y = 1$ d. $-1 < x < 0$ and $1 < x < \infty$ e. $\text{−}\infty < x < - 1$ and $0 < x < 1$ f. Not constant g. *y*-axis h. Even
31.
Function; a. Domain: all real numbers, range: $-1.5 \leq y \leq 1.5$ b. $x = 0$ c. $y = 0$ d. $\text{all real numbers}$ e. None f. Not constant g. Origin h. Odd
33.
Function; a. Domain: $\text{−}\infty < x < \infty,$ range: $-2 \leq y \leq 2$ b. $x = 0$ c. $y = 0$ d. $-2 < x < 2$ e. Not decreasing f. $\text{−}\infty < x < - 2$ and $2 < x < \infty$ g. Origin h. Odd
35.
Function; a. Domain: $-4 \leq x \leq 4,$ range: $-4 \leq y \leq 4$ b. $x = 1.2$ c. $y = 4$ d. Not increasing e. $0 < x < 4$ f. $-4 < x < 0$ g. No Symmetry h. Neither
37.
a\. $5x^{2} + x - 8;$ all real numbers b. $-5x^{2} + x - 8;$ all real numbers c. $5x^{3} - 40x^{2};$ all real numbers d. $\frac{x - 8}{5x^{2}};\ x \neq 0$
39.
a\. $-2x + 6;$ all real numbers b. $-2x^{2} + 2x + 12;$ all real numbers c. $\text{−}x^{4} + 2x^{3} + 12x^{2} - 18x - 27;$ all real numbers d. $- \frac{x + 3}{x + 1};\ x \neq \text{−}1,3$
41.
a\. $6 + \frac{2}{x};\ x \neq 0$ b. 6; $x \neq 0$ c. $\frac{6}{x} + \frac{1}{x^{2}};\ x \neq 0$ d. $6x + 1;\ x \neq 0$
43.
a\. $4x + 3;$ all real numbers b. $4x + 15;$ all real numbers
45.
a\. $x^{4} - 6x^{2} + 16;$ all real numbers b. $x^{4} + 14x^{2} + 46;$ all real numbers
47.
a\. $\frac{3x}{4 + x};x \neq 0,-4$ b. $\frac{4x + 2}{3};\ x \neq - \frac{1}{2}$
49.
a\. Yes, because there is only one winner for each year. b. No, because there are three teams that won more than once during the years 2001 to 2012.
51.
a\. $V(s) = s^{3}$ b. $V(11.8) \approx 1643;$ a cube of side length 11.8 each has a volume of approximately 1643 cubic units.
53.
a\. $N(x) = 15x$ b. i. $N(20) = 15(20) = 300;$ therefore, the vehicle can travel 300 mi on a full tank of gas. Ii. $N(15) = 225;$ therefore, the vehicle can travel 225 mi on 3/4 of a tank of gas. c. Domain: $0 \leq x \leq 20;$ range: $\lbrack 0,300\rbrack$ d. The driver had to stop at least once, given that it takes approximately 39 gal of gas to drive a total of 578 mi.
55.
a\. $A(t) = A\left( {r(t)} \right) = \pi \cdot \left( {6 - \frac{5}{t^{2} + 1}} \right)^{2}$ b. Exact: $\frac{121\pi}{4};$ approximately 95 cm2 c. $C(t) = C\left( {r(t)} \right) = 2\pi\left( {6 - \frac{5}{t^{2} + 1}} \right)$ d. Exact: $11\pi;$ approximately 35 cm
57.
a\. $S(x) = 8.5x + 750$ b. \$962.50, \$1090, \$1217.50 c. 77 skateboards
Section 1.2 Exercises
59.
a\. −1 b. Decreasing
61.
a\. 3/4 b. Increasing
63.
a\. 4/3 b. Increasing
65.
a\. 0 b. Horizontal
67.
$y = -6x + 9$
69.
$y = \frac{1}{3}x + 4$
71.
$y = \frac{1}{2}x$
73.
$y = \frac{3}{5}x - 3$
75.
a\. $\left( {m = 2,b = -3} \right)$ b.
77.
a\. $\left( {m = -6,b = 0} \right)$ b.
79.
a\. $\left( {m = 0,b = -6} \right)$ b.
81.
a\. $\left( {m = - \frac{2}{3},b = 2} \right)$ b.
83.
a\. 2 b. $\frac{5}{2},-1;$ c. −5 d. As $\left. x\rightarrow \pm \infty,y\rightarrow\infty \right.$ e. Neither
85.
a\. 2 b. $\pm \sqrt{2}$ c. −1 d. As $\left. x\rightarrow \pm \infty,y\rightarrow\infty \right.$ e. Even
87.
a\. 3 b. 0, $\pm \sqrt{3}$ c. 0 d. As $\left. x\rightarrow - \infty,y\rightarrow\infty \right.$ and $\left. x\rightarrow\infty,y\rightarrow - \infty \right.$ e. Odd
89.
91.
93.
95.
a\. $13,-3,5$ b.
97.
a\. $\frac{-3}{2},\frac{-1}{2},4$ b.
99.
True; $n = 3$
101.
False; $f(x) = x^{b},$ where $b$ is a real-valued constant, is a power function
103.
a\. $V(t) = -2733t + 20500$ b. $\left( {0,20,500} \right)$ means that the initial purchase price of the equipment is \$20,500; $\left( {7.5,0} \right)$ means that in 7.5 years the computer equipment has no value. c. \$6835 d. In approximately 6.4 years
105.
a\. $C = 0.75x + 125$ b. \$245 c. 167 cupcakes
107.
a\. $V(t) = -1500t + 26,000$ b. In 4 years, the value of the car is \$20,000.
109.
\$30,337.50
111.
96% of the total capacity
Section 1.3 Exercises
113.
$\frac{4\pi}{3}\ \text{rad}$
115.
$\frac{\text{−}\pi}{3}$
117.
$\frac{11\pi}{6}\ \text{rad}$
119.
$210\text{°}$
121.
$-540\text{°}$
123.
$-0.5$
125.
$- \frac{\sqrt{2}}{2}$
127.
$\frac{\sqrt{3} - 1}{2\sqrt{2}}$
129.
a\. $b = 5.7$ b. $\text{sin}\mspace{2mu} A = \frac{4}{7},\text{cos}\mspace{2mu} A = \frac{5.7}{7},\text{tan}\mspace{2mu} A = \frac{4}{5.7},\text{csc}\mspace{2mu} A = \frac{7}{4},\text{sec}\mspace{2mu} A = \frac{7}{5.7},\text{cot}\mspace{2mu} A = \frac{5.7}{4}$
131.
a\. $c = 151.7$ b. $\text{sin}\mspace{2mu} A = 0.5623,\text{cos}\mspace{2mu} A = 0.8273,\text{tan}\mspace{2mu} A = 0.6797,\text{csc}\mspace{2mu} A = 1.778,\text{sec}\mspace{2mu} A = 1.209,\text{cot}\mspace{2mu} A = 1.471$
133.
a\. $c = 85$ b. $\text{sin}\mspace{2mu} A = \frac{84}{85},\text{cos}\mspace{2mu} A = \frac{13}{85},\text{tan}\mspace{2mu} A = \frac{84}{13},\text{csc}\mspace{2mu} A = \frac{85}{84},\text{sec}\mspace{2mu} A = \frac{85}{13},\text{cot}\mspace{2mu} A = \frac{13}{84}$
135.
a\. $y = \frac{24}{25}$ b. $\text{sin}\mspace{2mu}\theta = \frac{24}{25},\text{cos}\mspace{2mu}\theta = \frac{7}{25},\text{tan}\mspace{2mu}\theta = \frac{24}{7},\text{csc}\mspace{2mu}\theta = \frac{25}{24},\text{sec}\mspace{2mu}\theta = \frac{25}{7},\text{cot}\mspace{2mu}\theta = \frac{7}{24}$
137.
a\. $x = \frac{\text{−}\sqrt{2}}{3}$ b. $\text{sin}\mspace{2mu}\theta = \frac{\sqrt{7}}{3},\text{cos}\mspace{2mu}\theta = \frac{\text{−}\sqrt{2}}{3},\text{tan}\mspace{2mu}\theta = \frac{\text{−}\sqrt{14}}{2},\text{csc}\mspace{2mu}\theta = \frac{3\sqrt{7}}{7},\text{sec}\mspace{2mu}\theta = \frac{-3\sqrt{2}}{2},\text{cot}\mspace{2mu}\theta = \frac{\text{−}\sqrt{14}}{7}$
139.
$\text{sec}^{2}x$
141.
$\text{sin}^{2}x$
143.
$\text{sec}^{2}\theta$
145.
$\frac{1}{\text{sin}\mspace{2mu} t}\left( = \text{csc}\mspace{2mu} t \right)$
155.
$\left\{ {\frac{\pi}{6},\frac{5\pi}{6}} \right\}$
157.
$\left\{ {\frac{\pi}{4},\frac{3\pi}{4},\frac{5\pi}{4},\frac{7\pi}{4}} \right\}$
159.
$\left\{ {\frac{2\pi}{3},\frac{5\pi}{3}} \right\}$
161.
$\left\{ {0,\pi,\frac{\pi}{3},\frac{5\pi}{3}} \right\}$
163.
$y = 4\mspace{2mu}\text{sin}\left( {\frac{\pi}{4}x} \right)$
165.
$y = \text{cos}\left( {2\pi x} \right)$
167.
a\. 1 b. $2\pi$ c. $\frac{\pi}{4}$ units to the right
169.
a\. $\frac{1}{2}$ b. $8\pi$ c. No phase shift
171.
a\. 3 b. $2$ c. $\frac{2}{\pi}$ units to the left
173.
Approximately 42 in.
175.
a\. 0.550 rad/sec b. 0.236 rad/sec c. 0.698 rad/min d. 1.697 rad/min
177.
$\approx 30.9{\ \text{in}}^{2}$
179.
a\. π/184; the voltage repeats every π/184 sec b. Approximately 59 periods
181.
a\. Amplitude = $10;\text{period} = \ 24$ b. $47.4\ \text{°}F$ c. 14 hours later, or 2 p.m. d.
Section 1.4 Exercises
183.
Not one-to-one
185.
Not one-to-one
187.
One-to-one
189.
a\. $f^{-1}(x) = \sqrt{x + 4}$ b. Domain $\text{:}\ x \geq -4,\text{range}\text{:}\ y \geq 0$
191.
a\. $f^{-1}(x) = \sqrt[3]{x - 1}$ b. Domain: all real numbers, range: all real numbers
193.
a\. $f^{-1}(x) = x^{2} + 1,$ b. Domain: $x \geq 0,$ range: $y \geq 1$
195.
197.
199.
These are inverses.
201.
These are not inverses.
203.
These are inverses.
205.
These are inverses.
207.
$\frac{\pi}{6}$
209.
$\frac{\pi}{4}$
211.
$\frac{\pi}{6}$
213.
$\frac{\sqrt{2}}{2}$
215.
$- \frac{\pi}{6}$
217.
a\. $x = f^{-1}(V) = \sqrt{0.04 - \frac{V}{500}}$ b. The inverse function determines the distance from the center of the artery at which blood is flowing with velocity *V*. c. 0.1 cm; 0.14 cm; 0.17 cm
219.
a\. \$31,250, \$66,667, \$107,143 b. $\left( {p = \frac{85C}{C + 75}} \right)$ c. 34 ppb
221.
a\. $\sim 92\text{°}$ b. $\sim 42\text{°}$ c. $\sim 27\text{°}$
223.
$x \approx 6.69,8.51;$ so, the temperature occurs on June 21 and August 15
225.
$\sim 1.5\ \text{sec}$
227.
$\text{tan}^{-1}\left( {\text{tan}(2.1)} \right) \approx - 1.0416;$ the expression does not equal 2.1 since $2.1 > 1.57 = \frac{\pi}{2}$—in other words, it is not in the restricted domain of $\text{tan}\mspace{2mu} x.\ \text{cos}^{-1}\left( {\text{cos}(2.1)} \right) = 2.1,$ since 2.1 is in the restricted domain of $\text{cos}\mspace{2mu} x.$
Section 1.5 Exercises
229.
a\. 125 b. 2.24 c. 9.74
231.
a\. 0.01 b. 10,000 c. 46.42
233.
d
235.
b
237.
e
239.
Domain: all real numbers, range: $\left( {2,\infty} \right),y = 2$
241.
Domain: all real numbers, range: $\left( {0,\infty} \right),y = 0$
243.
Domain: all real numbers, range: $\left( {\text{−}\infty,1} \right),y = 1$
245.
Domain: all real numbers, range: $\left( {-1,\infty} \right),y = -1$
247.
$8^{1\text{/}3} = 2$
249.
$5^{2} = 25$
251.
$e^{-3} = \frac{1}{e^{3}}$
253.
$e^{0} = 1$
255.
$\text{log}_{4}\left( \frac{1}{16} \right) = -2$
257.
$\text{log}_{9}1 = 0$
259.
$\text{log}_{64}4 = \frac{1}{3}$
261.
$\text{log}_{9}150 = y$
263.
$\text{log}_{4}0.125 = - \frac{3}{2}$
265.
Domain: $(1,\infty),$ range: $\left( {\text{−}\infty,\infty} \right),x = 1$
267.
Domain: $\left( {0,\infty} \right),$ range: $\left( {\text{−}\infty,\infty} \right),x = 0$
269.
Domain: $\left( {-1,\infty} \right),$ range: $\left( {\text{−}\infty,\infty} \right),x = -1$
271.
$2 + 3\text{log}_{3}a - \text{log}_{3}b$
273.
$\frac{3}{2} + \frac{1}{2}\text{log}_{5}x + \frac{3}{2}\text{log}_{5}y$
275.
$- \frac{3}{2} + \text{ln}\mspace{2mu} 6$
277.
$\frac{\text{ln}\mspace{2mu} 15}{3}$
279.
$\frac{3}{2}$
281.
$\text{log}\mspace{2mu} 7.21$
283.
$\frac{2}{3} + \frac{\text{log}\mspace{2mu} 11}{3\mspace{2mu}\text{log}\mspace{2mu} 7}$
285.
$x = \frac{1}{25}$
287.
$x = 4$
289.
$x = 3$
291.
$1 + \sqrt{5}$
293.
$\left( {\frac{\text{log}\mspace{2mu} 82}{\text{log}\mspace{2mu} 7} \approx 2.2646} \right)$
295.
$\left( {\frac{\text{log}\mspace{2mu} 211}{\text{log}\mspace{2mu} 0.5} \approx - 7.7211} \right)$
297.
$\left( {\frac{\text{log}\mspace{2mu} 0.452}{\text{log}\mspace{2mu} 0.2} \approx 0.4934} \right)$
299.
$\sim 17,491$
301.
Approximately \$131,653 is accumulated in 5 years.
303.
i\. a. pH = 8 b. Base ii. a. pH = 3 b. Acid iii. a. pH = 4 b. Acid
305.
a\. $\sim 333$ million b. 94 years from 2013, or in 2107
307.
a\. $k \approx 0.0578$ b. $\approx 92$ hours
309.
The San Francisco earthquake was $10^{3.4}\text{or}\ \approx 2512$ times more intense than the Japanese earthquake.
Review Exercises
311.
False
313.
False
315.
Domain: $x > 5,$ range: all real numbers
317.
Domain: $x > 2$ and $x < - 4,$ range: all real numbers
319.
Degree of 3, $y$-intercept: 0, zeros: 0, $\sqrt{3} - 1,-1 - \sqrt{3}$
321.
$\begin{array}{l}
{\cos^{2}x - \sin^{2}x = \cos 2x} \\
{= 1 - 2\sin^{2}x} \\
{= 2\cos^{2}x - 1}
\end{array}$
323.
$0, \pm 2\pi$
325.
4
327.
One-to-one; yes, the function has an inverse; inverse: $f^{-1}(x) = \frac{1}{y}$
329.
$x \geq - \frac{3}{2},f^{-1}(x) = - \frac{3}{2} + \frac{1}{2}\sqrt{4y - 7}$
331.
a\. $C(x) = 300 + 7x$ b. 100 shirts
333.
The population is less than 20,000 from December 8 through January 23 and more than 140,000 from May 29 through August 2
335.
78.51%
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