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  • Calculus for AP
  • PRECALCULUS REVIEW

Calculus for AP

Jon Rogawski & Ray Cannon

Chapter 1

PRECALCULUS REVIEW - all with Video Answers

Educators

+ 14 more educators

Section 1

Real Numbers, Functions, and Graphs

00:51

Problem 1

Use a calculator to find a rational number $r$ such that $\left|r-\pi^{2}\right|<10^{-4}$

Linh V.
Linh V.
Numerade Educator
04:23

Problem 2

Which of $(a)-(f)$ are true for $a=-3$ and $b=2 ?$

$$
\begin{array}{ll}{\text { (a) } a < b} & {\text { (b) }|a | < |b|} & {\text { (c) } a b > 0} \\ {\text { (d) } 3 a < 3 b} & {(\text { e) }-4 a<-4 b} & {\text { (f) } \frac{1}{a} < \frac{1}{b}}\end{array}
$$

EM
Evalyn M.
Numerade Educator
02:51

Problem 3

express the interval in terms of an inequality involving absolute value.

$$
[-2,2]
$$

SY
Srikar Y.
Numerade Educator
03:06

Problem 4

express the interval in terms of an inequality involving absolute value.

$$
(-4,4)
$$

Bahar T.
Bahar T.
Numerade Educator
10:16

Problem 5

express the interval in terms of an inequality involving absolute value.

$$
(0,4)
$$

Elyse G.
Elyse G.
Numerade Educator
04:44

Problem 6

express the interval in terms of an inequality involving absolute value.

$$
[-4,0]
$$

Milad G.
Milad G.
Numerade Educator
02:19

Problem 7

express the interval in terms of an inequality involving absolute value.

$$
[1,5]
$$

Jade M.
Jade M.
Numerade Educator
01:05

Problem 8

express the interval in terms of an inequality involving absolute value.

$$
(-2,8)
$$

Aymara G.
Aymara G.
Numerade Educator
01:41

Problem 9

write the inequality in the form $a < x < b$

$$
|x|<8
$$

Hasan S.
Hasan S.
Numerade Educator
04:12

Problem 10

write the inequality in the form $a < x < b$

$$
|x-12|<8
$$

Yogesh S.
Yogesh S.
Numerade Educator
00:46

Problem 11

write the inequality in the form $a < x < b$

$$
|2 x+1|<5
$$

Linh V.
Linh V.
Numerade Educator
01:21

Problem 12

write the inequality in the form $a < x < b$

$$
|3 x-4|<2
$$

PS
Pankhuri S.
Numerade Educator
00:32

Problem 13

express the set of numbers $x$ satisfying the given condition as an interval.

$$|x|<4$$

Linh V.
Linh V.
Numerade Educator
00:44

Problem 14

express the set of numbers $x$ satisfying the given condition as an interval.

$$|x| \leq 9$$

Aymara G.
Aymara G.
Numerade Educator
02:38

Problem 15

express the set of numbers $x$ satisfying the given condition as an interval.

$$
|x-4|<2
$$

Donitaa J.
Donitaa J.
Numerade Educator
00:32

Problem 16

express the set of numbers $x$ satisfying the given condition as an interval.

$$
|x+7|<2
$$

Aymara G.
Aymara G.
Numerade Educator
02:42

Problem 17

express the set of numbers $x$ satisfying the given condition as an interval.

$$
|4 x-1| \leq 8
$$

Dishary H.
Dishary H.
Numerade Educator
03:28

Problem 18

express the set of numbers $x$ satisfying the given condition as an interval.

$$
|3 x+5|<1
$$

Michelle L.
Michelle L.
Numerade Educator
04:49

Problem 19

describe the set as a union of finite or infinite intervals.

$$
\{x :|x-4|>2\}
$$

Benjamin M.
Benjamin M.
Numerade Educator
03:06

Problem 20

describe the set as a union of finite or infinite intervals.

$$
\{x :|2 x+4|>3\}
$$

GC
Gabi C.
Numerade Educator
01:10

Problem 21

describe the set as a union of finite or infinite intervals.

$$
\left\{x :\left|x^{2}-1\right|>2\right\}
$$

Carson M.
Carson M.
Numerade Educator
05:15

Problem 22

describe the set as a union of finite or infinite intervals.

$$
\left\{x :\left|x^{2}+2 x\right|>2\right\}
$$

Aymara G.
Aymara G.
Numerade Educator
02:46

Problem 23

describe the set as a union of finite or infinite intervals.

Match $(a)-(f)$ with $(1)-(v i)$
$$
\begin{array}{ll}{\text { (a) } a>3} & {\text { (b) }|a-5|<\frac{1}{3}} \\ {\text { (c) }\left|a-\frac{1}{3}\right|<5} & {\text { (d) }|a|>5} \\ {(\text { e) }|a-4|<3} & {\text { (f) } 1 \leq a \leq 5}\end{array} $$ $$
\begin{array}{l}{\text { (i) } a \text { lies to the right of } 3} \\ {\text { (ii) } a \text { lies between } 1 \text { and } 7} \\ {\text { (iii) The distance from } a \text { to } 5 \text { is less than } \frac{1}{3}}\end{array} $$ $$
\begin{array}{l}{\text { (iv) The distance from } a \text { to } 3 \text { is at most } 2 .} \\ {\text { (v) } a \text { is less than } 5 \text { units from } \frac{1}{3} \text { . }} \\ {\text { (vi) } a \text { lies either to the left of }-5 \text { or to the right of } 5 .}\end{array}
$$

Linh V.
Linh V.
Numerade Educator
01:21

Problem 24

Describe $\left\{x : \frac{x}{x+1}<0\right\}$ as an interval.

Aymara G.
Aymara G.
Numerade Educator
06:01

Problem 25

Describe $\left\{x : x^{2}+2 x<3\right\}$ as an interval. Hint: Plot $y=x^{2}+$ $2 x-3$

Justin L.
Justin L.
Numerade Educator
04:09

Problem 26

Describe the set of real numbers satisfying $|x-3|=|x-2|+1$ as a half-infinite interval.

Aymara G.
Aymara G.
Numerade Educator
01:06

Problem 27

Show that if $a>b,$ then $b^{-1}>a^{-1},$ provided that $a$ and $b$ have the same sign. What happens if $a>0$ and $b<0 ?$

Linh V.
Linh V.
Numerade Educator
01:31

Problem 28

Which $x$ satisfy both $|x-3|<2$ and $|x-5|<1 ?$

Aymara G.
Aymara G.
Numerade Educator
00:43

Problem 29

Show that if $|a-5|<\frac{1}{2}$ and $|b-8|<\frac{1}{2},$ then $|(a+b)-13|<1 .$ Hint: Use the triangle inequality.

Linh V.
Linh V.
Numerade Educator
02:39

Problem 30

Suppose that $|x-4|<1$
$$
\begin{array}{l}{\text { (a) What is the maximum possible value of }|x+4| ?} \\ {\text { (b) Show that }\left|x^{2}-16\right| \leq 9}\end{array}
$$

Aymara G.
Aymara G.
Numerade Educator
00:49

Problem 31

Suppose that $|a-6| \leq 2$ and $|b| \leq 3$
(a) What is the largest possible value of $|a+b| ?$
(b) What is the smallest possible value of $|a+b| ?$

Linh V.
Linh V.
Numerade Educator
01:02

Problem 32

Prove that $|x|-|y| \leq|x-y| .$ Hint: Apply the triangle inequality to $y$ and $x-y$

Aymara G.
Aymara G.
Numerade Educator
03:22

Problem 33

Express $r_{1}=0 . \overline{27}$ as a fraction. Hint: $100 r_{1}-r_{1}$ is an integer. Then express $r_{2}=0.2666 \ldots$ as a fraction.

Linh V.
Linh V.
Numerade Educator
01:00

Problem 34

Represent 1$/ 7$ and 4$/ 27$ as repeating decimals.

Aymara G.
Aymara G.
Numerade Educator
00:30

Problem 35

The text states: If the decimal expansions of numbers a and agree to k places, then $|a-b| \leq 10^{-k}$ . Show that the converse is false: For all $k$ there are numbers $a$ and $b$ whose decimal expansions do not agree at all but $|a-b| \leq 10^{-k} .$

Linh V.
Linh V.
Numerade Educator
03:29

Problem 36

Plot each pair of points and compute the distance between them:

$$
\begin{array}{ll}{\text { (a) }(1,4) \text { and }(3,2)} & {\text { (b) }(2,1) \text { and }(2,4)} \\ {\text { (c) }(0,0) \text { and }(-2,3)} & {\text { (d) }(-3,-3) \text { and }(-2,3)}\end{array}
$$

Aymara G.
Aymara G.
Numerade Educator
01:12

Problem 37

Find the equation of the circle with center $(2,4) :$

$$
\begin{array}{l}{\text { (a) with radius } r=3 \text { . }} \\ {\text { (b) that passes through }(1,-1) \text { . }}\end{array}
$$

Linh V.
Linh V.
Numerade Educator
01:31

Problem 38

Find all points with integer coordinates located at a distance 5 from the origin. Then find all points with integer coordinates located at a distance 5 from $(2,3) .$

Aymara G.
Aymara G.
Numerade Educator
00:19

Problem 39

Determine the domain and range of the function $$ f :\{r, s, t, u\} \rightarrow\{A, B, C, D, E\} $$ defined by $f(r)=A, f(s)=B, f(t)=B, f(u)=E$

Linh V.
Linh V.
Numerade Educator
01:03

Problem 40

Give an example of a function whose domain $D$ has three elements and whose range $R$ has two elements. Does a function exist whose domain $D$ has two elements and whose range $R$ has three elements?

Aymara G.
Aymara G.
Numerade Educator
00:22

Problem 41

find the domain and range of the function.

$$
f(x)=-x
$$

Linh V.
Linh V.
Numerade Educator
01:08

Problem 42

find the domain and range of the function.

$$
g(t)=t^{4}
$$

Aymara G.
Aymara G.
Numerade Educator
00:20

Problem 43

find the domain and range of the function.

$$
f(x)=x^{3}
$$

Linh V.
Linh V.
Numerade Educator
01:15

Problem 44

find the domain and range of the function.

$$
g(t)=\sqrt{2-t}
$$

Aymara G.
Aymara G.
Numerade Educator
00:30

Problem 45

find the domain and range of the function.

$$
f(x)=|x|
$$

Linh V.
Linh V.
Numerade Educator
01:16

Problem 46

find the domain and range of the function.

$$
h(s)=\frac{1}{s}
$$

Aymara G.
Aymara G.
Numerade Educator
00:36

Problem 47

find the domain and range of the function.

$$
f(x)=\frac{1}{x^{2}}
$$

Linh V.
Linh V.
Numerade Educator
01:49

Problem 48

find the domain and range of the function.

$$
g(t)=\cos \frac{1}{t}
$$

Aymara G.
Aymara G.
Numerade Educator
00:18

Problem 49

determine where $f(x)$ is increasing.

$$f(x)=|x+1|$$

Linh V.
Linh V.
Numerade Educator
00:39

Problem 50

determine where $f(x)$ is increasing.

$$
f(x)=x^{3}
$$

Aymara G.
Aymara G.
Numerade Educator
00:20

Problem 51

determine where $f(x)$ is increasing.

$$
f(x)=x^{4}
$$

Linh V.
Linh V.
Numerade Educator
02:20

Problem 52

determine where $f(x)$ is increasing.

$$
f(x)=\frac{1}{x^{4}+x^{2}+1}
$$

Aymara G.
Aymara G.
Numerade Educator
01:08

Problem 53

find the zeros of $f(x)$ and sketch its graph by plotting points. Use symmetry and increase/decrease information where appropriate.

$$
f(x)=x^{2}-4
$$

Linh V.
Linh V.
Numerade Educator
02:11

Problem 54

find the zeros of $f(x)$ and sketch its graph by plotting points. Use symmetry and increase/decrease information where appropriate.

$$
f(x)=2 x^{2}-4
$$

Aymara G.
Aymara G.
Numerade Educator
02:50

Problem 55

find the zeros of $f(x)$ and sketch its graph by plotting points. Use symmetry and increase/decrease information where appropriate.

$$
f(x)=x^{3}-4 x
$$

Linh V.
Linh V.
Numerade Educator
01:32

Problem 56

find the zeros of $f(x)$ and sketch its graph by plotting points. Use symmetry and increase/decrease information where appropriate.

$$
f(x)=x^{3}
$$

Aymara G.
Aymara G.
Numerade Educator
02:11

Problem 57

find the zeros of $f(x)$ and sketch its graph by plotting points. Use symmetry and increase/decrease information where appropriate.

$$
f(x)=2-x^{3}
$$

Linh V.
Linh V.
Numerade Educator
02:02

Problem 58

find the zeros of $f(x)$ and sketch its graph by plotting points. Use symmetry and increase/decrease information where appropriate.

$$
f(x)=\frac{1}{(x-1)^{2}+1}
$$

Aymara G.
Aymara G.
Numerade Educator
00:53

Problem 59

Which of the curves in Figure 26 is the graph of a function?

Linh V.
Linh V.
Numerade Educator
02:19

Problem 60

Determine whether the function is even, odd, or neither.

$$
\begin{array}{ll}{\text { (a) } f(x)=x^{5}} & {\text { (b) } g(t)=t^{3}-t^{2}} \\ {\text { (c) } F(t)=\frac{1}{t^{4}+t^{2}}}\end{array}
$$

Aymara G.
Aymara G.
Numerade Educator
02:07

Problem 61

Determine whether the function is even, odd, or neither.

$$
\begin{array}{l}{\text { (a) } f(t)=\frac{1}{t^{4}+t+1}-\frac{1}{t^{4}-t+1} \text { (b) } g(t)=2^{t}-2^{-t}} \\ {\text { (c) } G(\theta)=\sin \theta+\cos \theta \quad \text { (d) } H(\theta)=\sin \left(\theta^{2}\right)}\end{array}
$$

Linh V.
Linh V.
Numerade Educator
01:47

Problem 62

Write $f(x)=2 x^{4}-5 x^{3}+12 x^{2}-3 x+4$ as the sum of an even and an odd function.

Aymara G.
Aymara G.
Numerade Educator
00:44

Problem 63

Show that $f(x)=\ln \left(\frac{1-x}{1+x}\right)$ is an odd function.

Linh V.
Linh V.
Numerade Educator
01:45

Problem 64

State whether the function is increasing, decreasing, or neither.
$$
\begin{array}{l}{\text { (a) Surface area of a sphere as a function of its radius }} \\ {\text { (b) Temperature at a point on the equator as a function of time }} \\ {\text { (c) Price of an airline ticket as a function of the price of oil }} \\ {\text { (d) Pressure of the gas in a piston as a function of volume }}\end{array}
$$

Aymara G.
Aymara G.
Numerade Educator
00:35

Problem 65

let $f(x)$ be the function shown in Figure 27

Find the domain and range of $f(x) ?$

Linh V.
Linh V.
Numerade Educator
02:17

Problem 66

let $f(x)$ be the function shown in Figure 27.

Sketch the graphs of $f(x+2)$ and $f(x)+2$

Aymara G.
Aymara G.
Numerade Educator
02:42

Problem 67

let $f(x)$ be the function shown in Figure 27.

Sketch the graphs of $f(2 x), f\left(\frac{1}{2} x\right),$ and 2$f(x)$

Linh V.
Linh V.
Numerade Educator
01:17

Problem 68

let $f(x)$ be the function shown in Figure 27.

Sketch the graphs of $f(-x)$ and $-f(-x)$

Aymara G.
Aymara G.
Numerade Educator
00:41

Problem 69

let $f(x)$ be the function shown in Figure 27.

Extend the graph of $f(x)$ to $[-4,4]$ so that it is an even function.

Linh V.
Linh V.
Numerade Educator
00:51

Problem 70

let $f(x)$ be the function shown in Figure 27.

Extend the graph of $f(x)$ to $[-4,4]$ so that it is an odd function.

Aymara G.
Aymara G.
Numerade Educator
03:17

Problem 71

Suppose that $f(x)$ has domain $[4,8]$ and range $[2,6] .$ Find the domain and range of:

$$
\begin{array}{ll}{\text { (a) } f(x)+3} & {\text { (b) } f(x+3)} \\ {\text { (c) } f(3 x)} & {\text { (d) } 3 f(x)}\end{array}
$$

Linh V.
Linh V.
Numerade Educator
04:08

Problem 72

Let $f(x)=x^{2} .$ Sketch the graph over $[-2,2]$ of:

$$
\begin{array}{ll}{\text { (a) } f(x+1)} & {\text { (b) } f(x)+1} \\ {\text { (c) } f(5 x)} & {\text { (d) } 5 f(x)}\end{array}
$$

Aymara G.
Aymara G.
Numerade Educator
01:22

Problem 73

Suppose that the graph of $f(x)=\sin x$ is compressed horizontally by a factor of 2 and then shifted 5 units to the right.

$$
\begin{array}{l}{\text { (a) What is the equation for the new graph? }} \\ {\text { (b) What is the equation if you first shift by } 5 \text { and then compress by } 2 ?} \\ {\text { (c) } \mathrm{GU} \text { Verify your answers by plotting your equations. }}\end{array}
$$

Linh V.
Linh V.
Numerade Educator
02:31

Problem 74

Figure 28 shows the graph of $f(x)=|x|+1 .$ Match the functions $(\mathrm{a})-(\mathrm{e})$ with their graphs $(\mathrm{i})-(\mathrm{v})$.

$$
\begin{array}{ll}{\text { (a) } f(x-1)} & {\text { (b) }-f(x)} \\ {\text { (d) } f(x-1)-2} & {\text { (e) } f(x+1)}\end{array} \quad(\mathrm{c})-f(x)+2
$$

Aymara G.
Aymara G.
Numerade Educator
01:27

Problem 75

Sketch the graph of $f(2 x)$ and $f\left(\frac{1}{2} x\right),$ where $f(x)=|x|+1$ (Figure 28 ).

Linh V.
Linh V.
Numerade Educator
01:08

Problem 76

Find the function $f(x)$ whose graph is obtained by shifting the parabola $y=x^{2}$ three units to the right and four units down, as in Figure $29 .$

Aymara G.
Aymara G.
Numerade Educator
02:03

Problem 77

Define $f(x)$ to be the larger of $x$ and $2-x .$ Sketch the graph of $f(x) .$ What are its domain and range? Express $f(x)$ in terms of the absolute value function.

Linh V.
Linh V.
Numerade Educator
01:44

Problem 78

For each curve in Figure $30,$ state whether it is symmetric with respect to the $y$ -axis, the origin, both, or neither.

Aymara G.
Aymara G.
Numerade Educator
02:07

Problem 79

Show that the sum of two even functions is even and the sum of two odd functions is odd.

Linh V.
Linh V.
Numerade Educator
02:37

Problem 80

Suppose that $f(x)$ and $g(x)$ are both odd. Which of the following functions are even? Which are odd?
$$
(a)f(x) g(x) \quad \text { (b) } f(x)^{3}
$$ $$
(c)f(x)-g(x) \quad \text { (d) } \frac{f(x)}{g(x)}
$$

Aymara G.
Aymara G.
Numerade Educator
00:50

Problem 81

Prove that the only function whose graph is symmetric with respect to both the $y$ -axis and the origin is the function $f(x)=0$

Linh V.
Linh V.
Numerade Educator
00:54

Problem 82

Prove the triangle inequality by adding the two inequalities
$$-|a| \leq a \leq|a|, \quad-|b| \leq b \leq|b|$$

Aymara G.
Aymara G.
Numerade Educator
02:54

Problem 83

Show that a fraction $r=a / b$ in lowest terms has a finite decimal expansion if and only if $$b=2^{n} 5^{m} \quad$ for some $n, m \geq 0$$ Hint: Observe that $r$ has a finite decimal expansion when $10^{N} r$ is an integer for some $N \geq 0$ (and hence $b$ divides $10^{N} )$ .

Linh V.
Linh V.
Numerade Educator
02:12

Problem 84

Let $p=p_{1} \ldots p_{s}$ be an integer with digits $p_{1}, \ldots, p_{s} .$ Show that
$$\frac{p}{10^{s}-1}=0 . \overline{p_{1} \ldots p_{s}}$$ Use this to find the decimal expansion of $r=\frac{2}{11} .$ Note that $r=\frac{2}{11}=\frac{18}{10^{2}-1}$

Aymara G.
Aymara G.
Numerade Educator
01:19

Problem 85

A function $f(x)$ is symmetric with respect to the vertical line $x=a$ if $f(a-x)=f(a+x)$
$$
\begin{array}{l}{\text { (a) Draw the graph of a function that is symmetric with respect to }} \\ {x=2 \text { . }} \\ {\text { (b) Show that if } f(x) \text { is symmetric with respect to } x=a \text { , then } g(x)=} \\ {f(x+a) \text { is even. }}\end{array}
$$

Linh V.
Linh V.
Numerade Educator
01:13

Problem 86

Formulate a condition for $f(x)$ to be symmetric with respect to the point $(a, 0)$ on the $x$ -axis.

Aymara G.
Aymara G.
Numerade Educator

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