# Calculus for AP

## Educators

### Problem 1

In Exercises $1-6,$ use $E q .(1)$ to estimate $\Delta f=f(3.02)-f(3)$
$$f(x)=x^{2}$$

Robert L.

### Problem 2

In Exercises $1-6,$ use $E q .(1)$ to estimate $\Delta f=f(3.02)-f(3)$
$$f(x)=x^{4}$$

Michelle L.

### Problem 3

In Exercises $1-6,$ use $E q .(1)$ to estimate $\Delta f=f(3.02)-f(3)$
$$f(x)=x^{-1}$$

Robert L.

### Problem 4

In Exercises $1-6,$ use $E q .(1)$ to estimate $\Delta f=f(3.02)-f(3)$
$$f(x)=\frac{1}{x+1}$$

Michelle L.

### Problem 5

In Exercises $1-6,$ use $E q .(1)$ to estimate $\Delta f=f(3.02)-f(3)$
$$f(x)=\sqrt{x+6}$$

Robert L.

### Problem 6

In Exercises $1-6,$ use $E q .(1)$ to estimate $\Delta f=f(3.02)-f(3)$
$$f(x)=\tan \frac{\pi x}{3}$$

Michelle L.

### Problem 7

The cube root of 27 is $3 .$ How much larger is the cube root of 27.2$?$
Estimate using the Linear Approximation.

Robert L.

### Problem 8

Estimate $\ln \left(e^{3}+0.1\right)-\ln \left(e^{3}\right)$ using differentials.

Michelle L.

### Problem 9

In Exercises $9-12,$ use $E q .$ (1) to estimate $\Delta f .$ Use a calculator to compute both the error and the percentage error.
$$f(x)=\sqrt{1+x}, \quad a=3, \quad \Delta x=0.2$$

Robert L.

### Problem 10

In Exercises $9-12,$ use $E q .$ (1) to estimate $\Delta f .$ Use a calculator to compute both the error and the percentage error.
$$f(x)=2 x^{2}-x, \quad a=5, \quad \Delta x=-0.4$$

Michelle L.

### Problem 11

In Exercises $9-12,$ use $E q .$ (1) to estimate $\Delta f .$ Use a calculator to compute both the error and the percentage error.
$$f(x)=\frac{1}{1+x^{2}}, \quad a=3, \quad \Delta x=0.5$$

Robert L.

### Problem 12

In Exercises $9-12,$ use $E q .$ (1) to estimate $\Delta f .$ Use a calculator to compute both the error and the percentage error.
$$f(x)=\ln \left(x^{2}+1\right), \quad a=1, \quad \Delta x=0.1$$

Michelle L.

### Problem 13

In Exercises $13-16,$ estimate $\Delta$y using differentials $[E q .(3)]$
$$y=\cos x, \quad a=\frac{\pi}{6}, \quad d x=0.014$$

Robert L.

### Problem 14

In Exercises $13-16,$ estimate $\Delta$y using differentials $[E q .(3)]$
$$y=\tan ^{2} x, \quad a=\frac{\pi}{4}, \quad d x=-0.02$$

Michelle L.

### Problem 15

In Exercises $13-16,$ estimate $\Delta$y using differentials $[E q .(3)]$
$$y=\frac{10-x^{2}}{2+x^{2}}, \quad a=1, \quad d x=0.01$$

Robert L.

### Problem 16

In Exercises $13-16,$ estimate $\Delta$y using differentials $[E q .(3)]$
$$y=x^{1 / 3} e^{x-1}, \quad a=1, \quad d x=0.1$$

Michelle L.

### Problem 17

In Exercises $17-24$ , estimate using the Linear Approximation and find the error using a calculator.
$$\sqrt{26}-\sqrt{25}$$

Robert L.

### Problem 18

In Exercises $17-24$ , estimate using the Linear Approximation and find the error using a calculator.
$$16.5^{1 / 4}-16^{1 / 4}$$

Michelle L.

### Problem 19

In Exercises $17-24$ , estimate using the Linear Approximation and find the error using a calculator.
$$\frac{1}{\sqrt{101}}-\frac{1}{10}$$

Robert L.

### Problem 20

In Exercises $17-24$ , estimate using the Linear Approximation and find the error using a calculator.
$$\frac{1}{\sqrt{98}}-\frac{1}{10}$$

Michelle L.

### Problem 21

In Exercises $17-24$ , estimate using the Linear Approximation and find the error using a calculator.
$$9^{1 / 3}-2$$

Robert L.

### Problem 22

In Exercises $17-24$ , estimate using the Linear Approximation and find the error using a calculator.
$$\tan ^{-1}(1.05)-\frac{\pi}{4}$$

Michelle L.

### Problem 23

In Exercises $17-24$ , estimate using the Linear Approximation and find the error using a calculator.
$$e^{-0.1}-1$$

Robert L.

### Problem 24

In Exercises $17-24$ , estimate using the Linear Approximation and find the error using a calculator.
$$\ln (0.97)$$

Michelle L.

### Problem 25

Estimate $f(4.03)$ for $f(x)$ as in Figure $8 .$

Robert L.

### Problem 26

At a certain moment, an object in linear motion has velocity 100 $\mathrm{m} / \mathrm{s}$ . Estimate the distance traveled over the next quarter-second, and explain how this is an application of the Linear Approximation.

Michelle L.

### Problem 27

Which is larger: $\sqrt{2.1}-\sqrt{2}$ or $\sqrt{9.1}-\sqrt{9} ?$ Explain using the
Linear Approximation.

Robert L.

### Problem 28

Estimate $\sin 61^{\circ}-\sin 60^{\circ}$ using the Linear Approximation. Hint:
Express $\Delta \theta$ in radians.

Michelle L.

### Problem 29

Box office revenue at a multiplex cinema in Paris is $R(p)=$
$3600 p-10 p^{3}$ euros per showing when the ticket price is $p$ euros
Calculate $R(p)$ for $p=9$ and use the Linear Approximation to estit
mate $\Delta R$ if $p$ is raised or lowered by 0.5 euros.

Robert L.

### Problem 30

The stopping distance for an automobile is $F(s)=1.1 s+$
0.054$s^{2}$ ft, where $s$ is the speed in mph. Use the Linear Approximation
to estimate the change in stopping distance per additional mph when
$s=35$ and when $s=55 .$

Michelle L.

### Problem 31

A thin silver wire has length $L=18 \mathrm{cm}$ when the temperature is
$T=30^{\circ} \mathrm{C} .$ Estimate $\Delta L$ when $T$ decreases to $25^{\circ} \mathrm{C}$ if the coefficient of thermal expansion is $k=1.9 \times 10^{-1}($ see Example 3$)$ .

Robert L.

### Problem 32

At a certain moment, the temperature in a snake cage satisfies
$d T / d t=0.008^{\circ} \mathrm{C} / \mathrm{s} .$ Estimate the rise in temperature over the next
10 seconds.

Michelle L.

### Problem 33

The atmospheric pressure at altitude $h$ (kilometers) for $11 \leq h \leq$ 25 is approximately
$$P(h)=128 e^{-0.157 h}$$
$$\begin{array}{l}{\text { (a) Estimate } \Delta P \text { at } h=20 \text { when } \Delta h=0.5 .} \\ {\text { (b) Compute the actual change, and compute the percentage error in }} \\ {\text { the Linear Approximation. }}\end{array}$$

Robert L.

### Problem 34

The resistance $R$ of a copper wire at temperature $T=20^{\circ} \mathrm{C}$
is $R=15 \Omega .$ Estimate the resistance at $T=22^{\circ} \mathrm{C},$ assuming that
$d R / d\left.T\right|_{T=20}=0.06 \Omega^{\circ} \mathrm{C}$

Michelle L.

### Problem 35

Newton's Law of Gravitation shows that if a person weighs $w$ pounds on the surface of the earth, then his or her weight at distance $x$ from the center of the earth is
$$W(x)=\frac{w R^{2}}{x^{2}} \quad(\text { for } x \geq R)$$
where $R=3,960$ miles is the radius of the earth (Figure 9$)$
\begin{array}{l}{\text { (a) Show that the weight lost at altitude } h \text { miles above the earth's }} \\ {\text { surface is approximately } \Delta W \approx-(0.0005 w) h . \text { Hint: Use the Linear }} \\ {\text { Approximation with } d x=h \text { . }} \\ {\text { (b) Estimate the weight lost by a } 200-\text { -lb football player flying in a jet }} \\ {\text { at an altitude of } 7 \text { miles. }}\end{array}

Robert L.

### Problem 36

Using Exercise $35(\mathrm{a}),$ estimate the altitude at which a $130-\mathrm{lb}$ pilot
would weigh 129.5 $\mathrm{lb}$ .

Michelle L.

### Problem 37

A stone tossed vertically into the air with initial velocity $v \mathrm{cm} / \mathrm{s}$
reaches a maximum height of $h=v^{2} / 1960 \mathrm{cm} .$
$$\begin{array}{l}{\text { (a) Estimate } \Delta h \text { if } v=700 \mathrm{cm} / \mathrm{s} \text { and } \Delta v=1 \mathrm{cm} / \mathrm{s} \text { . }} \\ {\text { (b) Estimate } \Delta h \text { if } v=1,000 \mathrm{cm} / \mathrm{s} \text { and } \Delta v=1 \mathrm{cm} / \mathrm{s} \text { . }} \\ {\text { (c) In general, does a } 1 \mathrm{cm} / \mathrm{s} \text { increase in } v \text { lead to a greater change in }} \\ {h \text { at low or high initial velocities? Explain. }}\end{array}$$

Robert L.

### Problem 38

The side $s$ of a square carpet is measured at 6 $\mathrm{m} .$ Estimate the
maximum error in the area $A$ of the carpet if $s$ is accurate to within
2 centimeters.

Michelle L.

### Problem 39

In Exercises 39 and $40,$ use the following fact derived from Newton's
Laws. An object released at an angle $\theta$ with initial velocity $v$ ft/s travels
a horizontal distance
A player located 18.1 $\mathrm{ft}$ from the basket launches a successful jump
shot from a height of 10 $\mathrm{ft}($ level with the rim of the basket), at an angle
$\theta=34^{\circ}$ and initial velocity $v=25 \mathrm{ft} / \mathrm{s} .$
$$\begin{array}{l}{\text { (a) Show that } \Delta s \approx 0.255 \Delta \theta \text { ft for a small change of } \Delta \theta \text { . }} \\ {\text { (b) Is it likely that the shot would have been successful if the angle }} \\ {\text { had been off by } 2^{\circ} \text { ? }}\end{array}$$

Robert L.

### Problem 40

In Exercises 39 and $40,$ use the following fact derived from Newton's
Laws. An object released at an angle $\theta$ with initial velocity $v$ ft/s travels
a horizontal distance
Estimate $\Delta s$ if $\theta=34^{\circ}, v=25 \mathrm{ft}^{\prime} \mathrm{s},$ and $\Delta v=2$

Check back soon!

### Problem 41

The radius of a spherical ball is measured at $r=25 \mathrm{cm} .$ Estimate
the maximum error in the volume and surface area if $r$ is accurate to
within 0.5 $\mathrm{cm} .$

Robert L.

### Problem 42

The dosage $D$ of diphenhydramine for a dog of body mass $w$ kg
is $D=4.7 w^{2} / 3$ mg. Estimate the maximum allowable error in $w$ for a
cocker spaniel of mass $w=10$ kg if the percentage error in $D$ must be
less than 3$\% .$

Check back soon!

### Problem 43

The volume (in liters) and pressure $P$ (in atmospheres) of a cer-
tain gas satisfy $P V=24 .$ A measurement yields $V=4$ with a possible
error of $\pm 0.3$ L. Compute $P$ and estimate the maximum error in this
computation.

Robert L.

### Problem 44

In the notation of Exercise $43,$ assume that a measurement yields
$V=4 .$ Estimate the maximum allowable error in $V$ if $P$ must have an
error of less than 0.2 atm.

Michelle L.

### Problem 45

In Exercises $45-54,$ find the linearization at $x=a$
$$f(x)=x^{4}, \quad a=1$$

Robert L.

### Problem 46

In Exercises $45-54,$ find the linearization at $x=a$
$$f(x)=\frac{1}{x}, \quad a=2$$

Michelle L.

### Problem 47

In Exercises $45-54,$ find the linearization at $x=a$
$$f(\theta)=\sin ^{2} \theta, \quad a=\frac{\pi}{4}$$

Robert L.

### Problem 48

In Exercises $45-54,$ find the linearization at $x=a$
$$g(x)=\frac{x^{2}}{x-3}, \quad a=4$$

Michelle L.

### Problem 49

In Exercises $45-54,$ find the linearization at $x=a$
$$g(x)=\frac{x^{2}}{x-3}, \quad a=4$$

Robert L.

### Problem 50

In Exercises $45-54,$ find the linearization at $x=a$
$$y=(1+x)^{-1 / 2}, \quad a=3$$

Michelle L.

### Problem 51

In Exercises $45-54,$ find the linearization at $x=a$
$$y=\left(1+x^{2}\right)^{-1 / 2}, \quad a=0$$

Robert L.

### Problem 52

In Exercises $45-54,$ find the linearization at $x=a$
$$y=\tan ^{-1} x, \quad a=1$$

Michelle L.

### Problem 53

In Exercises $45-54,$ find the linearization at $x=a$
$$y=e^{\sqrt{x}}, \quad a=1$$

Robert L.

### Problem 54

In Exercises $45-54,$ find the linearization at $x=a$
$$y=e^{x} \ln x, \quad a=1$$

Michelle L.

### Problem 55

What is $f(2)$ if the linearization of $f(x)$ at $a=2$ is $L(x)=$ $2 x+4 ?$

Robert L.

### Problem 56

Compute the linearization of $f(x)=3 x-4$ at $a=0$ and $a=2$ . Prove more generally that a linear function coincides with its linearization at $x=a$ for all $a .$

Michelle L.

### Problem 57

Estimate $\sqrt{16.2}$ using the linearization $L(x)$ of $f(x)=\sqrt{x}$ at
$a=16 .$ Plot $f(x)$ and $L(x)$ on the same set of axes and determine
whether the estimate is too large or too small.

Robert L.

### Problem 58

Estimate 1/ $\sqrt{15}$ using a suitable linearization of $f(x)=$
1$/ \sqrt{x} .$ Plot $f(x)$ and $L(x)$ on the same set of axes and determine
whether the estimate is too large or too small. Use a calculator to com-
pute the percentage error.

Michelle L.

### Problem 59

In Exercises $59-67$, approximate using linearization and use a calculator to compute the percentage error.
$$\frac{1}{\sqrt{17}}$$

Robert L.

### Problem 60

In Exercises $59-67$, approximate using linearization and use a calculator to compute the percentage error.
$$\frac{1}{101}$$

Michelle L.

### Problem 61

In Exercises $59-67$, approximate using linearization and use a calculator to compute the percentage error.
$$\frac{1}{(10.03)^{2}}$$

Robert L.

### Problem 62

In Exercises $59-67$, approximate using linearization and use a calculator to compute the percentage error.
$$(17)^{1 / 4}$$

Michelle L.

### Problem 63

In Exercises $59-67$, approximate using linearization and use a calculator to compute the percentage error.
$$(64.1)^{1 / 3}$$

Robert L.

### Problem 64

In Exercises $59-67$, approximate using linearization and use a calculator to compute the percentage error.
$$(1.2)^{5 / 3}$$

Michelle L.

### Problem 65

In Exercises $59-67$, approximate using linearization and use a calculator to compute the percentage error.
$$\cos ^{-1}(0.52)$$

Robert L.

### Problem 66

In Exercises $59-67$, approximate using linearization and use a calculator to compute the percentage error.
$$\ln 1.07$$

Michelle L.

### Problem 67

In Exercises $59-67$, approximate using linearization and use a calculator to compute the percentage error.
$$e^{-0.012}$$

Robert L.

### Problem 68

Compute the linearization $L(x)$ of $f(x)=x^{2}-x^{3 / 2}$ at
$a=4 .$ Then plot $f(x)-L(x)$ and find an interval $I$ around $a=4$
such that $|f(x)-L(x)| \leq 0.1$ for $x \in I$

Michelle L.

### Problem 69

Show that the Linear Approximation to $f(x)=\sqrt{x}$ at $x=9$
yields the estimate $\sqrt{9+h}-3 \approx \frac{1}{6} h .$ Set $K=0.001$ and show that
$\left|f^{\prime \prime}(x)\right| \leq K$ for $x \geq 9 .$ Then verify numerically that the error $E$ sat-
isfies $\mathrm{Eq} .(5)$ for $h=10^{-n},$ for $1 \leq n \leq 4$

Check back soon!

### Problem 70

The Linear Approximation to $f(x)=\tan x$ at $x=\frac{\pi}{4}$ yields the estimate $\tan \left(\frac{\pi}{4}+h\right)-1 \approx 2 h .$ Set $K=6.2$ and show, using a plot,
that $\left|f^{\prime \prime}(x)\right| \leq K$ for $x \in\left[\frac{\pi}{4}, \frac{\pi}{4}+0.1\right] .$ Then verify numerically that the error $E$ satisfies $\mathrm{Eq} .(5)$ for $h=10^{-n},$ for $1 \leq n \leq 4$

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### Problem 71

Compute $d y / d x$ at the point $P=(2,1)$ on the curve $y^{3}+3 x y=7$
and show that the linearization at $P$ is $L(x)=-\frac{1}{3} x+\frac{5}{3} .$ Use $L(x)$ to
estimate the $y$ -coordinate of the point on the curve where $x=2.1 .$

Check back soon!

### Problem 72

Apply the method of Exercise 71 to $P=(0.5,1)$ on $y^{5}+y-$
$2 x=1$ to estimate the $y$ -coordinate of the point on the curve where
$x=0.55 .$

Michelle L.

### Problem 73

Apply the method of Exercise 71 to $P=(-1,2)$ on $y^{4}+7 x y=2$
to estimate the solution of $y^{4}-7.7 y=2$ near $y=2$

Check back soon!

### Problem 74

Show that for any real number $k,(1+\Delta x)^{k} \approx 1+k \Delta x$ for small
$\Delta x .$ Estimate $(1.02)^{0.7}$ and $(1.02)^{-0.3} .$

Michelle L.

### Problem 75

Let $\Delta f=f(5+h)-f(5),$ where $f(x)=x^{2}$ . Verify directly that
$E=\left|\Delta f-f^{\prime}(5) h\right|$ satisfies $(5)$ with $K=2$

Robert L.
Let $\Delta f=f(1+h)-f(1)$ where $f(x)=x^{-1} .$ Show directly
that $E=\left|\Delta f-f^{\prime}(1) h\right|$ is equal to $h^{2} /(1+h) .$ Then prove that $E \leq$
2$h^{2}$ if $-\frac{1}{2} \leq h \leq \frac{1}{2}$ . Hint: In this case, $\frac{1}{2} \leq 1+h \leq \frac{3}{2}$