Calculus for AP

Jon Rogawski & Ray Cannon

Chapter 2

LIMITS

Educators

ZZ

Problem 1

A ball dropped from a state of rest at time $t=0$ travels a distance $s(t)=4.9 t^{2} \mathrm{m}$ in $t$ seconds.
(a) How far does the ball travel during the time interval $[2,2.5] ?$
(b) Compute the average velocity over $[2,2.5].$
(c) Compute the average velocity for the time intervals in the table and estimate the ball's instantaneous velocity at $t=2$.

Foster W.
Numerade Educator

Problem 2

A wrench released from a state of rest at time $t=0$ travels a distance $s(t)=4.9 t^{2} \mathrm{m}$ in $t$ seconds. Estimate the instantaneous velocity at $t=3$.

Dishary H.
Numerade Educator

Problem 3

Let $v=20 \sqrt{T}$ as in Example $2 .$ Estimate the instantaneous rate of change of $v$ with respect to $T$ when $T=300 \mathrm{K}$.

Foster W.
Numerade Educator

Problem 4

Compute $\Delta y / \Delta x$ for the interval $[2,5],$ where $y=4 x-9 .$ What is the instantaneous rate of change of $y$ with respect to $x$ at $x=2 ?$

Dishary H.
Numerade Educator

Problem 5

A stone is tossed vertically into the air from ground level with an initial velocity of 15 $\mathrm{m} / \mathrm{s} .$ Its height at time $t$ is $h(t)=$ $15 t-4.9 t^{2} \mathrm{m}$.

Compute the stone's average velocity over the time interval $[0.5,2.5]$ and indicate the corresponding secant line on a sketch of the graph of $h(t)$ .

Foster W.
Numerade Educator

Problem 6

A stone is tossed vertically into the air from ground level with an initial velocity of 15 $\mathrm{m} / \mathrm{s} .$ Its height at time $t$ is $h(t)=$ $15 t-4.9 t^{2} \mathrm{m}$.

Compute the stone's average velocity over the time intervals $[1,1.01],[1,1.001],[1,1.0001]$ and $[0.99,1],[0.999,1],[0.9999,1],$ and then estimate the instantaneous velocity at $t=1$.

Dishary H.
Numerade Educator

Problem 7

With an initial deposit of $\$ 100,$ the balance in a bank account after $t$ years is $f(t)=100(1.08)^{t}$ dollars.
(a) What are the units of the rate of change of $f(t) ?$
(b) Find the average rate of change over $[0,0.5]$ and $[0,1]$.
(c) Estimate the instantaneous rate of change at $t=0.5$ by computing the average rate of change over intervals to the left and right of $t=0.5$.

Foster W.
Numerade Educator

Problem 8

The position of a particle at time $t$ is $s(t)=t^{3}+t$ . Compute the average velocity over the time interval $[1,4]$ and estimate the instantaneous velocity at $t=1$.

Dishary H.
Numerade Educator

Problem 9

Figure 8 shows the estimated number $N$ of Internet users in Chile, based on data from the United Nations Statistics Division.
(a) Estimate the rate of change of $N$ at $t=2003.5$.
(b) Does the rate of change increase or decrease as $t$ increases? Explain graphically.
(c) Let $R$ be the average rate of change over $[2001,2005] .$ Compute $R.$
(d) Is the rate of change at $t=2002$ greater than or less than the average rate $R ?$ Explain graphically.

Foster W.
Numerade Educator

Problem 10

The atmospheric temperature $T($ in $^{\circ} \mathrm{C})$ at altitude $h$ meters above a certain point on earth is $T=15-0.0065 h$ for $h \leq 12,000 \mathrm{m} .$ What are the average and instantaneous rates of change of $T$ with respect to $h ?$ Why are they the same? Sketch the graph of $T$ for $h \leq 12,000 .$

Dishary H.
Numerade Educator

Problem 11

Estimate the instantaneous rate of change at the point indicated.
$P(x)=3 x^{2}-5 ; \quad x=2$

Foster W.
Numerade Educator

Problem 12

Estimate the instantaneous rate of change at the point indicated.
$f(t)=12 t-7 ; \quad t=-4$

Dishary H.
Numerade Educator

Problem 13

Estimate the instantaneous rate of change at the point indicated.
$y(x)=\frac{1}{x+2} ; \quad x=2$

Foster W.
Numerade Educator

Problem 14

Estimate the instantaneous rate of change at the point indicated.
$y(t)=\sqrt{3 t+1} ; \quad t=1$

Dishary H.
Numerade Educator

Problem 15

Estimate the instantaneous rate of change at the point indicated.
$f(x)=e^{x} ; \quad x=0$

Foster W.
Numerade Educator

Problem 16

Estimate the instantaneous rate of change at the point indicated.
$f(x)=e^{x} ; \quad x=e$

Dishary H.
Numerade Educator

Problem 17

Estimate the instantaneous rate of change at the point indicated.
$f(x)=\ln x ; \quad x=3$

Foster W.
Numerade Educator

Problem 18

Estimate the instantaneous rate of change at the point indicated.
$f(x)=\tan ^{-1} x ; \quad x=\frac{\pi}{4}$

Dishary H.
Numerade Educator

Problem 19

The height (in centimeters) at time $t$ (in seconds) of a small mass oscillating at the end of a spring is $h(t)=8 \cos (12 \pi t) .$
(a) Calculate the mass's average velocity over the time intervals $[0,0.1]$ and $[3,3.5] .$
(b) Estimate its instantaneous velocity at $t=3$.

Foster W.
Numerade Educator

Problem 20

The number $P(t)$ of $E .$coll cells at time $t$ (hours) in a petri dish is plotted in Figure 9.
(a) Calculate the average rate of change of $P(t)$ over the time interval $[1,3]$ and draw the corresponding secant line.
(b) Estimate the slope $m$ of the line in Figure $9 .$ What does $m$ represent?

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Problem 21

Assume that the period $T$ (in seconds) of a pendulum (the time required for a complete back-and-forth cycle $)$ is $T=\frac{3}{2} \sqrt{L},$ where $L$ is the pendulum's length (in meters).
(a) What are the units for the rate of change of $T$ with respect to $L ?$ Explain what this rate measures.
(b) Which quantities are represented by the slopes of lines $A$ and $B$ in Figure 10$?$
(c) Estimate the instantaneous rate of change of $T$ with respect to $L$ when $L=3 \mathrm{m} .$

Foster W.
Numerade Educator

Problem 22

The graphs in Figure 11 represent the positions of moving particles as functions of time.
(a) Do the instantaneous velocities at times $t_{1}, t_{2}, t_{3}$ in (A) form an increasing or a decreasing sequence?
(b) Is the particle speeding up or slowing down in $(\mathrm{A}) ?$
(c) Is the particle speeding up or slowing down in $(\mathrm{B}) ?$

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Problem 23

An advertising campaign boosted sales of Crunchy Crust frozen pizza to a peak level of $S_{0}$ dollars per month. A marketing study showed that after $t$ months, monthly sales declined to
$$S(t)=S_{0} g(t), \quad \text { where } g(t)=\frac{1}{\sqrt{1+t}}.$$
Do sales decline more slowly or more rapidly as time increases? Answer by referring to a sketch the graph of $g(t)$ together with several tangent lines.

Foster W.
Numerade Educator

Problem 24

The fraction of a city's population infected by a flu virus is plotted as a function of time (in weeks) in Figure 12.
(a) Which quantities are represented by the slopes of lines $A$ and $B ?$ Estimate these slopes.
(b) Is the flu spreading more rapidly at $t=1,2,$ or 3$?$
(c) Is the flu spreading more rapidly at $t=4,5,$ or 6$?$

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Problem 25

The graphs in Figure 13 represent the positions $s$ of moving particles as functions of time $t .$ Match each graph with a description:
(a) Speeding up
(b) Speeding up and then slowing down
(c) Slowing down
(d) Slowing down and then speeding up

Foster W.
Numerade Educator

Problem 26

An epidemiologist finds that the percentage $N(t)$ of susceptible children who were infected on day $t$ during the first three weeks of a measles outbreak is given, to a reasonable approximation, by the formula (Figure 14)
$$N(t)=\frac{100 t^{2}}{t^{3}+5 t^{2}-100 t+380}$$
(a) Draw the secant line whose slope is the average rate of change in infected children over the intervals $[4,6]$ and $[12,14] .$ Then compute these average rates (in units of percent per day).
(b) Is the rate of decline greater at $t=8$ or $t=16 ?$
(c) Estimate the rate of change of $N(t)$ on day 12 .

Dishary H.
Numerade Educator

Problem 27

The fungus Fusarium exosporium infects a field of flax plants through the roots and causes the plants to wilt. Eventually, the entire field is infected. The percentage $f(t)$ of infected plants as a function of time $t$ (in days) since planting is shown in Figure 15.
(a) What are the units of the rate of change of $f(t)$ with respect to $t ?$ What does this rate measure?
(b) Use the graph to rank (from smallest to largest) the average infection rates over the intervals $[0,12],[20,32],$ and $[40,52] .$
(c) Use the following table to compute the average rates of infection over the intervals $[30,40],[40,50],[30,50] .$
(d) Draw the tangent line at $t=40$ and estimate its slope.

Foster W.
Numerade Educator

Problem 28

Let $v=20 \sqrt{T}$ as in Example $2 .$ Is the rate of change of $v$ with respect to $T$ greater at low temperatures or high temperatures? Explain in terms of the graph.

Dishary H.
Numerade Educator

Problem 29

If an object in linear motion (but with changing velocity) covers $\Delta s$ meters in $\Delta t$ seconds, then its average velocity is $v_{0}=\Delta s / \Delta t \mathrm{m} / \mathrm{s} .$ Show that it would cover the same distance if it traveled at constant velocity $v_{0}$ over the same time interval. This justifies our calling $\Delta s / \Delta t$ the average velocity.

Foster W.
Numerade Educator

Problem 30

Sketch the graph of $f(x)=x(1-x)$ over $[0,1] .$ Refer to the graph and, without making any computations, find:
(a) The average rate of change over $[0,1]$
(b) The (instantaneous) rate of change at $x=\frac{1}{2}$
(c) The values of $x$ at which the rate of change is positive

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Problem 31

Which graph in Figure 16 has the following property: For all $x,$ the average rate of change over $[0, x]$ is greater than the instantaneous rate of change at $x .$ Explain.

Foster W.
Numerade Educator

Problem 32

The height of a projectile fired in the air vertically with initial velocity 25 $\mathrm{m} / \mathrm{s}$ is
$$h(t)=25 t-4.9 t^{2} \mathrm{m}.$$
(a) Compute $h(1) .$ Show that $h(t)-h(1)$ can be factored with $(t-1)$ as a factor.
(b) Using part (a), show that the average velocity over the interval $[1, t]$ is $20.1-4.9 t.$
(c) Use this formula to find the average velocity over several intervals $[1, t]$ with $t$ close to $1 .$ Then estimate the instantaneous velocity at time $t=1 .$

ZZ
Zafar Z.
Numerade Educator

Problem 33

Let $Q(t)=t^{2} .$ As in the previous exercise, find a formula for the average rate of change of $Q$ over the interval $[1, t]$ and use it to estimate the instantaneous rate of change at $t=1 .$ Repeat for the interval $[2, t]$ and estimate the rate of change at $t=2$.

Foster W.
Numerade Educator

Problem 34

Show that the average rate of change of $f(x)=x^{3}$ over $[1, x]$ is equal to
$$x^{2}+x+1.$$
Use this to estimate the instantaneous rate of change of $f(x)$ at $x=1.$

Dishary H.
Numerade Educator

Problem 35

Find a formula for the average rate of change of $f(x)=x^{3}$ over $[2, x]$ and use it to estimate the instantaneous rate of change at $x=2$.

Foster W.
Numerade Educator

Problem 36

Let $T=\frac{3}{2} \sqrt{L}$ as in Exercise 21. The numbers in the second column of Table 4 are increasing, and those in the last column are decreasing. Explain why in terms of the graph of $T$ as a function of $L .$ Also, explain graphically why the instantaneous rate of change at $L=3$ lies between 0.4329 and $0.4331 .$

Dishary H.
Numerade Educator