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

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

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

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

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

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

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

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

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

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

Estimate the instantaneous rate of change at the point indicated.

$P(x)=3 x^{2}-5 ; \quad x=2$

Foster W.

Numerade Educator

Estimate the instantaneous rate of change at the point indicated.

$f(t)=12 t-7 ; \quad t=-4$

Dishary H.

Numerade Educator

Estimate the instantaneous rate of change at the point indicated.

$y(x)=\frac{1}{x+2} ; \quad x=2$

Foster W.

Numerade Educator

Estimate the instantaneous rate of change at the point indicated.

$y(t)=\sqrt{3 t+1} ; \quad t=1$

Dishary H.

Numerade Educator

Estimate the instantaneous rate of change at the point indicated.

$f(x)=e^{x} ; \quad x=0$

Foster W.

Numerade Educator

Estimate the instantaneous rate of change at the point indicated.

$f(x)=e^{x} ; \quad x=e$

Dishary H.

Numerade Educator

Estimate the instantaneous rate of change at the point indicated.

$f(x)=\ln x ; \quad x=3$

Foster W.

Numerade Educator

Estimate the instantaneous rate of change at the point indicated.

$f(x)=\tan ^{-1} x ; \quad x=\frac{\pi}{4}$

Dishary H.

Numerade Educator

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

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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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

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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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

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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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

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

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

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

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

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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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

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 $t=1 .$

Zafar Z.

Numerade Educator

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

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

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

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