# Calculus for AP

## 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.

### 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.

### 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.

### 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.

### 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.

### 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.
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? Check back soon! ### 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$(\mathrm{B}) ?$Check back soon! ### 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$?$Check back soon! ### 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 Check back soon! ### 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 .\$ 