# Calculus for AP

## Educators  ### Problem 1

Let $f(x)=5 x^{2} .$ Show that $f(3+h)=5 h^{2}+30 h+45 .$ Then show that
\begin{equation}\frac{f(3+h)-f(3)}{h}=5 h+30\end{equation}
and compute $f^{\prime}(3)$ by taking the limit as $h \rightarrow 0.$ Foster W.

### Problem 2

Let $f(x)=2 x^{2}-3 x-5$ . Show that the secant line through $(2, f(2))$ and $(2+h, f(2+h))$ has slope $2 h+5 .$ Then use this formula to compute the slope of:
\begin{equation}\begin{array}{l}{\text { (a) The secant line through }(2, f(2)) \text { and }(3, f(3))} \\ {\text { (b) The tangent line at } x=2 \text { (by taking a limit) }}\end{array}\end{equation} Bahar T.

### Problem 3

Compute $f^{\prime}(a)$ in two ways, using $E q .(1)$ and $E q_{ .}(2).$ Foster W.

### Problem 4

Compute $f^{\prime}(a)$ in two ways, using $E q .(1)$ and $E q_{ .}(2).$ Bahar T.

### Problem 5

Compute $f^{\prime}(a)$ in two ways, using $E q .(1)$ and $E q_{ .}(2).$ Foster W.

### Problem 6

Compute $f^{\prime}(a)$ in two ways, using $E q .(1)$ and $E q_{ .}(2).$ Bahar T.

### Problem 7

Refer to Figure II.
\begin{equation} \begin{array}{l}{\text {Find the slope of the secant line through }(2, f(2)) \text { and }} \\ {(2.5, f(2.5)) . \text { Is it larger or smaller than } f^{\prime}(2) ? \text { Explain. }}\end{array} \end{equation} Foster W.

### Problem 8

Refer to Figure II.
\begin{equation}\begin{array}{l}{\text { Estimate } \frac{f(2+h)-f(2)}{h} \text { for } h=-0.5 . \text { What does this }} \\ {\text { quantity represent? Is it larger or smaller than } f^{\prime}(2) ? \text { Explain. }}\end{array}\end{equation} Bahar T.

### Problem 9

Refer to Figure II.
\begin{equation}\begin{array}{l}{\text { Estimate } f^{\prime}(1) \text { and } f^{\prime}(2) \text {.}} \end{array}\end{equation} Foster W.

### Problem 10

Refer to Figure II.
\begin{equation}\begin{array}{l}{\text { Find a value of } h \text { for which } \frac{f(2+h)-f(2)}{h}=0.}\end{array}\end{equation} Bahar T.

### Problem 11

Refer to Figure 12.
\begin{equation}\begin{array}{l}{\text { Determine } f^{\prime}(a) \text { for } a=1,2,4,7.} \end{array}
\end{equation} Foster W.

### Problem 12

Refer to Figure 12.
\begin{equation}\begin{array}{l}{\text { For which values of } x \text { is } f^{\prime}(x)<0 ?}\end{array}\end{equation} Bahar T.

### Problem 13

Refer to Figure 12.
\begin{equation}\begin{array}{l}{\text { Which is larger, } f^{\prime}(5.5) \text { or } f^{\prime}(6.5) ?}\end{array}\end{equation} Foster W.

### Problem 14

Refer to Figure 12.
\begin{equation}\begin{array}{l}{\text { Show that } f^{\prime}(3) \text { does not exist. }}\end{array}\end{equation} Bahar T.

### Problem 15

use the limit definition to calculate the derivative of the linear function.
\begin{equation}f(x)=7 x-9\end{equation} Foster W.

### Problem 16

use the limit definition to calculate the derivative of the linear function.
\begin{equation}f(x)=12\end{equation} Bahar T.

### Problem 17

use the limit definition to calculate the derivative of the linear function.
\begin{equation}g(t)=8-3 t\end{equation} Foster W.

### Problem 18

use the limit definition to calculate the derivative of the linear function.
\begin{equation}k(z)=14 z+12\end{equation} Bahar T.

### Problem 19

Find an equation of the tangent line at $x=3,$ assuming that $f(3)=5$ and $f^{\prime}(3)=2 ?$ Foster W.

### Problem 20

Find $f(3)$ and $f^{\prime}(3),$ assuming that the tangent line to $y=f(x)$ at $a=3$ has equation $y=5 x+2 .$ Bahar T.

### Problem 21

Describe the tangent line at an arbitrary point on the "curve" $y=2 x+8 .$ Foster W.

### Problem 22

Suppose that $f(2+h)-f(2)=3 h^{2}+5 h .$ Calculate:
\begin{equation}\begin{array}{l}{\text { (a) The slope of the secant line through }(2, f(2)) \text { and }(6, f(6))} \\ {\text { (b) } f^{\prime}(2)}\end{array}\end{equation} Bahar T.

### Problem 23

Let $f(x)=\frac{1}{x} .$ Does $f(-2+h)$ equal $\frac{1}{-2+h}$ or $\frac{1}{-2}+\frac{1}{h} ?$ Compute the difference quotient at $a=-2$ with $h=0.5.$ Foster W.

### Problem 24

Let $f(x)=\sqrt{x} .$ Does $f(5+h)$ equal $\sqrt{5+h}$ or $\sqrt{5}+\sqrt{h} ?$ Compute the difference quotient at $a=5$ with $h=1.$ Bahar T.

### Problem 25

Let $f(x)=1 / \sqrt{x} .$ Compute $f^{\prime}(5)$ by showing that
\begin{equation}\frac{f(5+h)-f(5)}{h}=-\frac{1}{\sqrt{5} \sqrt{5+h}(\sqrt{5+h}+\sqrt{5})}\end{equation} Foster W.

### Problem 26

Find an equation of the tangent line to the graph of $f(x)=1 / \sqrt{x}$ at $x=9 .$ Bahar T.

### Problem 27

Use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line. Foster W.

### Problem 28

Use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line. Bahar T.

### Problem 29

Use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line. Foster W.

### Problem 30

Use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line. Bahar T.

### Problem 31

Use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line. Foster W.

### Problem 32

Use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line. Bahar T.

### Problem 33

Use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line. Foster W.

### Problem 34

Use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line. Bahar T.

### Problem 35

Use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line. Foster W.

### Problem 36

Use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line. Bahar T.

### Problem 37

Use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line. Foster W.

### Problem 38

Use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line. Bahar T.

### Problem 39

Use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line. Foster W.

### Problem 40

Use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line. Bahar T.

### Problem 41

Use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line. Foster W.

### Problem 42

Use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line. Bahar T.

### Problem 43

Use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line. Foster W.

### Problem 44

Use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line. Bahar T.

### Problem 45

Figure 13 displays data collected by the biologist Julian Huxley $(1887-1975)$ on the average antler weight $W$ of male red deer as a function of age $t .$ Estimate the derivative at $t=4 .$ For which values of $t$ is the slope of the tangent line equal to zero? For which values is it negative? Foster W.

### Problem 46

Figure 14$(\mathrm{A})$ shows the graph of $f(x)=\sqrt{x}$ . The close-up in Figure 14$(\mathrm{B})$ shows that the graph is nearly a straight line near $x=16$. Estimate the slope of this line and take it as an estimate for $f^{\prime}(16).$ Then compute $f^{\prime}(16)$ and compare with your estimate. Bahar T.

### Problem 47

Let $f(x)=\frac{4}{1+2^{x}}.$
\begin{equation}\begin{array}{l}{\text { (a) Plot } f(x) \text { over }[-2,2] . \text { Then zoom in near } x=0 \text { until the graph }} \\ {\text { appears straight, and estimate the slope } f^{\prime}(0) \text {.}} \\ {\text { (b) Use (a) to find an approximate equation to the tangent line at } x=0 \text {.}} \\ {\text { Plot this line and } f(x) \text { on the same set of axes. }}\end{array}\end{equation} Foster W.

### Problem 48

Let $f(x)=\cot x .$ Estimate $f^{\prime}\left(\frac{\pi}{2}\right)$ graphically by zooming in on a plot of $f(x)$ near $x=\frac{\pi}{2}$.

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### Problem 49

Determine the intervals along the $x$-axis on which the derivative in Figure 15 is positive. Foster W.

### Problem 50

Sketch the graph of $f(x)=\sin x$ on $[0, \pi]$ and guess the value of $f^{\prime}\left(\frac{\pi}{2}\right).$ Then calculate the difference quotient at $x=\frac{\pi}{2}$ for two small positive and negative values of $h$ . Are these calculations consistent with your guess?

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### Problem 51

Each limit represents a derivative $f^{\prime}(a).$ Find $f(x)$ and $a.$
\begin{equation}\lim _{h \rightarrow 0} \frac{(5+h)^{3}-125}{h}\end{equation} Foster W.

### Problem 52

Each limit represents a derivative $f^{\prime}(a).$ Find $f(x)$ and $a.$
\begin{equation}\lim _{x \rightarrow 5} \frac{x^{3}-125}{x-5}\end{equation} Bahar T.

### Problem 53

Each limit represents a derivative $f^{\prime}(a).$ Find $f(x)$ and $a.$
\begin{equation}\lim _{h \rightarrow 0} \frac{\sin \left(\frac{\pi}{6}+h\right)-0.5}{h}\end{equation} Foster W.

### Problem 54

Each limit represents a derivative $f^{\prime}(a).$ Find $f(x)$ and $a.$
\begin{equation}\lim _{x \rightarrow \frac{1}{4}} \frac{x^{-1}-4}{x-\frac{1}{4}}\end{equation} Bahar T.

### Problem 55

Each limit represents a derivative $f^{\prime}(a).$ Find $f(x)$ and $a.$
\begin{equation}\lim _{h \rightarrow 0} \frac{5^{2+h}-25}{h}\end{equation} Foster W.

### Problem 56

Each limit represents a derivative $f^{\prime}(a).$ Find $f(x)$ and $a.$
\begin{equation}\lim _{h \rightarrow 0} \frac{5^{h}-1}{h}\end{equation} Bahar T.

### Problem 57

Apply the method of Example 6 to $f(x)=\sin x$ to determine $f^{\prime}\left(\frac{\pi}{4}\right)$ accurately to four decimal places. Foster W.

### Problem 58

Apply the method of Example 6 to $f(x)=\cos x$ to determine $f^{\prime}\left(\frac{\pi}{5}\right)$ accurately to four decimal places. Use a graph of $f(x)$ to explain how the method works in this case.

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### Problem 59

For each graph in Figure $16,$ determine whether $f^{\prime}(1)$ is larger or smaller than the slope of the secant line between $x=1$ and $x=1+h$ for $h>0 .$ Explain. Foster W.

### Problem 60

Refer to the graph of $f(x)=2^{x}$ in Figure 17.
\begin{equation}
\begin{array}{c}{\text { (a) Explain graphically why, for } h>0}, \end{array}
\end{equation}
\begin{equation}\frac{f(-h)-f(0)}{-h} \leq f^{\prime}(0) \leq \frac{f(h)-f(0)}{h} \end{equation}
\begin{equation}
\begin{array}{l}{\text { (b) Use (a) to show that } 0.69314 \leq f^{\prime}(0) \leq 0.69315 .} \\ {\text { (c) Similarly, compute } f^{\prime}(x) \text { to four decimal places for } x=1,2,3,4 \text { . }} \\ {\text { (d) Now compute the ratios } f^{\prime}(x) / f^{\prime}(0) \text { for } x=1,2,3,4 . \text { Can you }} \\ {\text { guess an approximate formula for } f^{\prime}(x) ?}\end{array}
\end{equation}

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### Problem 61

Sketch the graph of $f(x)=x^{5 / 2}$ on $[0,6].$
\begin{equation}
\begin{array}{c}{\text { (a) Use the sketch to justify the inequalities for } h>0 \text { : }}\end{array}
\end{equation}
\begin{equation}
\frac{f(4)-f(4-h)}{h} \leq f^{\prime}(4) \leq \frac{f(4+h)-f(4)}{h}
\end{equation}
\begin{equation}
\begin{array}{l}{\text { (b) Use (a) to compute } f^{\prime}(4) \text { to four decimal places. }} \\ {\text { (c) Use a graphing utility to plot } f(x) \text { and the tangent line at } x=4 \text { , }} \\ {\text { using your estimate for } f^{\prime}(4) \text { . }}\end{array}
\end{equation} Foster W.

### Problem 62

Verify that $P=\left(1, \frac{1}{2}\right)$ lies on the graphs of both $f(x)=1 /\left(1+x^{2}\right)$ and $L(x)=\frac{1}{2}+m(x-1)$ for every slope $m .$ Plot $f(x)$ and $L(x)$ on the same axes for several values of m until you find a value of $m$ for which $y=L(x)$ appears tangent to the graph of $f(x).$ What is your estimate for $f^{\prime}(1) ?$

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### Problem 63

Use a plot of $f(x)=x^{x}$ to estimate the value $c$ such that $f^{\prime}(c)=0 .$ Find $c$ to sufficient accuracy so that
\begin{equation}
\end{equation} Foster W.

### Problem 64

Plot $f(x)=x^{x}$ and $y=2 x+a$ on the same set of axes for several values of $a$ until the line becomes tangent to the graph. Then estimate the value $c$ such that $f^{\prime}(c)=2.$

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### Problem 65

In Exercises $65-71$, estimate derivatives using the symmetric difference quotient (SDQ), deffned as the average of the difference quotients at hand $-h :$
\begin{equation}
\end{equation}
The SDQ usually gives a better approximation to the derivative than the difference quotient.
\begin{equation}
\begin{array}{l}{\text {The vapor pressure of water at temperature } T \text { (in kelvins) is the }} \\ {\text { atmospheric pressure } P \text { at which no net evaporation takes place. Use }} \\ {\text { the following table to estimate } P^{\prime}(T) \text { for } T=303,313,323,333,343} \\ {\text { by computing the SDQ given by Eq. (4) with } h=10 \text {.}}\end{array}
\end{equation}
\begin{equation}
\begin{array}{cccccc}\hline T(\mathrm{K}) & {293} & {303} & {313} & {323} & {333} & {343} & {353} \\ \hline P(\text { atm }) & {0.0278} & {0.0482} & {0.0808} & {0.1311} & {0.2067} & {0.3173} & {0.4754} \\ \hline\end{array}
\end{equation} Foster W.

### Problem 66

In Exercises $65-71$, estimate derivatives using the symmetric difference quotient (SDQ), deffned as the average of the difference quotients at hand $-h :$
\begin{equation}
\end{equation}
The SDQ usually gives a better approximation to the derivative than the difference quotient.
\begin{equation}
\begin{array}{l}{\text {Use the SDQ with } h=1 \text { year to estimate } P^{\prime}(T) \text { in the years }} \\ {2000,2002,2004,2006, \text { where } P(T) \text { is the U.S. ethanol production }} \\ {\text { (Figure } 18 ) \text { . Express vour answer in the correct units. }}\end{array}
\end{equation} Bahar T.

### Problem 67

In Exercises $65-71$, estimate derivatives using the symmetric difference quotient (SDQ), deffned as the average of the difference quotients at hand $-h :$
The SDQ usually gives a better approximation to the derivative than the difference quotient.
In Exercises $67-68,$ traffic speed $S$ along a certain road (in kmh) varies as a function of traffic density $q$ (number of cars per $k m$ of road). Use the following data to answer the questions:
\begin{array}{|c|c|c|c|c|c|}\hline q \ (\text {density}) & {60} & {70} & {80} & {90} & {100} \\ \hline S \ (\text {speed}) & {72.5} & {67.5} & {63.5} & {60} & {65} \\ \hline\end{array}
\begin{equation}\begin{array}{l}{\text { Estimate } S^{\prime}(80).} \end{array}\end{equation} Foster W.

### Problem 68

In Exercises $65-71$, estimate derivatives using the symmetric difference quotient (SDQ), deffned as the average of the difference quotients at hand $-h :$
The SDQ usually gives a better approximation to the derivative than the difference quotient.
In Exercises $67-68,$ traffic speed $S$ along a certain road (in kmh) varies as a function of traffic density $q$ (number of cars per $k m$ of road). Use the following data to answer the questions:
\begin{array}{|c|c|c|c|c|c|}\hline q \ (\text {density}) & {60} & {70} & {80} & {90} & {100} \\ \hline S \ (\text {speed}) & {72.5} & {67.5} & {63.5} & {60} & {65} \\ \hline\end{array}
\begin{array}{l}{\text {} \text { Explain why } V=q S, \text { called traffic volume, is equal to }} \\ {\text { the number of cars passing a point per hour. Use the data to estimate }} \\ {V^{\prime}(80) .}\end{array}

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### Problem 69

In Exercises $65-71$, estimate derivatives using the symmetric difference quotient (SDQ), deffned as the average of the difference quotients at hand $-h :$
\begin{equation}
\end{equation}
The SDQ usually gives a better approximation to the derivative than the difference quotient.
Exercises $69-71 :$ The current (in amperes) at timet (inseconds) flowing in the circuit in Figure 19 is given by Kirchhoff's Law:
\begin{equation}i(t)=C v^{\prime}(t)+R^{-1} v(t)\end{equation}
where $v(t)$ is the voltage (in volts), C the capacitance (infarads), and R the resistance (in ohms, $\Omega ).$
\begin{equation}\begin{aligned} \text { Calculate the current at } t &=3 \text { if } \\ v(t) &=0.5 t+4 \mathrm{V} \\ \text { where } C=0.01 \mathrm{F} \text { and } R &=100 \Omega \end{aligned}\end{equation} Foster W.

### Problem 70

In Exercises $65-71$, estimate derivatives using the symmetric difference quotient (SDQ), deffned as the average of the difference quotients at hand $-h :$
\begin{equation}
\end{equation}
The SDQ usually gives a better approximation to the derivative than the difference quotient.
Exercises $69-71 :$ The current (in amperes) at timet (inseconds) flowing in the circuit in Figure 19 is given by Kirchhoff's Law:
\begin{equation}i(t)=C v^{\prime}(t)+R^{-1} v(t)\end{equation}
where $v(t)$ is the voltage (in volts), C the capacitance (infarads), and R the resistance (in ohms, $\Omega ).$
\begin{equation}
\begin{array}{l}{\text { Use the following data to estimate } v^{\prime}(10) \text { (by an SDQ). Then estimate }} \\ {\text { } i(10), \text { assuming } C=0.03 \text { and } R=1,000 .}\end{array}
\end{equation}
\begin{equation}
\begin{array}{|c|c|c|c|c|c|}\hline t & {9.8} & {9.9} & {10} & {10.1} & {10.2} \\ \hline v(t) & {256.52} & {257.32} & {258.11} & {258.9} & {259.69} \\ \hline\end{array}
\end{equation}

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### Problem 71

In Exercises $65-71$, estimate derivatives using the symmetric difference quotient (SDQ), deffned as the average of the difference quotients at hand $-h :$
\begin{equation}
\end{equation}
The SDQ usually gives a better approximation to the derivative than the difference quotient.
Exercises $69-71 :$ The current (in amperes) at timet (inseconds) flowing in the circuit in Figure 19 is given by Kirchhoff's Law:
\begin{equation}i(t)=C v^{\prime}(t)+R^{-1} v(t)\end{equation}
where $v(t)$ is the voltage (in volts), C the capacitance (infarads), and R the resistance (in ohms, $\Omega ).$

\begin{equation}
\begin{array}{l}{\text {Assume that } R=200 \Omega \text { but } C \text { is unknown. Use the following data }} \\ {\text { to estimate } v^{\prime}(4) \text { (by an SDQ and deduce an approximate value for the }} \\ {\text { capacitance } C .}\end{array}
\end{equation}
\begin{equation}
\begin{array}{|c|c|c|c|c|c|}\hline t & {3.8} & {3.9} & {4} & {4.1} & {4.2} \\ \hline v(t) & {388.8} & {404.2} & {420} & {436.2} & {452.8} \\ \hline i(t) & {32.34} & {33.22} & {34.1} & {34.98} & {35.86} \\ \hline\end{array}
\end{equation} Foster W.

### Problem 72

The SDQ usually approximates the derivative much more closely than does the ordinary difference quotient. Let $f(x)=2^{x}$ and $a=0$ . Compute the SDQ with $h=0.001$ and the ordinary difference quotients with $h=\pm 0.001 .$ Compare with the actual value, which is $f^{\prime}(0)=\ln 2 .$

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### Problem 73

Explain how the symmetric difference quotient defined by Eq. (4) can be interpreted as the slope of a secant line. Foster W.

### Problem 74

Which of the two functions in Figure 20 satisfies the inequality
\begin{equation}\frac{f(a+h)-f(a-h)}{2 h} \leq \frac{f(a+h)-f(a)}{h}\end{equation}
for $h>0 ?$ Explain in terms of secant lines.

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### Problem 75

Show that if $f(x)$ is a quadratic polynomial, then the SDQ at $x=a($ for any $h \neq 0)$ is equal to $f^{\prime}(a)$ . Explain the graphical meaning of this result. Foster W.
Let $f(x)=x^{-2} .$ Compute $f^{\prime}(1)$ by taking the limit of the SDQs $($ with $a=1)$ as $h \rightarrow 0 .$ 