Calculus for AP

Jon Rogawski & Colin Adams

Chapter 3

DIFFERENTIATION - all with Video Answers

Educators

Section 1

Definition of the Derivative

Problem 1

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

Foster Wisusik

Foster Wisusik

Numerade Educator

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 Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 3

Compute $f^{\prime}(a)$ in two ways, using $E q .(1)$ and $E q .(2).$
$$f(x)=x^{2}+9 x, \quad a=0$$

Foster Wisusik

Numerade Educator

Problem 4

Compute $f^{\prime}(a)$ in two ways, using $E q .(1)$ and $E q .(2).$
$$f(x)=x^{2}+9 x, \quad a=2$$

Bahar Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 5

Compute $f^{\prime}(a)$ in two ways, using $E q .(1)$ and $E q .(2).$
$$f(x)=3 x^{2}+4 x+2, \quad a=-1$$

Claire Rochford

Claire Rochford

Numerade Educator

Problem 6

Compute $f^{\prime}(a)$ in two ways, using $E q .(1)$ and $E q .(2).$
$$f(x)=x^{3}, \quad a=2$$

Bahar Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 7

Compute $f^{\prime}(a)$ in two ways, using $E q .(1)$ and $E q .(2).$
$$f(x)=x^{3}+2 x, \quad a=1$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 8

Compute $f^{\prime}(a)$ in two ways, using $E q .(1)$ and $E q .(2).$
$$f(x)=\frac{1}{x}, \quad a=2$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 9

Refer to Figure 12.
\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}

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 10

Refer to Figure 12.
\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}

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 11

Refer to Figure 12.
\begin{equation}
\begin{array}{l}{\text { Estimate } f^{\prime}(1) \text { and } f^{\prime}(2) \text { . }}\end{array}
\end{equation}

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 12

Refer to Figure 12.
\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}

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 13

Refer to Figure 13.
$$\begin{array}{l}{\text { Determine } f^{\prime}(a) \text { for } a=1,2,4,7}. \end{array}$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 14

Refer to Figure 13.
$$\begin{array}{l}{\text { For which values of } x \text { is } f^{\prime}(x)<0 ?} \end{array}$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 15

Refer to Figure 13.
$$\begin{array}{l}{\text { Which is larger, } f^{\prime}(5.5) \text { or } f^{\prime}(6.5) ?} \end{array}$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 16

Refer to Figure 13.
$$\begin{array}{l}{\text { Show that } f^{\prime}(3) \text { does not exist. }} \end{array}$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 17

Use the limit definition to calculate the derivative of the linear function.
$$f(x)=7 x-9$$

Foster Wisusik

Numerade Educator

Problem 18

Use the limit definition to calculate the derivative of the linear function.
$$f(x)=12$$

Bahar Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 19

Use the limit definition to calculate the derivative of the linear function.
$$g(t)=8-3 t$$

Foster Wisusik

Numerade Educator

Problem 20

Use the limit definition to calculate the derivative of the linear function.
$$k(z)=14 z+12$$

Bahar Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 21

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

Foster Wisusik

Numerade Educator

Problem 22

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 Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 23

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

Foster Wisusik

Numerade Educator

Problem 24

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 Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 25

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 Wisusik

Numerade Educator

Problem 26

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 Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 27

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

Foster Wisusik

Numerade Educator

Problem 28

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

Bahar Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 29

Use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line.
$$f(x)=2 x^{2}+10 x, \quad a=3$$

Foster Wisusik

Numerade Educator

Problem 30

Use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line.
$$f(x)=4-x^{2}, \quad a=-1$$

Bahar Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 31

Use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line.
$$f(t)=t-2 t^{2}, \quad a=3$$

Foster Wisusik

Numerade Educator

Problem 32

Use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line.
$$f(x)=8 x^{3}, \quad a=1$$

Bahar Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 33

Use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line.
$$f(x)=x^{3}+x, \quad a=0$$

Foster Wisusik

Numerade Educator

Problem 34

Use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line.
$$f(t)=2 t^{3}+4 t, \quad a=4$$

Bahar Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 35

Use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line.
$$f(x)=x^{-1}, \quad a=8$$

Foster Wisusik

Numerade Educator

Problem 36

Use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line.
$$f(x)=x+x^{-1}, \quad a=4$$

Bahar Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 37

Use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line.
$$f(x)=\frac{1}{x+3}, \quad a=-2$$

Foster Wisusik

Numerade Educator

Problem 38

Use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line.
$$f(t)=\frac{2}{1-t}, \quad a=-1$$

Bahar Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 39

Use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line.
$$f(x)=\sqrt{x+4}, \quad a=1$$

Foster Wisusik

Numerade Educator

Problem 40

Use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line.
$$f(t)=\sqrt{3 t+5}, \quad a=-1$$

Bahar Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 41

Use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line.
$$f(x)=\frac{1}{\sqrt{x}}, \quad a=4$$

Foster Wisusik

Numerade Educator

Problem 42

Use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line.
$$f(x)=\frac{1}{\sqrt{2 x+1}}, \quad a=4$$

Bahar Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 43

Use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line.
$$f(t)=\sqrt{t^{2}+1}, \quad a=3$$

Foster Wisusik

Numerade Educator

Problem 44

Use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line.
$$f(x)=x^{-2}, \quad a=-1$$

Bahar Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 45

Use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line.
$$f(x)=\frac{1}{x^{2}+1}, \quad a=0$$

Foster Wisusik

Numerade Educator

Problem 46

Use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line.
$$f(t)=t^{-3}, \quad a=1$$

Bahar Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 47

Figure 14 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 Wisusik

Numerade Educator

Problem 48

Figure 15$(\mathrm{A})$ shows the graph of $f(x)=\sqrt{x}$ . The close-up in Figure 15$(\mathrm{B})$ 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.

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 49

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

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 50

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

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 51

Determine the intervals along the $x$ -axis on which the derivative in Figure 16 is positive.

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 52

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?

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 53

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

Foster Wisusik

Numerade Educator

Problem 54

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

Bahar Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 55

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

Foster Wisusik

Numerade Educator

Problem 56

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

Bahar Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 57

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

Foster Wisusik

Numerade Educator

Problem 58

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

Bahar Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 59

Apply the method of Example 7 to $f(x)=\sin x$ to determine $f^{\prime}\left(\frac{\pi}{4}\right)$ accurately to four decimal places.

Foster Wisusik

Numerade Educator

Problem 60

Apply the method of Example 7 to $f(x)=\cos x$ to determine $f^{\prime}\left(\frac{\pi}{5}\right)$ accurately to four decimal places. Use a graph of $f$ to explain how the method works in this case.

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 61

For each graph in Figure 17 , 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.

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 62

Refer to the graph of $f(x)=2^{x}$ in Figure 18.

\begin{equation}
\begin{array}{c}{\text { (a) Explain graphically why, for } h>0,} \end{array}
\end{equation}
$$\frac{f(-h)-f(0)}{-h} \leq f^{\prime}(0) \leq \frac{f(h)-f(0)}{h}$$
\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}

Carson Merrill

Numerade Educator

Problem 63

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 :} \\ {\frac{f(4)-f(4-h)}{h} \leq f^{\prime}(4) \leq \frac{f(4+h)-f(4)}{h}}\end{array}
\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 } y=f(x) \text { and the tangent line at }} \\ {x=4, \text { utilizing your estimate for } f^{\prime}(4) \text { . }}\end{array}
\end{equation}

Foster Wisusik

Numerade Educator

Problem 64

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 $y=f(x)$ and $y=L(x)$ 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 $f .$ What is your estimate for $f^{\prime}(1) ?$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 65

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
$$\left|\frac{f(c+h)-f(c)}{h}\right| \leq 0.006 \quad \text { for } \quad h=\pm 0.001$$

Foster Wisusik

Numerade Educator

Problem 66

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 $f^{\prime}(c)=2.$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 67

Estimate derivatives using the symmetric difference quotient $(S D Q),$ defined as the average of the difference quotients at $h$ and $-h :$
$$\begin{array}{r}{\frac{1}{2}\left(\frac{f(a+h)-f(a)}{h}+\frac{f(a-h)-f(a)}{-h}\right)} \\ {=\frac{f(a+h)-f(a-h)}{2 h}}\end{array}$$
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 .}\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}

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 68

estimate derivatives using the symmetric difference quotient $(S D Q),$ defined as the average of the difference quotients at $h$ and $-h :$
$$\begin{array}{r}{\frac{1}{2}\left(\frac{f(a+h)-f(a)}{h}+\frac{f(a-h)-f(a)}{-h}\right)} \\ {=\frac{f(a+h)-f(a-h)}{2 h}}\end{array}$$
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 }} \\ {2005,2007,2009,2011, \text { where } P(T) \text { is the U.S. ethanol production }} \\ {\text { (Figure 19). Express your answer in the correct units. }}\end{array}
\end{equation}

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 69

estimate derivatives using the symmetric difference quotient $(S D Q),$ defined as the average of the difference quotients at $h$ and $-h :$
$$\begin{array}{r}{\frac{1}{2}\left(\frac{f(a+h)-f(a)}{h}+\frac{f(a-h)-f(a)}{-h}\right)} \\ {=\frac{f(a+h)-f(a-h)}{2 h}}\end{array}$$
The SDQ usually gives a better approximation to the derivative than the difference quotient.
In Exercises $69-70,$ traffic speed $S$ along a certain road (in kilometers per hour) varies as a function of traffic density $q$ (number of cars per kilometer of road). Use the following data to answer the questions:
\begin{equation}
\begin{array}{l}{\text { Estimate } S^{\prime}(80)} \end{array}
\end{equation}

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 70

estimate derivatives using the symmetric difference quotient $(S D Q),$ defined as the average of the difference quotients at $h$ and $-h :$
$$\begin{array}{r}{\frac{1}{2}\left(\frac{f(a+h)-f(a)}{h}+\frac{f(a-h)-f(a)}{-h}\right)} \\ {=\frac{f(a+h)-f(a-h)}{2 h}}\end{array}$$
The SDQ usually gives a better approximation to the derivative than the difference quotient.
In Exercises $69-70,$ traffic speed $S$ along a certain road (in kilometers per hour) varies as a function of traffic density $q$ (number of cars per kilometer of road). Use the following data to answer the questions:
\begin{equation}
\begin{array}{l}{\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}
\end{equation}

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 71

estimate derivatives using the symmetric difference quotient $(S D Q),$ defined as the average of the difference quotients at $h$ and $-h :$
$$\begin{array}{r}{\frac{1}{2}\left(\frac{f(a+h)-f(a)}{h}+\frac{f(a-h)-f(a)}{-h}\right)} \\ {=\frac{f(a+h)-f(a-h)}{2 h}}\end{array}$$
The SDQ usually gives a better approximation to the derivative than the difference quotient.
Exercises $71-73 :$ The current (in amperes) at timet (in seconds) flowing
in the circuit in Figure 20 is given by Kirchhoff's Law:
$$i(t)=C v^{\prime}(t)+R^{-1} v(t)$$
where $v(t)$ is the voltage (in volts, $V ), C$ the capacitance (in farads, F), 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}

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 72

estimate derivatives using the symmetric difference quotient $(S D Q),$ defined as the average of the difference quotients at $h$ and $-h :$
$$\begin{array}{r}{\frac{1}{2}\left(\frac{f(a+h)-f(a)}{h}+\frac{f(a-h)-f(a)}{-h}\right)} \\ {=\frac{f(a+h)-f(a-h)}{2 h}}\end{array}$$
The SDQ usually gives a better approximation to the derivative than the difference quotient.
Exercises $71-73 :$ The current (in amperes) at timet (in seconds) flowing
in the circuit in Figure 20 is given by Kirchhoff's Law:
$$i(t)=C v^{\prime}(t)+R^{-1} v(t)$$
where $v(t)$ is the voltage (in volts, $V ), C$ the capacitance (in farads, F), 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=1000 .}\end{array}
\end{equation}
$$\begin{array}{|c|c|c|c|c|c|}\hline t & {9.8} & {9.9} & {10} & {10.1} & {10.2} \\ \hline v(t) & {256.52} & {257.32} & {258.11} & {258.9} & {259.69} \\ \hline\end{array}$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 73

estimate derivatives using the symmetric difference quotient $(S D Q),$ defined as the average of the difference quotients at $h$ and $-h :$
$$\begin{array}{r}{\frac{1}{2}\left(\frac{f(a+h)-f(a)}{h}+\frac{f(a-h)-f(a)}{-h}\right)} \\ {=\frac{f(a+h)-f(a-h)}{2 h}}\end{array}$$
The SDQ usually gives a better approximation to the derivative than the difference quotient.
Exercises $71-73 :$ The current (in amperes) at timet (in seconds) flowing
in the circuit in Figure 20 is given by Kirchhoff's Law:
$$i(t)=C v^{\prime}(t)+R^{-1} v(t)$$
where $v(t)$ is the voltage (in volts, $V ), C$ the capacitance (in farads, F), 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}
$$\begin{array}{|c|c|c|c|c|c|}\hline t & {3.8} & {3.9} & {4} & {4.1} & {4.2} \\ \hline v(t) & {388.8} & {404.2} & {420} & {436.2} & {452.8} \\ \hline i(t) & {32.34} & {33.22} & {34.1} & {34.98} & {35.86} \\ \hline\end{array}$$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 74

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 $f^{\prime}(0)=\ln 2 .$

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 75

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

Foster Wisusik

Numerade Educator

Problem 76

Which of the two functions in Figure 21 satisfies the inequality
$$\frac{f(a+h)-f(a-h)}{2 h} \leq \frac{f(a+h)-f(a)}{h}$$
for $h>0 ?$ Explain in terms of secant lines.

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 77

Show that if $f$ 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 Wisusik

Numerade Educator

Problem 78

Let $f(x)=x^{-2} .$ Compute $f^{\prime}(1)$ by taking the limit of the SDQs (with $a=1 )$ as $h \rightarrow 0 .$

Bahar Tehranipoor

Bahar Tehranipoor

Numerade Educator