Calculus: Early Transcendentals

Jon Rogawski, Colin Adams, Robert Franzosa

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:
(a) The slope of the secant line through ( $2, f(2))$ and $(3, f(3))$
(b) The slope of the tangent line at $x=2$ (by taking a limit)

Bahar Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 3

In Exercises $3-8,$ compute $f^{\prime}(a)$ in two ways, using Eq. (1) and Eq. (2).
$$
f(x)=x^{2}+9 x, \quad a=0
$$

Lucas Finney

Numerade Educator

Problem 4

In Exercises $3-8,$ compute $f^{\prime}(a)$ in two ways, using Eq. (1) and Eq. (2).
$$
f(x)=x^{2}+9 x, \quad a=2
$$

Lucas Finney

Numerade Educator

Problem 5

In Exercises $3-8,$ compute $f^{\prime}(a)$ in two ways, using Eq. (1) and Eq. (2).
$$
f(x)=3 x^{2}+4 x+2, \quad a=-1
$$

Lucas Finney

Numerade Educator

Problem 6

In Exercises $3-8,$ compute $f^{\prime}(a)$ in two ways, using Eq. (1) and Eq. (2).
$$
f(x)=x^{3}, \quad a=2
$$

Lucas Finney

Numerade Educator

Problem 7

In Exercises $3-8,$ compute $f^{\prime}(a)$ in two ways, using Eq. (1) and Eq. (2).
$$
f(x)=x^{3}+2 x, \quad a=1
$$

Lucas Finney

Numerade Educator

Problem 8

In Exercises $3-8,$ compute $f^{\prime}(a)$ in two ways, using Eq. (1) and Eq. (2).
$$
f(x)=\frac{1}{x}, \quad a=2
$$

Lucas Finney

Numerade Educator

Problem 9

In Exercises 9-12, refer to Figure $13 .$
$[Z$ Find the slope of the secant line through $(2, f(2))$ and (2.5, $f(2.5)$ ). Is it greater than or less than $f^{\prime}(2)$ ? Explain.

Lucas Finney

Numerade Educator

Problem 10

In Exercises 9-12, refer to Figure $13 .$
Estimate $\frac{f(2+h)-f(2)}{h}$ for $h=-0.5 .$ What does this quantity represent? Is it greater than or less than $f^{\prime}(2) ?$ Explain.

Lucas Finney

Numerade Educator

Problem 11

In Exercises 9-12, refer to Figure $13 .$
Estimate $f^{\prime}(1)$ and $f^{\prime}(2)$.

Lucas Finney

Numerade Educator

Problem 12

In Exercises 9-12, refer to Figure $13 .$
Find a value of $h$ for which $\frac{f(2+h)-f(2)}{h}=0$.

Lucas Finney

Numerade Educator

Problem 13

In Exercises $13-16,$ refer to Figure $14 .$
Determine $f^{\prime}(a)$ for $a=1,2,4,7$

Lucas Finney

Numerade Educator

Problem 14

Determine $f^{\prime}(a)$ for $a=1,2,4,7$
For which values of $x$ is $f^{\prime}(x)<0 ?$

Lucas Finney

Numerade Educator

Problem 15

In Exercises $13-16,$ refer to Figure $14 .$
Which is larger, $f^{\prime}(5.5)$ or $f^{\prime}(6.5) ?$

Lucas Finney

Numerade Educator

Problem 16

In Exercises $13-16,$ refer to Figure $14 .$
Show that $f^{\prime}(3)$ does not exist.

Lucas Finney

Numerade Educator

Problem 17

In Exercises $17-20$, use the limit definition to calculate the derivative of the linear function.
$$
f(x)=7 x-9
$$

Lucas Finney

Numerade Educator

Problem 18

In Exercises $17-20$, use the limit definition to calculate the derivative of the linear function.]$$
f(x)=12
$$

Lucas Finney

Numerade Educator

Problem 19

In Exercises $17-20$, use the limit definition to calculate the derivative of the linear function.
$$
g(t)=8-3 t
$$

Lucas Finney

Numerade Educator

Problem 20

In Exercises $17-20$, use the limit definition to calculate the derivative of the linear function.
$$
k(z)=14 z+12
$$

Lucas Finney

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 graph of $y=2 x+8$

Lucas Finney

Numerade Educator

Problem 24

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

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

In Exercises $29-46,$ use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line.
$$
f(x)=2 x^{2}+10 x, \quad a=3
$$

Lucas Finney

Numerade Educator

Problem 30

In Exercises $29-46,$ use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line.
$$
f(x)=4-x^{2}, \quad a=-1
$$

Lucas Finney

Numerade Educator

Problem 31

In Exercises $29-46,$ use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line.
$$
f(t)=t-2 t^{2}, \quad a=3
$$

Lucas Finney

Numerade Educator

Problem 32

In Exercises $29-46,$ use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line.
$$
f(x)=8 x^{3}, \quad a=1
$$

Lucas Finney

Numerade Educator

Problem 33

In Exercises $29-46,$ use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line.
$$
f(x)=x^{3}+x, \quad a=0
$$

Lucas Finney

Numerade Educator

Problem 34

In Exercises $29-46,$ use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line.
$$
f(t)=2 t^{3}+4 t, \quad a=4
$$

Lucas Finney

Numerade Educator

Problem 35

In Exercises $29-46,$ use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line.
$$
f(x)=x^{-1}, \quad a=8
$$

Lucas Finney

Numerade Educator

Problem 36

In Exercises $29-46,$ use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line.
$$
f(x)=x+x^{-1}, \quad a=4
$$

Lucas Finney

Numerade Educator

Problem 37

In Exercises $29-46,$ use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line.
$$
f(x)=\frac{1}{x+3}, \quad a=-2
$$

Lucas Finney

Numerade Educator

Problem 38

In Exercises $29-46,$ use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line.
$$
f(t)=\frac{2}{1-t}, \quad a=-1
$$

Lucas Finney

Numerade Educator

Problem 39

In Exercises $29-46,$ use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line.
$$
f(x)=\sqrt{x+4}, \quad a=1
$$

Lucas Finney

Numerade Educator

Problem 40

In Exercises $29-46,$ use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line.
$$
f(t)=\sqrt{3 t+5}, \quad a=-1
$$

Lucas Finney

Numerade Educator

Problem 41

In Exercises $29-46,$ use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line.
$$
f(x)=\frac{1}{\sqrt{x}}, \quad a=4
$$

Lucas Finney

Numerade Educator

Problem 42

In Exercises $29-46,$ use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line.
$$
f(x)=\frac{1}{\sqrt{2 x+1}}, \quad a=4
$$

Lucas Finney

Numerade Educator

Problem 43

In Exercises $29-46,$ use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line.
$$
f(t)=\sqrt{t^{2}+1}, \quad a \leq 3
$$

Lucas Finney

Numerade Educator

Problem 44

In Exercises $29-46,$ use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line.
$$
f(x)=x^{-2}, \quad a=-1
$$

Lucas Finney

Numerade Educator

Problem 45

In Exercises $29-46,$ use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line.
$$
f(x)=\frac{1}{x^{2}+1}, \quad a=0
$$

Lucas Finney

Numerade Educator

Problem 46

In Exercises $29-46,$ use the limit definition to compute $f^{\prime}(a)$ and find an equation of the tangent line.
$$
f(t)=t^{-3}, \quad a=1
$$

Lucas Finney

Numerade Educator

Problem 47

Show that $f$ is not differentiable at $x=1$ and has a comer in its graph there.
$$
f(x)=\left\{\begin{array}{ll}
1 & x \leq 1 \\
x^{2} & x>1
\end{array}\right.
$$

Lucas Finney

Numerade Educator

Problem 48

Show that $f$ is not differentiable at $x=0$ and has a corner in its graph there.
$$
f(x)=\left\{\begin{array}{ll}
x^{3} & x \leq 0 \\
x & x>0
\end{array}\right.
$$

Lucas Finney

Numerade Educator

Problem 49

In Exercises $49-51,$ sketch a graph of $f$ and identify the points c such that $f^{\prime}(c)$ does not exist. In which cases is there a corner at $c ?$
$$
f(x)=|x+3|
$$

Lucas Finney

Numerade Educator

Problem 50

In Exercises $49-51,$ sketch a graph of $f$ and identify the points c such that $f^{\prime}(c)$ does not exist. In which cases is there a corner at $c ?$
$$
f(x)=x^{2 / 5}
$$

Lucas Finney

Numerade Educator

Problem 51

In Exercises $49-51,$ sketch a graph of $f$ and identify the points c such that $f^{\prime}(c)$ does not exist. In which cases is there a corner at $c ?$
$$
f(x)=\left|x^{2}-4\right|
$$

Lucas Finney

Numerade Educator

Problem 52

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 $f^{\prime}(16)$ with the limit definition and compare with your estimate.

Lucas Finney

Numerade Educator

Problem 53

GU Let $f(x)=\frac{4}{1+2^{x}} .$ Plot $f$ over $[-2,2] .$ Then zoom in near $x=0$ until the graph appears straight, and estimate the slope $f^{\prime}(0)$.

Lucas Finney

Numerade Educator

Problem 54

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

Carson Merrill

Numerade Educator

Problem 55

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 56

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 57

In Exercises $57-62,$ each limit represents a derivative $f^{\prime}(a)$. Find $f(x)$ and $a$.
$$
\lim _{h \rightarrow 0} \frac{(5+h)^{3}-125}{h}
$$

Lucas Finney

Numerade Educator

Problem 58

In Exercises $57-62,$ each limit represents a derivative $f^{\prime}(a)$. Find $f(x)$ and $a$.
$$
\lim _{x \rightarrow 5} \frac{x^{3}-125}{x-5}
$$

Lucas Finney

Numerade Educator

Problem 59

In Exercises $57-62,$ each limit represents a derivative $f^{\prime}(a)$. Find $f(x)$ and $a$.
$$
\lim _{h \rightarrow 0} \frac{\sin \left(\frac{\pi}{6}+h\right)-0.5}{h}
$$

Lucas Finney

Numerade Educator

Problem 60

In Exercises $57-62,$ each limit represents a derivative $f^{\prime}(a)$. Find $f(x)$ and $a$.
$$
\lim _{x \rightarrow \frac{1}{4}} \frac{x^{-1}-4}{x-\frac{1}{4}}
$$

Lucas Finney

Numerade Educator

Problem 61

In Exercises $57-62,$ each limit represents a derivative $f^{\prime}(a)$. Find $f(x)$ and $a$.
$$
\lim _{h \rightarrow 0} \frac{5^{2+h}-25}{h}
$$

Lucas Finney

Numerade Educator

Problem 62

In Exercises $57-62,$ each limit represents a derivative $f^{\prime}(a)$. Find $f(x)$ and $a$.
$$
\lim _{h \rightarrow 0} \frac{5^{h}-1}{h}
$$

Lucas Finney

Numerade Educator

Problem 63

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 Wisusik

Numerade Educator

Problem 64

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 $f$ to explain how the method works in this case.

Subham Jyoti Mishra

Subham Jyoti Mishra

Numerade Educator

Problem 65

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 66

$\quad$ Refer to the graph of $f(x)=2^{x}$ in Figure 18 .
(a) Explain graphically why, for $h>0$, $$ \frac{f(-h)-f(0)}{-h} \leq f^{\prime}(0) \leq \frac{f(h)-f(0)}{h} $$
(b) Use (a) to show that $0.69314 \leq f^{\prime}(0) \leq 0.69315$.
(c) Similarly, compute $f^{\prime}(x)$ to four decimal places for $x=1,2,3,4$.
(d) Now compute the ratios $f^{\prime}(x) / f^{\prime}(0)$ for $x=1,2,3,4$. Can you guess an approximate formula for $f^{\prime}(x) ?$

Carson Merrill

Numerade Educator

Problem 67

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

Carson Merrill

Numerade Educator

Problem 68

GU) Verify that $P=\left(1, \frac{1}{2}\right)$ lies on the graphs of both $f(x)=1 /\left(1+x^{2}\right) \quad$ and $\quad L(x)=\frac{1}{2}+m(x-1)$ 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 $f^{\prime}(1) ?$

Carson Merrill

Numerade Educator

Problem 69

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

Lucas Finney

Numerade Educator

Problem 70

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

Lucas Finney

Numerade Educator

Problem 71

The vapor pressure of water at temperature $T$ (in kelvins) is the atmospheric pressure $P$ at which no net evaporation takes place. In Exercises $71-72$, use the following table to estimate the indicated derivatives using the difference quotient approximation.. Estimate $P^{\prime}(T)$ for $T=293,313,333$. (Include proper units on the derivative.)

Carson Merrill

Numerade Educator

Problem 72

The vapor pressure of water at temperature $T$ (in kelvins) is the atmospheric pressure $P$ at which no net evaporation takes place. In Exercises $71-72$, use the following table to estimate the indicated derivatives using the difference quotient approximation.
Estimate $P^{\prime}(T)$ for $T=303,323,343$. (Include proper units on the derivative.)

Carson Merrill

Numerade Educator

Problem 73

Let $P(t)$ represent the U.S. ethanol production as shown in Figure 19. $\ln$ Exercises $73-74$, estimate the indicated derivatives using the difference quotient approximation.
Estimate $P^{\prime}(t)$ for $t=1997,2001,2005,2009$. (Include proper units on the derivative.)

Carson Merrill

Numerade Educator

Problem 74

Let $P(t)$ represent the U.S. ethanol production as shown in Figure 19. $\ln$ Exercises $73-74$, estimate the indicated derivatives using the difference quotient approximation.
Estimate $P^{\prime}(t)$ for $t=1999,2003,2007,2011$. (Include proper units on the derivative.)

Carson Merrill

Numerade Educator

Problem 75

With $P(T)$ as in Exercises 71 and 72 , estimate $P^{\prime}(T)$ for $T=303$, $313,333,343,$ now using the $\mathrm{SDQ}$

Carson Merrill

Numerade Educator

Problem 76

With $P(t)$ as in Exercises 73 and 74 , estimate $P^{\prime}(t)$ for $t=1999$, $2001,2005,2011,$ now using the $\mathrm{SDQ}$

Carson Merrill

Numerade Educator

Problem 77

In Exercises $77-78,$ traffic speed $S$ along Kaman Road (in kilometers per hour) varies as a function of traffic density $q$ (number of cars per kilometer of road) according to the data:
Estimate $S^{\prime}(80)$ using the $\mathrm{SDQ}$. (Include proper units on the derivative.)

Carson Merrill

Numerade Educator

Problem 78

In Exercises $77-78,$ traffic speed $S$ along Kaman Road (in kilometers per hour) varies as a function of traffic density $q$ (number of cars per kilometer of road) according to the data:
$\quad$ Explain why $V=q S$, called traffic volume, is equal to the number of cars passing a point per hour. Use the data and the SDQ to estimate $V^{\prime}(80)$. (Include proper units on the derivative.)

Carson Merrill

Numerade Educator

Problem 79

Exercises $79-81:$ The current (in amperes) at time t (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, $\underline{F}$ ), and $R$ the resistance (in ohms, $\Omega) .$
Calculate the current at $t=3$ if
$$
v(t)=0.5 t+4 \mathrm{~V}
$$
where $C=0.01 \mathrm{~F}$ and $R=100 \Omega$

Linda Hand

Numerade Educator

Problem 80

Exercises $79-81:$ The current (in amperes) at time t (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, $\underline{F}$ ), and $R$ the resistance (in ohms, $\Omega) .$
Use the following data and the SDQ to estimate $v^{\prime}(10)$. Then estimate $i(10),$ assuming $C=0.03$ and $R=1000$.

Carson Merrill

Numerade Educator

Problem 81

Exercises $79-81:$ The current (in amperes) at time t (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, $\underline{F}$ ), and $R$ the resistance (in ohms, $\Omega) .$
Assume that $R=200 \Omega$ but $C$ is unknown. Use the following data and the $\mathrm{SDQ}$ to estimate $v^{\prime}(4),$ and deduce an approximate value for the capacitance $C$.

Carson Merrill

Numerade Educator

Problem 82

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 $\mathrm{SDQ}$ with $h=0.001$ and the ordinary difference quotients with $h=\pm 0.001 .$ Compare with the actual value, which is $f^{\prime}(0)=\ln 2 .$

Carson Merrill

Numerade Educator

Problem 83

(a) Show that the symmetric difference quotient $\frac{f(x+h)-f(x-h)}{2 h}$ is the slope of the secant line to the graph of $f$ from $x-h$ to $x+h$. (Include an illustration.)
(b) Prove that the symmetric difference quotient is the average of the slopes of the secant lines from $x$ to $x+h$ and from $x-h$ to $x$.

Linda Hand

Numerade Educator

Problem 84

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 85

Show that if $f$ is a quadratic polynomial, then the $\mathrm{SDQ}$ 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 86

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

Bahar Tehranipoor

Bahar Tehranipoor

Numerade Educator