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Calculus Volume 1

Gilbert Strang

Chapter 3

Derivatives

Educators


Problem 1

For the following exercises, use Equation 3.3 to find the slope of the secant line between the values $x_{1}$ and $x_{2}$ for each function $y=f(x)$
$$
f(x)=4 x+7 ; x_{1}=2, x_{2}=5
$$

Erin O.
Numerade Educator

Problem 2

For the following exercises, use Equation 3.3 to find the slope of the secant line between the values $x_{1}$ and $x_{2}$ for each function $y=f(x)$
$$
f(x)=8 x-3 ; x_{1}=-1, x_{2}=3
$$

Amy J.
Numerade Educator

Problem 3

For the following exercises, use Equation 3.3 to find the slope of the secant line between the values $x_{1}$ and $x_{2}$ for each function $y=f(x)$
$$
f(x)=x^{2}+2 x+1 ; x_{1}=3, x_{2}=3.5
$$

Brian K.
Numerade Educator

Problem 4

For the following exercises, use Equation 3.3 to find the slope of the secant line between the values $x_{1}$ and $x_{2}$ for each function $y=f(x)$
$$
f(x)=-x^{2}+x+2 ; x_{1}=0.5, x_{2}=1.5
$$

Amy J.
Numerade Educator

Problem 5

For the following exercises, use Equation 3.3 to find the slope of the secant line between the values $x_{1}$ and $x_{2}$ for each function $y=f(x)$
$$
f(x)=\frac{4}{3 x-1} ; x_{1}=1, x_{2}=3
$$

Brian K.
Numerade Educator

Problem 6

For the following exercises, use Equation 3.3 to find the slope of the secant line between the values $x_{1}$ and $x_{2}$ for each function $y=f(x)$
$$
f(x)=\frac{x-7}{2 x+1} ; x_{1}=-2, x_{2}=0
$$

Amy J.
Numerade Educator

Problem 7

For the following exercises, use Equation 3.3 to find the slope of the secant line between the values $x_{1}$ and $x_{2}$ for each function $y=f(x)$
$$
f(x)=\sqrt{x} ; x_{1}=1, x_{2}=16
$$

Brian K.
Numerade Educator

Problem 8

For the following exercises, use Equation 3.3 to find the slope of the secant line between the values $x_{1}$ and $x_{2}$ for each function $y=f(x)$
$$
f(x)=\sqrt{x-9} ; x_{1}=10, x_{2}=13
$$

Amy J.
Numerade Educator

Problem 9

For the following exercises, use Equation 3.3 to find the slope of the secant line between the values $x_{1}$ and $x_{2}$ for each function $y=f(x)$
$$
f(x)=x^{1 / 3}+1 ; x_{1}=0, x_{2}=8
$$

Brian K.
Numerade Educator

Problem 10

For the following exercises, use Equation 3.3 to find the slope of the secant line between the values $x_{1}$ and $x_{2}$ for each function $y=f(x)$
$$
f(x)=6 x^{2 / 3}+2 x^{1 / 3} ; x_{1}=1, x_{2}=27
$$

Amy J.
Numerade Educator

Problem 11

For the following functions,
a. use Equation 3.4 to find the slope of the tangent line $m_{\text { tan }}=f^{\prime}(a),$ and
b. find the equation of the tangent line to $f$ at $x=a$ .
$$
f(x)=3-4 x, a=2
$$

Brian K.
Numerade Educator

Problem 12

For the following functions,
a. use Equation 3.4 to find the slope of the tangent line $m_{\text { tan }}=f^{\prime}(a),$ and
b. find the equation of the tangent line to $f$ at $x=a$ .
$$
f(x)=\frac{x}{5}+6, a=-1
$$

Amy J.
Numerade Educator

Problem 13

For the following functions,
a. use Equation 3.4 to find the slope of the tangent line $m_{\text { tan }}=f^{\prime}(a),$ and
b. find the equation of the tangent line to $f$ at $x=a$ .
$$
f(x)=x^{2}+x, a=1
$$

Brian K.
Numerade Educator

Problem 14

For the following functions,
a. use Equation 3.4 to find the slope of the tangent line $m_{\text { tan }}=f^{\prime}(a),$ and
b. find the equation of the tangent line to $f$ at $x=a$ .
$$
f(x)=1-x-x^{2}, a=0
$$

Amy J.
Numerade Educator

Problem 15

For the following functions,
a. use Equation 3.4 to find the slope of the tangent line $m_{\text { tan }}=f^{\prime}(a),$ and
b. find the equation of the tangent line to $f$ at $x=a$ .
$$
f(x)=\frac{7}{x}, a=3
$$

Brian K.
Numerade Educator

Problem 16

For the following functions,
a. use Equation 3.4 to find the slope of the tangent line $m_{\text { tan }}=f^{\prime}(a),$ and
b. find the equation of the tangent line to $f$ at $x=a$ .
$$
f(x)=\sqrt{x+8}, a=1
$$

Amy J.
Numerade Educator

Problem 17

For the following functions,
a. use Equation 3.4 to find the slope of the tangent line $m_{\text { tan }}=f^{\prime}(a),$ and
b. find the equation of the tangent line to $f$ at $x=a$ .
$$
f(x)=2-3 x^{2}, a=-2
$$

Brian K.
Numerade Educator

Problem 18

For the following functions,
a. use Equation 3.4 to find the slope of the tangent line $m_{\text { tan }}=f^{\prime}(a),$ and
b. find the equation of the tangent line to $f$ at $x=a$ .
$$
f(x)=\frac{-3}{x-1}, a=4
$$

Amy J.
Numerade Educator

Problem 19

For the following functions,
a. use Equation 3.4 to find the slope of the tangent line $m_{\text { tan }}=f^{\prime}(a),$ and
b. find the equation of the tangent line to $f$ at $x=a$ .
$$
f(x)=\frac{2}{x+3}, a=-4
$$

Brian K.
Numerade Educator

Problem 20

For the following functions,
a. use Equation 3.4 to find the slope of the tangent line $m_{\text { tan }}=f^{\prime}(a),$ and
b. find the equation of the tangent line to $f$ at $x=a$ .
$$
f(x)=\frac{3}{x^{2}}, a=3
$$

Amy J.
Numerade Educator

Problem 21

For the following functions $y=f(x),$ find $f^{\prime}(a)$ using Equation $3.3 .$
$$
f(x)=5 x+4, a=-1
$$

Brian K.
Numerade Educator

Problem 22

For the following functions $y=f(x),$ find $f^{\prime}(a)$ using Equation $3.3 .$
$$
f(x)=-7 x+1, a=3
$$

Amy J.
Numerade Educator

Problem 23

Consider the below function,
$$
f(x)=x^{2}+9 x, a=2
$$
Find the slope of the tangent line $f^{\prime}(a)$ and the equation of the tangent line at $x=a$

Brian K.
Numerade Educator

Problem 24

For the following functions $y=f(x),$ find $f^{\prime}(a)$ using Equation $3.3 .$
$$
f(x)=3 x^{2}-x+2, a=1
$$

Amy J.
Numerade Educator

Problem 25

For the following functions $y=f(x),$ find $f^{\prime}(a)$ using Equation $3.3 .$
$$
f(x)=\sqrt{x}, a=4
$$

Brian K.
Numerade Educator

Problem 26

For the following functions $y=f(x),$ find $f^{\prime}(a)$ using Equation $3.3 .$
$$
f(x)=\sqrt{x-2}, a=6
$$

Amy J.
Numerade Educator

Problem 27

For the following functions $y=f(x),$ find $f^{\prime}(a)$ using Equation $3.3 .$
$$
f(x)=\frac{1}{x}, a=2
$$

Brian K.
Numerade Educator

Problem 28

For the following functions $y=f(x),$ find $f^{\prime}(a)$ using Equation $3.3 .$
$$
f(x)=\frac{1}{x-3}, a=-1
$$

Amy J.
Numerade Educator

Problem 29

For the following functions $y=f(x),$ find $f^{\prime}(a)$ using Equation $3.3 .$
$$
f(x)=\frac{1}{x^{3}}, a=1
$$

Brian K.
Numerade Educator

Problem 30

For the following functions $y=f(x),$ find $f^{\prime}(a)$ using Equation $3.3 .$
$$
f(x)=\frac{1}{\sqrt{x}}, a=4
$$

Amy J.
Numerade Educator

Problem 31

For the following exercises, given the function $y=f(x)$
a. find the slope of the secant line $P Q$ for each point $Q(x, f(x))$ with $x$ value given in the table.
b. Use the answers from a. to estimate the value of the slope of the tangent line at $P .$
c. Use the answer from b. to find the equation of the tangent line to $f$ at point $P .$
ITI $f(x)=x^{2}+3 x+4, P(1,8)$ (Round to decimal places.)

Brian K.
Numerade Educator

Problem 32

For the following exercises, given the function $y=f(x)$
a. find the slope of the secant line $P Q$ for each point $Q(x, f(x))$ with $x$ value given in the table.
b. Use the answers from a. to estimate the value of the slope of the tangent line at $P .$
c. Use the answer from b. to find the equation of the tangent line to $f$ at point $P .$
$$
[\mathrm{T}] f(x)=\frac{x+1}{x^{2}-1}, P(0,-1)
$$

Amy J.
Numerade Educator

Problem 33

For the following exercises, given the function $y=f(x)$
a. find the slope of the secant line $P Q$ for each point $Q(x, f(x))$ with $x$ value given in the table.
b. Use the answers from a. to estimate the value of the slope of the tangent line at $P .$
c. Use the answer from b. to find the equation of the tangent line to $f$ at point $P .$
[T] $f(x)=10 e^{0.5 x}, P(0,10)$ (Round to 4 decimal places.)

Brian K.
Numerade Educator

Problem 34

For the following exercises, given the function $y=f(x)$
a. find the slope of the secant line $P Q$ for each point $Q(x, f(x))$ with $x$ value given in the table.
b. Use the answers from a. to estimate the value of the slope of the tangent line at $P .$
c. Use the answer from b. to find the equation of the tangent line to $f$ at point $P .$
$$
[\mathrm{T}] f(x)=\tan (x), P(\pi, 0)
$$

Amy J.
Numerade Educator

Problem 35

IT] For the following position functions $y=s(t), \quad$ an object is moving along a straight line, where $t$ is in seconds and $s$ is in meters. Find
a. the simplified expression for the average velocity from $t=2$ to $t=2+h$
b. the average velocity between $t=2$ and $t=2+h, \quad$ where $(\mathrm{i}) h=0.1, \quad$ (ii) $h=0.01$ (iii) $h=0.001,$ and (iv) $h=0.0001 ;$ and
c. use the answer from a. to estimate the instantaneous velocity at $t=2$ second.
$$
s(t)=\frac{1}{3} t+5
$$

Brian K.
Numerade Educator

Problem 36

IT] For the following position functions $y=s(t), \quad$ an object is moving along a straight line, where $t$ is in seconds and $s$ is in meters. Find
a. the simplified expression for the average velocity from $t=2$ to $t=2+h$
b. the average velocity between $t=2$ and $t=2+h, \quad$ where $(\mathrm{i}) h=0.1, \quad$ (ii) $h=0.01$ (iii) $h=0.001,$ and (iv) $h=0.0001 ;$ and
c. use the answer from a. to estimate the instantaneous velocity at $t=2$ second.
$$
s(t)=t^{2}-2 t
$$

Amy J.
Numerade Educator

Problem 37

IT] For the following position functions $y=s(t), \quad$ an object is moving along a straight line, where $t$ is in seconds and $s$ is in meters. Find
a. the simplified expression for the average velocity from $t=2$ to $t=2+h$
b. the average velocity between $t=2$ and $t=2+h, \quad$ where $(\mathrm{i}) h=0.1, \quad$ (ii) $h=0.01$ (iii) $h=0.001,$ and (iv) $h=0.0001 ;$ and
c. use the answer from a. to estimate the instantaneous velocity at $t=2$ second.
$$
s(t)=2 t^{3}+3
$$

Brian K.
Numerade Educator

Problem 38

IT] For the following position functions $y=s(t), \quad$ an object is moving along a straight line, where $t$ is in seconds and $s$ is in meters. Find
a. the simplified expression for the average velocity from $t=2$ to $t=2+h$
b. the average velocity between $t=2$ and $t=2+h, \quad$ where $(\mathrm{i}) h=0.1, \quad$ (ii) $h=0.01$ (iii) $h=0.001,$ and (iv) $h=0.0001 ;$ and
c. use the answer from a. to estimate the instantaneous velocity at $t=2$ second.
$$
s(t)=\frac{16}{t^{2}}-\frac{4}{t}
$$

Amy J.
Numerade Educator

Problem 39

Use the following graph to evaluate a. $f^{\prime}(1)$ and b. $f^{\prime}(6) .$

Brian K.
Numerade Educator

Problem 40

Use the following graph to evaluate a. $f^{\prime}(-3)$ and b. $f^{\prime}(1.5) .$

Amy J.
Numerade Educator

Problem 41

For the following exercises, use the limit definition of derivative to show that the derivative does not exist at $x=a$ for each of the given functions.
$$
f(x)=x^{1 / 3}, x=0
$$

Brian K.
Numerade Educator

Problem 42

For the following exercises, use the limit definition of derivative to show that the derivative does not exist at $x=a$ for each of the given functions.
$$
f(x)=x^{2 / 3}, x=0
$$

Carson M.
Numerade Educator

Problem 43

For the following exercises, use the limit definition of derivative to show that the derivative does not exist at $x=a$ for each of the given functions.
$$
f(x)=\left\{\begin{array}{l}{1, x<1} \\ {x, x \geq 1}\end{array}, x=1\right.
$$

Brian K.
Numerade Educator

Problem 44

For the following exercises, use the limit definition of derivative to show that the derivative does not exist at $x=a$ for each of the given functions.
$$
f(x)=\frac{|x|}{x}, x=0
$$

Amy J.
Numerade Educator

Problem 45

[T] The position in feet of a race car along a straight track after $t$ seconds is modeled by the function
$$
s(t)=8 t^{2}-\frac{1}{16} t^{3}
$$
a. Find the average velocity of the vehicle over the following time intervals to four decimal places:
$$
\begin{array}{l}{\text { i. }[4,4.1]} \\ {\text { ii. }[4,4.01]} \\ {\text { iii. }[4,4.001]} \\ {\text { iv. }[4,4,0001]}\end{array}
$$
b. Use a. to draw a conclusion about the instantaneous velocity of the vehicle at $t=4$ seconds.

Brian K.
Numerade Educator

Problem 46

[T] The distance in feet that a ball rolls down an incline is modeled by the function $s(t)=14 t^{2}, \quad$ where $t$ is seconds after the ball begins rolling.
a. Find the average velocity of the ball over the following time intervals:
$$
\begin{array}{ll}{\text { i. }} & {[5,5.1]} \\ {\text { ii. }} & {[5,5.01]} \\ {\text { iii. }} & {[5,5.001]} \\ {\text { iv. }} & {[5,5.0001]}\end{array}
$$
b. Use the answers from a. to draw a conclusion about the instantaneous velocity of the ball at $t=5$
seconds.

Amy J.
Numerade Educator

Problem 47

Two vehicles start out traveling side by side along a straight road. Their position functions, shown in the
following graph, are given by $s=f(t)$ and $s=g(t)$ where $s$ is measured in feet and $t$ is measured in seconds.
a. Which vehicle has traveled farther at $t=2$ seconds?
b. What is the approximate velocity of each vehicle at $t=3$ seconds?
c. Which vehicle is traveling faster at $t=4$ seconds?
d. What is true about the positions of the vehicles at $t-4$ seconds?

Brian K.
Numerade Educator

Problem 48

[T] The total cost $C(x),$ in hundreds of dollars, to produce $x$ jars of mayonnaise is given by
$C(x)=0.000003 x^{3}+4 x+300$
a. Calculate the average cost per jar over the following intervals:
i. $[100,100.1]$
ii. $[100,100.01]$
iii. $[100,100.001]$
iv. $[100,100.0001]$
b. Use the answers from a. to estimate the average cost to produce 100 jars of mayonnaise.

Amy J.
Numerade Educator

Problem 49

[T] For the function $f(x)=x^{3}-2 x^{2}-11 x+12$ do the following.
a. Use a graphing calculator to graph $f$ in an appropriate viewing window.
b. Use the ZOOM feature on the calculator to approximate the two values of $x=a$ for which
$m_{\text { tan }}=f^{\prime}(a)=0 .$

Norman A.
Numerade Educator

Problem 50

[T] For the function $f(x)=\frac{x}{1+x^{2}}, \quad$ do the following.
a. Use a graphing calculator to graph $f$ in an appropriate viewing window.
b. Use the ZOOM feature on the calculator to approximate the values of $x=a$ for which
$m_{\text { tan }}=f^{\prime}(a)=0$

Amy J.
Numerade Educator

Problem 51

Suppose that $N(x)$ computes the number of gallons of gas used by a vehicle traveling $x$ miles. Suppose the vehicle gets 30 $\mathrm{mpg.}$
a. Find a mathematical expression for $N(x)$
b. What is $N(100) ?$ Explain the physical meaning.
c. What is $N^{\prime}(100) ?$ Explain the physical meaning.

Brian K.
Numerade Educator

Problem 52

ITI For the function $f(x)=x^{4}-5 x^{2}+4,$ do the following.
a. Use a graphing calculator to graph $f$ in an appropriate viewing window.
b. Use the nDeriv function, which numerically finds the derivative, on a graphing calculator to estimate
$f^{\prime}(-2), f^{\prime}(-0.5), f^{\prime}(1.7),$ and $f^{\prime}(2.718)$ .

Amy J.
Numerade Educator

Problem 53

$[\mathrm{T}]$ For the function $f(x)=\frac{x^{2}}{x^{2}+1}, \quad$ do the following.
a. Use a graphing calculator to graph $f$ in an appropriate viewing window.
b. Use the nDeriv function on a graphing calculator to find $f^{\prime}(-4), f^{\prime}(-2), f^{\prime}(2),$ and $f^{\prime}(4)$ .

Norman A.
Numerade Educator