University Calculus: Early Transcendentals 4th

Joel Hass, Christopher Heil, Przemyslaw Bogacki

Chapter 3

Derivatives

Educators


Problem 1

Use the grid and a straight edge to make a rough estimate of the slope of the curve (in $y$ -units per $x$ -unit) at the points $P_{1}$ and $P_{2}$
(GRAPH CAN'T COPY)

Check back soon!

Problem 2

Use the grid and a straight edge to make a rough estimate of the slope of the curve (in $y$ -units per $x$ -unit) at the points $P_{1}$ and $P_{2}$
(GRAPH CAN'T COPY)

Check back soon!

Problem 3

Use the grid and a straight edge to make a rough estimate of the slope of the curve (in $y$ -units per $x$ -unit) at the points $P_{1}$ and $P_{2}$
(GRAPH CAN'T COPY)

Check back soon!

Problem 4

Use the grid and a straight edge to make a rough estimate of the slope of the curve (in $y$ -units per $x$ -unit) at the points $P_{1}$ and $P_{2}$
(GRAPH CAN'T COPY)

Check back soon!

Problem 5

Find an equation for the tangent line to the curve at the given point. Then sketch the curve and tangent line together.
$$y=4-x^{2}, \quad(-1,3)$$

Check back soon!

Problem 6

Find an equation for the tangent line to the curve at the given point. Then sketch the curve and tangent line together.
$$y=(x-1)^{2}+1$$

Christopher S.
Numerade Educator

Problem 7

Find an equation for the tangent line to the curve at the given point. Then sketch the curve and tangent line together.
$$y=2 \sqrt{x}$$

Christopher S.
Numerade Educator

Problem 8

Find an equation for the tangent line to the curve at the given point. Then sketch the curve and tangent line together.
$$y=\frac{1}{x^{2}}, \quad(-1,1)$$

Check back soon!

Problem 9

Find an equation for the tangent line to the curve at the given point. Then sketch the curve and tangent line together.
$$y=x^{3}, \quad(-2,-8)$$

Check back soon!

Problem 10

Find an equation for the tangent line to the curve at the given point. Then sketch the curve and tangent line together.
$$y=\frac{1}{x^{3}}, \quad\left(-2,-\frac{1}{8}\right)$$

Check back soon!

Problem 11

Find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there.
$$f(x)=x^{2}+1, \quad(2,5)$$

Check back soon!

Problem 12

Find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there.
$$f(x)=x-2 x^{2}, \quad(1,-1)$$

Check back soon!

Problem 13

Find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there.
$$g(x)=\frac{x}{x-2}$$

Christopher S.
Numerade Educator

Problem 14

Find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there.
$$g(x)=\frac{8}{x^{2}}$$

Christopher S.
Numerade Educator

Problem 15

Find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there.
$$h(t)=t^{3}, \quad(2,8)$$

Check back soon!

Problem 16

Find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there.
$$h(t)=t^{3}+3 t$$

Christopher S.
Numerade Educator

Problem 17

Find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there.
$$f(x)=\sqrt{x}, \quad(4,2)$$

Check back soon!

Problem 18

Find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there.
$$f(x)=\sqrt{x+1}$$

Christopher S.
Numerade Educator

Problem 19

Find the slope of the curve at the point indicated.
$$y=5 x-3 x^{2}, \quad x=1$$

Check back soon!

Problem 20

Find the slope of the curve at the point indicated.
$$y=x^{3}-2 x+7, x=-2$$

Check back soon!

Problem 21

Find the slope of the curve at the point indicated.
$$y=\frac{1}{x-1}, \quad x=3$$

Check back soon!

Problem 22

Find the slope of the curve at the point indicated.
$$y=\frac{x-1}{x+1}, x=0$$

Check back soon!

Problem 23

Growth of yeast cells In a controlled laboratory experiment, yeast cells are grown in an automated cell culture system that counts the number $P$ of cells present at hourly intervals. The number after $t$ hours is shown in the accompanying figure.
a. Explain what is meant by the derivative $P^{\prime}(5) .$ What are its units?
b. Which is larger, $P^{\prime}(2)$ or $P^{\prime}(3) ?$ Give a reason for your answer.
c. The quadratic curve capturing the trend of the data points (see Section 1.4 ) is given by $P(t)=6.10 t^{2}-9.28 t+16.43$ Find the instantaneous rate of growth when $t=5$ hours.
(FIGURE CAN'T COPY)

Check back soon!

Problem 24

Effectiveness of a drug On a scale from 0 to 1 , the effectiveness E of a pain-killing drug $t$ hours after entering the bloodstream is displayed in the accompanying figure.
a. At what times does the effectiveness appear to be increasing? What is true about the derivative at those times?
b. At what time would you estimate that the drug reaches its maximum effectiveness? What is true about the derivative at that time? What is true about the derivative as time increases
in the 1 hour before your estimated time?

Check back soon!

Problem 25

At what points do the graphs of the functions have a horizontal tangent lines?
$$f(x)=x^{2}+4 x-1$$

Check back soon!

Problem 26

At what points do the graphs of the functions have a horizontal tangent lines?
$$g(x)=x^{3}-3 x$$

Check back soon!

Problem 27

Find equations of all lines having slope -1 that are tangent to the curve $y=1 /(x-1)$

Check back soon!

Problem 28

Find an equation of the straight line having slope $1 / 4$ that is tangent to the curve $y=\sqrt{x}$

Check back soon!

Problem 29

Object dropped from a tower An object is dropped from the top of a $100-\mathrm{m}$ -high tower. Its height above ground after $t$ sec is $100-4.9 t^{2} \mathrm{m} .$ How fast is it falling 2 sec after it is dropped?

Check back soon!

Problem 30

Speed of a rocket At $t$ sec after liftoff, the height of a rocket is
$3 t^{2} \mathrm{ft} .$ How fast is the rocket climbing 10 sec after liftoff?

Christopher S.
Numerade Educator

Problem 31

Circle's changing area What is the rate of change of the area of a circle $\left(A=\pi r^{2}\right)$ with respect to the radius when the radius is $r=3 ?$

Check back soon!

Problem 32

Ball's changing volume What is the rate of change of the volume of a ball $\left(V=(4 / 3) \pi r^{3}\right)$ with respect to the radius when the radius is $r=2 ?$

Check back soon!

Problem 33

Show that the line $y=m x+b$ is its own tangent line at any point $\left(x_{1}, m x_{0}+b\right)$

Christopher S.
Numerade Educator

Problem 34

Find the slope of the tangent line to the curve $y=1 / \sqrt{x}$ at the point where $x=4$

Check back soon!

Problem 35

Does the graph of
$$
f(x)=\left\{\begin{array}{ll}
x^{2} \sin (1 / x), & x \neq 0 \\
0, & x=0
\end{array}\right.
$$
have a tangent line at the origin? Give reasons for your answer.

Check back soon!

Problem 36

Does the graph of
$$
g(x)=\left\{\begin{array}{ll}
x \sin (1 / x), & x \neq 0 \\
0, & x=0
\end{array}\right.
$$
have a tangent line at the origin? Give reasons for your answer.

Check back soon!

Problem 37

However, $y=x^{2 / 3}$ has no vertical tangent line at $x=0$ (see next figure):
$$
\begin{aligned}
\lim _{h \rightarrow 0} \frac{g(0+h)-g(0)}{h} &=\lim _{h \rightarrow 0} \frac{h^{2 / 3}-0}{h} \\
&=\lim _{h \rightarrow 0} \frac{1}{h^{1 / 3}}
\end{aligned}
$$
does not exist, because the limit is $\infty$ from the right and $-\infty$ from the Ieft.
Does the graph of
$$
f(x)=\left\{\begin{aligned}
-1, & x<0 \\
0, & x=0 \\
1, & x>0
\end{aligned}\right.
$$
have a vertical tangent line at the origin? Give reasons for your answer.
(GRAPH CAN'T COPY)

Check back soon!

Problem 38

We say that a continuous curve $y=f(x)$ has a vertical tangent line at the point where $x=x_{0}$ if the limit of the difference quotient is $\infty$ or - $\infty .$ For example, $y=x^{1 / 3}$ has a vertical tangent line at $x=0$ (see accompanying figure):
$$
\begin{aligned}
\lim _{h \rightarrow 0} \frac{f(0+h)-f(0)}{h} &=\lim _{h \rightarrow 0} \frac{h^{1 / 3}-0}{h} \\
&=\lim _{h \rightarrow 0} \frac{1}{h^{2 / 3}}=\infty
\end{aligned}
$$
However, $y=x^{2 / 3}$ has no vertical tangent line at $x=0$ (see next figure):
$$
\begin{aligned}
\lim _{h \rightarrow 0} \frac{g(0+h)-g(0)}{h} &=\lim _{h \rightarrow 0} \frac{h^{2 / 3}-0}{h} \\
&=\lim _{h \rightarrow 0} \frac{1}{h^{1 / 3}}
\end{aligned}
$$
does not exist, because the limit is $\infty$ from the right and $-\infty$ from the Ieft.
Does the graph of
$$
U(x)=\left\{\begin{array}{ll}
0, & x<0 \\
1, & x \geq 0
\end{array}\right.
$$
have a vertical tangent line at the point (0,1)$?$ Give reasons for your answer

Check back soon!

Problem 39

Graph the curves
a. Where do the graphs appear to have vertical tangent lines?
b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to Exercises 37 and 38
$$y=x^{2 / 3}$$

Check back soon!

Problem 40

Graph the curves
a. Where do the graphs appear to have vertical tangent lines?
b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to Exercises 37 and 38
$$y=x^{4 / 5}$$

Check back soon!

Problem 41

Graph the curves
a. Where do the graphs appear to have vertical tangent lines?
b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to Exercises 37 and 38
$$y=x^{1 / 3}$$

Check back soon!

Problem 42

Graph the curves
a. Where do the graphs appear to have vertical tangent lines?
b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to Exercises 37 and 38
$$y=x^{3 / 5}$$

Check back soon!

Problem 43

Graph the curves
a. Where do the graphs appear to have vertical tangent lines?
b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to Exercises 37 and 38
$$y=4 x^{2 / 5}-2 x$$

Check back soon!

Problem 44

Graph the curves
a. Where do the graphs appear to have vertical tangent lines?
b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to Exercises 37 and 38
$$y=x^{5 / 3}-5 x^{2 / 3}$$

Check back soon!

Problem 45

Graph the curves
a. Where do the graphs appear to have vertical tangent lines?
b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to Exercises 37 and 38
$$y=x^{2 / 3}-(x-1)^{1 / 3}$$

Check back soon!

Problem 46

Graph the curves
a. Where do the graphs appear to have vertical tangent lines?
b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to Exercises 37 and 38
$$y=x^{1 / 3}+(x-1)^{1 / 3}$$

Check back soon!

Problem 47

Graph the curves
a. Where do the graphs appear to have vertical tangent lines?
b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to Exercises 37 and 38
$$y=\left\{\begin{array}{ll}
-\sqrt{|x|}, & x \leq 0 \\
\sqrt{x}, & x>0
\end{array}\right.$$

Check back soon!

Problem 48

Graph the curves
a. Where do the graphs appear to have vertical tangent lines?
b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to Exercises 37 and 38
$$y=\sqrt{|4-x|}$$

Check back soon!

Problem 49

Use a CAS to perform the following steps for the functions.
a. Plot $y=f(x)$ over the interval $\left(x_{0}-1 / 2\right) \leq x \leq\left(x_{0}+3\right)$
b. Holding $x_{0}$ fixed, the difference quotient
$$
q(h)=\frac{f\left(x_{0}+h\right)-f\left(x_{0}\right)}{h}
$$
at $x_{0}$ becomes a function of the step size $h .$ Enter this function into your CAS workspace.
c. Find the limit of $q$ as $h \rightarrow 0$
d. Define the secant lines $y=f\left(x_{0}\right)+q \cdot\left(x-x_{0}\right)$ for $h=3,2$ and $1 .$ Graph them, together with $f$ and the tangent line, over the interval in part (a).
$$f(x)=x^{3}+2 x, \quad x_{0}=0$$

Check back soon!

Problem 50

Use a CAS to perform the following steps for the functions.
a. Plot $y=f(x)$ over the interval $\left(x_{0}-1 / 2\right) \leq x \leq\left(x_{0}+3\right)$
b. Holding $x_{0}$ fixed, the difference quotient
$$
q(h)=\frac{f\left(x_{0}+h\right)-f\left(x_{0}\right)}{h}
$$
at $x_{0}$ becomes a function of the step size $h .$ Enter this function into your CAS workspace.
c. Find the limit of $q$ as $h \rightarrow 0$
d. Define the secant lines $y=f\left(x_{0}\right)+q \cdot\left(x-x_{0}\right)$ for $h=3,2$ and $1 .$ Graph them, together with $f$ and the tangent line, over the interval in part (a).
$$f(x)=x+\frac{3}{x}, x_{0}=1$$

Check back soon!

Problem 51

Use a CAS to perform the following steps for the functions.
a. Plot $y=f(x)$ over the interval $\left(x_{0}-1 / 2\right) \leq x \leq\left(x_{0}+3\right)$
b. Holding $x_{0}$ fixed, the difference quotient
$$
q(h)=\frac{f\left(x_{0}+h\right)-f\left(x_{0}\right)}{h}
$$
at $x_{0}$ becomes a function of the step size $h .$ Enter this function into your CAS workspace.
c. Find the limit of $q$ as $h \rightarrow 0$
d. Define the secant lines $y=f\left(x_{0}\right)+q \cdot\left(x-x_{0}\right)$ for $h=3,2$ and $1 .$ Graph them, together with $f$ and the tangent line, over the interval in part (a).
$$f(x)=x+\sin (2 x), \quad x_{0}=\pi / 2$$

Check back soon!

Problem 52

Use a CAS to perform the following steps for the functions.
a. Plot $y=f(x)$ over the interval $\left(x_{0}-1 / 2\right) \leq x \leq\left(x_{0}+3\right)$
b. Holding $x_{0}$ fixed, the difference quotient
$$
q(h)=\frac{f\left(x_{0}+h\right)-f\left(x_{0}\right)}{h}
$$
at $x_{0}$ becomes a function of the step size $h .$ Enter this function into your CAS workspace.
c. Find the limit of $q$ as $h \rightarrow 0$
d. Define the secant lines $y=f\left(x_{0}\right)+q \cdot\left(x-x_{0}\right)$ for $h=3,2$ and $1 .$ Graph them, together with $f$ and the tangent line, over the interval in part (a).
$$f(x)=\cos x+4 \sin (2 x), \quad x_{0}=\pi$$

Check back soon!