Calculus: Early Transcendental Functions

Robert Smith, Roland Minton

Chapter 2

Differentiation - all with Video Answers

Educators

+ 2 more educators

Section 1

Tangent Lines and Velocity

Problem 1

Use Definition 1.1 to find an equation of the tangent line to $y=f(x)$ at $x=a$. Graph $y=f(x)$ and the tangent line to verify that you have the correct equation.
$f(x)=x^{2}-2, a=1$

Lucas Finney

Lucas Finney

Numerade Educator

Problem 2

Use Definition 1.1 to find an equation of the tangent line to $y=f(x)$ at $x=a$. Graph $y=f(x)$ and the tangent line to verify that you have the correct equation.
$f(x)=x^{2}-2, a=0$

Lucas Finney

Numerade Educator

Problem 3

Use Definition 1.1 to find an equation of the tangent line to $y=f(x)$ at $x=a$. Graph $y=f(x)$ and the tangent line to verify that you have the correct equation.
$f(x)=x^{2}-3 x, a=-2$

Lucas Finney

Numerade Educator

Problem 4

Use Definition 1.1 to find an equation of the tangent line to $y=f(x)$ at $x=a$. Graph $y=f(x)$ and the tangent line to verify that you have the correct equation.
$f(x)=x^{3}+x, a=1$

Lucas Finney

Numerade Educator

Problem 5

Use Definition 1.1 to find an equation of the tangent line to $y=f(x)$ at $x=a$. Graph $y=f(x)$ and the tangent line to verify that you have the correct equation.
$f(x)=\frac{2}{x+1}, a=1$

Lucas Finney

Numerade Educator

Problem 6

Use Definition 1.1 to find an equation of the tangent line to $y=f(x)$ at $x=a$. Graph $y=f(x)$ and the tangent line to verify that you have the correct equation.
$f(x)=\frac{x}{x-1}, a=0$

Lucas Finney

Numerade Educator

Problem 7

Use Definition 1.1 to find an equation of the tangent line to $y=f(x)$ at $x=a$. Graph $y=f(x)$ and the tangent line to verify that you have the correct equation.
$f(x)=\sqrt{x+3}, a=-2$

Lucas Finney

Numerade Educator

Problem 8

Use Definition 1.1 to find an equation of the tangent line to $y=f(x)$ at $x=a$. Graph $y=f(x)$ and the tangent line to verify that you have the correct equation.
$f(x)=\sqrt{x+3}, a=1$

Lucas Finney

Numerade Educator

Problem 9

Compute the slope of the secant line between the points at (a) $x=1$ and $x=2,$ (b) $x=2$ and $x=3$, (c) $x=1.5$ and $x=2,$ (d) $x=2$ and $x=2.5,(\mathrm{e}) x=1.9$ and $x=2,(f) x=2$ and $x=2.1,$ and $(g)$ use parts $(a)-(f)$ and other calculations as needed to estimate the slope of the tangent line at $x=2$.
$f(x)=x^{3}-x$

Andrew Bassila

Numerade Educator

Problem 10

Compute the slope of the secant line between the points at (a) $x=1$ and $x=2,$ (b) $x=2$ and $x=3$, (c) $x=1.5$ and $x=2,$ (d) $x=2$ and $x=2.5,(\mathrm{e}) x=1.9$ and $x=2,(f) x=2$ and $x=2.1,$ and $(g)$ use parts $(a)-(f)$ and other calculations as needed to estimate the slope of the tangent line at $x=2$.
$f(x)=\sqrt{x^{2}+1}$

Andrew Bassila

Numerade Educator

Problem 11

Compute the slope of the secant line between the points at (a) $x=1$ and $x=2,$ (b) $x=2$ and $x=3$, (c) $x=1.5$ and $x=2,$ (d) $x=2$ and $x=2.5,(\mathrm{e}) x=1.9$ and $x=2,(f) x=2$ and $x=2.1,$ and $(g)$ use parts $(a)-(f)$ and other calculations as needed to estimate the slope of the tangent line at $x=2$.
$f(x)=\frac{x-1}{x+1}$

Stanton Flemons

Stanton Flemons

Numerade Educator

Problem 12

Compute the slope of the secant line between the points at (a) $x=1$ and $x=2,$ (b) $x=2$ and $x=3$, (c) $x=1.5$ and $x=2,$ (d) $x=2$ and $x=2.5,(\mathrm{e}) x=1.9$ and $x=2,(f) x=2$ and $x=2.1,$ and $(g)$ use parts $(a)-(f)$ and other calculations as needed to estimate the slope of the tangent line at $x=2$.
$f(x)=e^{x}$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 13

List the points $A, B, C$ and $D$ in order of increasing slope of the tangent line.

Claire Rochford

Claire Rochford

Numerade Educator

Problem 14

List the points $A, B, C$ and $D$ in order of increasing slope of the tangent line.

Claire Rochford

Claire Rochford

Numerade Educator

Problem 15

Use the position function $s$ (in meters) to find the velocity at time $t=a$ seconds.
$s(t)=-4.9 t^{2}+5$
(a) $a=1 ;$ (b) $a=2$

Sanchit Jain

Numerade Educator

Problem 16

Use the position function $s$ (in meters) to find the velocity at time $t=a$ seconds.
$s(t)=4 t-4.9 t^{2},$ (a) $a=0 ;$ (b) $a=1$

Sanchit Jain

Numerade Educator

Problem 17

Use the position function $s$ (in meters) to find the velocity at time $t=a$ seconds.
$s(t)=\sqrt{t+16},$ (a) $a=0$
(b) $a=2$

Sanchit Jain

Numerade Educator

Problem 18

Use the position function $s$ (in meters) to find the velocity at time $t=a$ seconds.
$s(t)=4 / t,$ (a) $a=2$
(b) $a=4$

Sanchit Jain

Numerade Educator

Problem 19

The function represents the position in feet of an object at time $t$ seconds. Find the average velocity between
(a) $t=0$ and $t=2,$
(b) $t=1$ and $t=2,(\mathrm{c}) t=1.9$ and $t=2$
(d) $t=1.99$ and $t=2,$ and
(e) estimate the instantaneous velocity at $t=2$
$s(t)=16 t^{2}+10$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 20

The function represents the position in feet of an object at time $t$ seconds. Find the average velocity between
(a) $t=0$ and $t=2,$
(b) $t=1$ and $t=2,(\mathrm{c}) t=1.9$ and $t=2$
(d) $t=1.99$ and $t=2,$ and
(e) estimate the instantaneous velocity at $t=2$
$s(t)=3 t^{3}+t$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 21

The function represents the position in feet of an object at time $t$ seconds. Find the average velocity between
(a) $t=0$ and $t=2,$
(b) $t=1$ and $t=2,(\mathrm{c}) t=1.9$ and $t=2$
(d) $t=1.99$ and $t=2,$ and
(e) estimate the instantaneous velocity at $t=2$
$s(t)=\sqrt{t^{2}+8 t}$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 22

The function represents the position in feet of an object at time $t$ seconds. Find the average velocity between
(a) $t=0$ and $t=2,$
(b) $t=1$ and $t=2,(\mathrm{c}) t=1.9$ and $t=2$
(d) $t=1.99$ and $t=2,$ and
(e) estimate the instantaneous velocity at $t=2$
$s(t)=3 \sin (t-2)$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 23

Use graphical and numerical evidence to explain why a tangent line to the graph of $y=f(x)$ at $x=a$ does not exist.
$f(x)=|x-1|$ at $a=1$

Lucas Finney

Numerade Educator

Problem 24

Use graphical and numerical evidence to explain why a tangent line to the graph of $y=f(x)$ at $x=a$ does not exist.
$f(x)=\frac{4 x}{x-1}$ at $a=1$

Lucas Finney

Numerade Educator

Problem 25

Use graphical and numerical evidence to explain why a tangent line to the graph of $y=f(x)$ at $x=a$ does not exist.
$f(x)=\left\{\begin{array}{ll}x^{2}-1 & \text { if } x<0 \\ x+1 & \text { if } x \geq 0\end{array}\right.$ at $a=0$

Lucas Finney

Numerade Educator

Problem 26

Use graphical and numerical evidence to explain why a tangent line to the graph of $y=f(x)$ at $x=a$ does not exist.
$f(x)=\left\{\begin{array}{ll}-2 x & \text { if } x<0 \\ x^{2}-4 x & \text { if } x>0\end{array}\right.$ at $a=0$

Lucas Finney

Numerade Educator

Problem 27

Sketch in a plausible tangent line at the given point, or state that there is no tangent line.
$y=\sin x$ at $x=\pi$

Lucas Finney

Numerade Educator

Problem 28

Sketch in a plausible tangent line at the given point, or state that there is no tangent line.
$y=\tan ^{-1} x$ at $x=0$

Lucas Finney

Numerade Educator

Problem 29

Sketch in a plausible tangent line at the given point, or state that there is no tangent line.
$y=|x|$ at $x=0$

Lucas Finney

Numerade Educator

Problem 30

Sketch in a plausible tangent line at the given point, or state that there is no tangent line.
$y=x$ at $x=1$

Lucas Finney

Numerade Educator

Problem 31

Interpret $(a)-(c),$ as in example $1.6 .$
Suppose that $f(t)$ represents the balance in dollars of a bank account $t$ years after January 1,2000 .
(a) $\frac{f(4)-f(2)}{2}=21,034$
(b) $2[f(4)-f(3.5)]=25,036$
and (c) $\lim _{h \rightarrow 0} \frac{f(4+h)-f(4)}{h}=30,000$.

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 32

Interpret $(a)-(c),$ as in example $1.6 .$
Suppose that $f(m)$ represents the value of a car in dollars that has been driven $m$ thousand miles.
(a) $\frac{f(40)-f(38)}{2}=-2103$,
(b) $f(40)-f(39)=-2040$
and (c) $\lim _{h \rightarrow 0} \frac{f(40+h)-f(40)}{h}=-2000$.

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 33

Sometimes an incorrect method accidentally produces a correct answer. For quadratic functions (but definitely not most other functions), the average velocity between $t=r$ and $t=s$ equals the average of the velocities at $t=r$ and $t=s .$ To show this, assume that $f(t)=a t^{2}+b t+c$ is the distance function. Show that the average velocity between $t=r$ and $t=s$ equals $a(s+r)+b$. Show that the velocity at $t=r$ is $2 a r+b$ and the velocity at $t=s$ is $2 a s+b$. Finally, show that $a(s+r)+b=\frac{(2 a r+b)+(2 a s+b)}{2}$

Carson Merrill

Numerade Educator

Problem 34

Find a cubic function [try $\left.f(t)=t^{3}+\cdots\right]$ and numbers $r$ and $s$ such that the average velocity between $t=r$ and $t=s$ is different from the average of the velocities at $t=r$ and $t=s$

Andrew Bassila

Numerade Educator

Problem 35

(a) Find all points at which the slope of the tangent line to $y=x^{3}+3 x+1$ equals 5
(b) Show that the slope of the tangent line to $y=x^{3}+3 x+1$ cannot equal 1 at any point.

Aman Gupta

Numerade Educator

Problem 36

(a) Show that the graphs of $y=x^{2}+1$ and $y=x$ do not intersect.
(b) Find the value of $x$ such that the tangent lines to $y=x^{2}+1$ and $y=x$ are parallel.

Claire Rochford

Claire Rochford

Numerade Educator

Problem 37

(a) Find an equation of the tangent line to $y=x^{3}+3 x+1$ at
$$
x=1
$$
(b) Show that the tangent line in part
(a) intersects $y=x^{3}+3 x+1$ at more than one point.
(c) Show that for any number $c$ the tangent line to $y=x^{2}+1$ at $x=c$ only intersects $y=x^{2}+1$ at one point.

Aman Gupta

Numerade Educator

Problem 38

Show that $\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}=\lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a} .$ (Hint:
Let $h=x-a .)$

Andrew Bassila

Numerade Educator

Problem 39

The table shows the freezing temperature of water in degrees Celsius at various pressures. Estimate the slope of the tangent line at $p=1$ and interpret the result. Estimate the slope of the tangent line at $p=3$ and interpret the result.
$$
\begin{array}{|c|c|c|c|c|c|}
\hline p(\mathrm{~atm}) & 0 & 1 & 2 & 3 & 4 \\
\hline{ }^{\circ} \mathrm{C} & 0 & -7 & -20 & -16 & -11 \\
\hline
\end{array}
$$

Andrew Bassila

Numerade Educator

Problem 40

The table shows the range of a soccer kick launched at $30^{\circ}$ above the horizontal at various initial speeds. Estimate the slope of the tangent line at $v=50$ and interpret the result.
$$
\begin{array}{|c|c|c|c|c|c|}
\hline \text { Distance (yd) } & 19 & 28 & 37 & 47 & 58 \\
\hline \text { Speed (mph) } & 30 & 40 & 50 & 60 & 70 \\
\hline
\end{array}
$$

Andrew Bassila

Numerade Educator

Problem 41

The graph shows the elevation of a person on a climb up a cliff as a function of time. When did the climber reach the top? When was the hiker going the fastest on the way up? When was the hiker going the fastest on the way down? What do you think occurred at places where the graph is level?

Claire Rochford

Claire Rochford

Numerade Educator

Problem 42

The graph shows the amount of water in a city water tank as a function of time. When was the tank the fullest? the emptiest? When was the tank filling up at the fastest rate? When was the tank emptying at the fastest rate? What time of day do you think the level portion represents?

Claire Rochford

Claire Rochford

Numerade Educator

Problem 43

Suppose a hot cup of coffee is left in a room for 2 hours. Sketch a reasonable graph of what the temperature would look like as a function of time. Then sketch a graph of what the rate of change of the temperature would look like.

Andrew Bassila

Numerade Educator

Problem 44

Sketch a graph representing the height of a bungee-jumper. Sketch the graph of the person's velocity (use $+$ for upward velocity and $-$ for downward velocity).

Andrew Bassila

Numerade Educator