# Calculus Early Transcendentals

## Educators BS MP
MM
+ 16 more educators

### Problem 1

If $f(x)=x+\sqrt{2-x}$ and $g(u)=u+\sqrt{2-u},$ is it true that $f=g ?$ Melinda M.

### Problem 2

If
$$f(x)=\frac{x^{2}-x}{x-1} \quad \text { and } \quad g(x)=x$$
is it true that $f=g ?$

Check back soon!

### Problem 3

The graph of a function $f$ is given.
(a) State the value of $f(1) .$
(b) Estimate the value of $f(-1)$
(c) For what values of $x$ is $f(x)=1 ?$
(d) Estimate the value of $x$ such that $f(x)=0$
(e) State the domain and range of $f$
(f) On what interval is $f$ increasing?

BS
Brent S.

### Problem 4

The graphs of $f$ and $g$ are given.
(a) State the values of $f(-4)$ and $g(3)$
(b) For what values of $x$ is $f(x)=g(x) ?$
(c) Estimate the solution of the equation $f(x)=-1$
(d) On what interval is $f$ decreasing?
(e) State the domain and range of $f$
(f) State the domain and range of $g .$

Check back soon!

### Problem 5

Figure 1 was recorded by an instrument operated by the Cali- fornia Department of Mines and Geology at the University Hospital of the University of Southern California in Los Angeles. Use it to estimate the range of the vertical ground acceleration function at USC during the Northridge earthquake. Suzanne W.

### Problem 6

In this section we discussed examples of ordinary, everyday functions: Population is a function of time, postage cost is a function of weight, water temperature is a function of time. Give three other examples of functions from everyday life that are described verbally. What can you say about the domain and range of each of your functions? If possible, sketch a rough graph of each function.

MP
Matt P.

### Problem 7

$7-10$ Determine whether the curve is the graph of a function of $x .$ If it is, state the domain and range of the function.

MM
Marcus M.

### Problem 8

$7-10$ Determine whether the curve is the graph of a function of $x .$ If it is, state the domain and range of the function.

SB
Shannon B.

### Problem 9

$7-10$ Determine whether the curve is the graph of a function of $x .$ If it is, state the domain and range of the function. Suzanne W.

### Problem 10

$7-10$ Determine whether the curve is the graph of a function of $x .$ If it is, state the domain and range of the function. Suzanne W.

### Problem 11

The graph shown gives the weight of a certain person as a function of age. Describe in words how this person's weight varies over time. What do you think happened when this person was 30 years old? Zach F.

### Problem 12

The graph shows the height of the water in a bathtub as a function of time. Give a verbal description of what you think happened.

Check back soon!

### Problem 13

You put some ice cubes in a glass, fill the glass with cold water, and then let the glass sit on a table. Describe how the temperature of the water changes as time passes. Then sketch a rough graph of the temperature of the water as a function of the elapsed time.

Check back soon!

### Problem 14

Three runners compete in a 100 -meter race. The graph depicts the distance run as a function of time for each runner. Describe in words what the graph tells you about this race. Who won the race? Did each runner finish the race?

KV
Kurt V.

### Problem 15

The graph shows the power consumption for a day in September in San Francisco. (P is measured in megawatts; $t$ is measured in hours starting at midnight.)
(a) What was the power consumption at 6 AM? At 6 PM?
(b) When was the power consumption the lowest? When was it the highest? Do these times seem reasonable?

Check back soon!

### Problem 16

Sketch a rough graph of the number of hours of daylight as a function of the time of year.

Check back soon!

### Problem 17

Sketch a rough graph of the outdoor temperature as a function of time during a typical spring day.

RM
Rohini M.

### Problem 18

Sketch a rough graph of the market value of a new car as a function of time for a period of 20 years. Assume the car is well maintained. Serkan O.

### Problem 19

Sketch the graph of the amount of a particular brand of coffee sold by a store as a function of the price of the coffee.

Check back soon!

### Problem 20

You place a frozen pie in an oven and bake it for an hour. Then you take it out and let it cool before eating it. Describe how the temperature of the pie changes as time passes. Then sketch a rough graph of the temperature of the pie as a function of time.

SL
Samuel L.

### Problem 21

A homeowner mows the lawn every Wednesday afternoon. Sketch a rough graph of the height of the grass as a function of time over the course of a four-week period.

Check back soon!

### Problem 22

An airplane takes off from an airport and lands an hour later at another airport, 400 miles away. If $t$ represents the time in minutes since the plane has left the terminal building, let $x(t)$ be the horizontal distance traveled and $y(t)$ be the altitude of the plane.
(a) Sketch a possible graph of $x(t) .$
(b) Sketch a possible graph of $y(t) .$
(c) Sketch a possible graph of the ground speed.
(d) Sketch a possible graph of the vertical velocity. Robert H.

### Problem 23

The number $N$ (in millions) of US cellular phone subscribers is shown in the table. (Midyear estimates are given.)
$$\begin{array}{|c|c|c|c|c|c|}\hline t & {1996} & {1998} & {2000} & {2002} & {2004} & {2006} \\ \hline N & {44} & {69} & {109} & {141} & {182} & {233} \\ \hline\end{array}$$
(a) Use the data to sketch a rough graph of $N$ as a function of $t$
(b) Use your graph to estimate the number of cell-phone subscribers at midyear in 2001 and $2005 .$ Christy G.

### Problem 24

Temperature readings $T$ (In$^{\circ} \mathrm{F}$) were recorded every two hours from midnight to $2 : 00 \mathrm{PM}$ in Phoenix on September $10,2008$ The time $t$ was measured in hours from midnight.
$$\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline t & {0} & {2} & {4} & {6} & {8} & {10} & {12} & {14} \\ \hline T & {82} & {75} & {74} & {75} & {84} & {90} & {93} & {94} \\ \hline\end{array}$$
(a) Use the readings to sketch a rough graph of $T$ as a function of $t$ .
(b) Use your graph to estimate the temperature at $9 : 00$ AM.

GD
Gilbert D.

### Problem 25

If $f(x)=3 x^{2}-x+2$ find $f(2), \quad f(-2), f(a), f(-a)$ $f(a+1), 2 f(a), f(2 a), f\left(a^{2}\right),[f(a)]^{2}$ and $f(a+h)$

Check back soon!

### Problem 26

A spherical balloon with radius $r$ inches has volume $V(r)=\frac{4}{3} \pi r^{3} .$ Find a function that represents the amount of air required to inflate the balloon from a radius of $r$ inches to a radius of $r+1$ inches.

Check back soon!

### Problem 27

$27-30$ Evaluate the difference quotient for the given function. Simplify your answer.
$$f(x)=4+3 x-x^{2}, \quad \frac{f(3+h)-f(3)}{h}$$ Suzanne W.

### Problem 28

$27-30$ Evaluate the difference quotient for the given function. Simplify your answer.
$$f(x)=x^{3}, \quad \frac{f(a+h)-f(a)}{h}$$ Suzanne W.

### Problem 29

$27-30$ Evaluate the difference quotient for the given function. Simplify your answer.
$$f(x)=\frac{1}{x}, \quad \frac{f(x)-f(a)}{x-a}$$ Sabrina C.

### Problem 30

$27-30$ Evaluate the difference quotient for the given function. Simplify your answer.
$$f(x)=\frac{x+3}{x+1}, \quad \frac{f(x)-f(1)}{x-1}$$ Suzanne W.

### Problem 31

$31-37$ Find the domain of the function.
$$f(x)=\frac{x+4}{x^{2}-9}$$ Suzanne W.

### Problem 32

$31-37$ Find the domain of the function.
$$f(x)=\frac{2 x^{3}-5}{x^{2}+x-6}$$ Suzanne W.

### Problem 33

$31-37$ Find the domain of the function.
$$f(t)=\sqrt{2 t-1}$$ Suzanne W.

### Problem 34

$31-37$ Find the domain of the function.
$$g(t)=\sqrt{3-t}-\sqrt{2+t}$$ Suzanne W.

### Problem 35

$31-37$ Find the domain of the function.
$$h(x)=\frac{1}{\sqrt{x^{2}-5 x}}$$ Suzanne W.

### Problem 36

$31-37$ Find the domain of the function.
$$f(u)=\frac{u+1}{1+\frac{1}{u+1}}$$ Suzanne W.

### Problem 37

$31-37$ Find the domain of the function.
$$F(p)=\sqrt{2-\sqrt{p}}$$ Shreyas K.

### Problem 38

Find the domain and range and sketch the graph of the function $h(x)=\sqrt{4-x^{2}}$

Check back soon!

### Problem 39

$39-50$ Find the domain and sketch the graph of the function.
$$f(x)=2-0.4 x$$ Suzanne W.

### Problem 40

$39-50$ Find the domain and sketch the graph of the function.
$$F(x)=x^{2}-2 x+1$$ Suzanne W.

### Problem 41

$39-50$ Find the domain and sketch the graph of the function.
$$f(t)=2 t+t^{2}$$ Suzanne W.

### Problem 42

$39-50$ Find the domain and sketch the graph of the function.
$$H(t)=\frac{4-t^{2}}{2-t}$$ Suzanne W.

### Problem 43

$39-50$ Find the domain and sketch the graph of the function.
$$g(x)=\sqrt{x-5}$$ Suzanne W.

### Problem 44

$39-50$ Find the domain and sketch the graph of the function.
$$F(x)=|2 x+1|$$ Suzanne W.

### Problem 45

$39-50$ Find the domain and sketch the graph of the function.
$$G(x)=\frac{3 x+|x|}{x}$$ Suzanne W.

### Problem 46

$39-50$ Find the domain and sketch the graph of the function.
$$g(x)=|x|-x$$ Suzanne W.

### Problem 47

$39-50$ Find the domain and sketch the graph of the function.
$$f(x)=\left\{\begin{array}{ll}{x+2} & {\text { if } x<0} \\ {1-x} & {\text { if } x \geqslant 0}\end{array}\right.$$ Suzanne W.

### Problem 48

$39-50$ Find the domain and sketch the graph of the function.
$$f(x)=\left\{\begin{array}{ll}{3-\frac{1}{2} x} & {\text { if } x \leqslant 2} \\ {2 x-5} & {\text { if } x>2}\end{array}\right.$$ Suzanne W.

### Problem 49

$39-50$ Find the domain and sketch the graph of the function.
$$f(x)=\left\{\begin{array}{ll}{x+2} & {\text { if } x \leqslant-1} \\ {x^{2}} & {\text { if } x>-1}\end{array}\right.$$ Suzanne W.

### Problem 50

$39-50$ Find the domain and sketch the graph of the function.
$$f(x)=\left\{\begin{array}{ll}{x+9} & {\text { if } x<-3} \\ {-2 x} & {\text { if }|x| \leqslant 3} \\ {-6} & {\text { if } x>3}\end{array}\right.$$ Suzanne W.

### Problem 51

\begin{equation}
\begin{array}{l}{51-56 \text { Find an expression for the function whose graph is the }} \\ {\text { given curve. }}\end{array}
\end{equation}
The line segment joining the points $(1,-3)$ and $(5,7)$ Suzanne W.

### Problem 52

\begin{equation}
\begin{array}{l}{51-56 \text { Find an expression for the function whose graph is the }} \\ {\text { given curve. }}\end{array}
\end{equation}
The line segment joining the points $(-5,10)$ and $(7,-10)$ Suzanne W.

### Problem 53

\begin{equation}
\begin{array}{l}{51-56 \text { Find an expression for the function whose graph is the }} \\ {\text { given curve. }}\end{array}
\end{equation}
The bottom half of the parabola $x+(y-1)^{2}=0$ Suzanne W.

### Problem 54

\begin{equation}
\begin{array}{l}{51-56 \text { Find an expression for the function whose graph is the }} \\ {\text { given curve. }}\end{array}
\end{equation}
The top half of the circle $x^{2}+(y-2)^{2}=4$ Suzanne W.

### Problem 55

$51-56$ Find an expression for the function whose graph is the given curve. Suzanne W.

### Problem 56

$51-56$ Find an expression for the function whose graph is the given curve. Suzanne W.

### Problem 57

\begin{equation}
\begin{array}{l}{57-61 \text { Find a formula for the described function and state its }} \\ {\text { domain. }}\end{array}
\end{equation}
A rectangle has perimeter 20 $\mathrm{m} .$ Express the area of the rect- angle as a function of the length of one of its sides. Jill S.

### Problem 58

\begin{equation}
\begin{array}{l}{57-61 \text { Find a formula for the described function and state its }} \\ {\text { domain. }}\end{array}
\end{equation}
A rectangle has area 16 $\mathrm{m}^{2} .$ Express the perimeter of the rectangle as a function of the length of one of its sides. Suzanne W.

### Problem 59

\begin{equation}
\begin{array}{l}{57-61 \text { Find a formula for the described function and state its }} \\ {\text { domain. }}\end{array}
\end{equation}
Express the area of an equilateral triangle as a function of the length of a side. Suzanne W.

### Problem 60

\begin{equation}
\begin{array}{l}{57-61 \text { Find a formula for the described function and state its }} \\ {\text { domain. }}\end{array}
\end{equation}
Express the surface area of a cube as a function of its volume. Suzanne W.

### Problem 61

\begin{equation}
\begin{array}{l}{57-61 \text { Find a formula for the described function and state its }} \\ {\text { domain. }}\end{array}
\end{equation}
An open rectangular box with volume 2 $\mathrm{m}^{3}$ has a square base. Express the surface area of the box as a function of the length of a side of the base. Suzanne W.

### Problem 62

A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 30 $\mathrm{ft}$ , express the area $A$ of the window as a function of the width $x$ of the window.

Check back soon!

### Problem 63

A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 12 in. by 20 in. by cutting out equal squares of side $x$ at each corner and then folding up the sides as in the figure. Express the volume $V$ of the box as a function of $X .$

KK
Kristine K.