Problem 1

(a) Write an equation that defines the exponential function

with base $a>0$ .

(b) What is the domain of this function?

(c) If $a \neq 1,$ what is the range of this function?

(d) Sketch the general shape of the graph of the exponential function for each of the following cases.

(i) $a>1$

Check back soon!

Problem 2

(a) How is the number $e$ defined?

(b) What is an approximate value for $e ?$

(c) What is the natural exponential function?

Check back soon!

Problem 3

Graph the given functions on a common screen. How are these graphs related?

$$y=2^{x}, \quad y=e^{x}, \quad y=5^{x}, \quad y=20^{x}$$

Check back soon!

Problem 4

Graph the given functions on a common screen. How are these graphs related?

$$y=e^{x}, \quad y=e^{-x}, \quad y=8^{x}, \quad y=8^{-x}$$

Check back soon!

Problem 5

Graph the given functions on a common screen. How are these graphs related?

$$y=3^{x}, \quad y=10^{x}, \quad y=\left(\frac{1}{3}\right)^{x}, \quad y=\left(\frac{1}{10}\right)^{x}$$

Check back soon!

Problem 6

Graph the given functions on a common screen. How are these graphs related?

$$y=0.9^{x}, \quad y=0.6^{x}, \quad y=0.3^{x}, \quad y=0.1^{x}$$

Check back soon!

Problem 7

Make a rough sketch of the graph of the function. Do

not use a calculator. Just use the graphs given in Figures 3 and

9 and, if necessary, the transformations of Section 1.2 .

$y=10^{x+2}$

Check back soon!

Problem 8

Make a rough sketch of the graph of the function. Do

not use a calculator. Just use the graphs given in Figures 3 and

9 and, if necessary, the transformations of Section 1.2 .

$$y=(0.5)^{x}-2$$

Check back soon!

Problem 9

Make a rough sketch of the graph of the function. Do

not use a calculator. Just use the graphs given in Figures 3 and

9 and, if necessary, the transformations of Section 1.2 .

$$y=-2^{-x}$$

Check back soon!

Problem 10

Make a rough sketch of the graph of the function. Do

not use a calculator. Just use the graphs given in Figures 3 and

9 and, if necessary, the transformations of Section 1.2 .

$$y=e^{|x|}$$

Check back soon!

Problem 11

Make a rough sketch of the graph of the function. Do

not use a calculator. Just use the graphs given in Figures 3 and

9 and, if necessary, the transformations of Section 1.2 .

$$y=1-\frac{1}{2} e^{-x}$$

Check back soon!

Problem 12

Make a rough sketch of the graph of the function. Do

not use a calculator. Just use the graphs given in Figures 3 and

9 and, if necessary, the transformations of Section 1.2 .

$$y=2\left(1-e^{x}\right)$$

Check back soon!

Problem 13

Starting with the graph of $y=e^{x},$ write the equation of the

graph that results from

(a) shifting 2 units downward

(b) shifting 2 units to the right

(c) reflecting about the $x$ -axis

(d) reflecting about the $y$ -axis

(e) reflecting about the $x$ -axis and then about the $y$ -axis

Check back soon!

Problem 14

Starting with the graph of $y=e^{x},$ find the equation of the

graph that results from

(a) reflecting about the line $y=4$

(b) reflecting about the line $x=2$

Check back soon!

Problem 15

Find the domain of each function.

$$f(x)=\frac{1-e^{x^{2}}}{1-e^{1-x^{2}}}$$

Check back soon!

Problem 15

Find the domain of each function.

$$f(x)=\frac{1-e^{x^{2}}}{1-e^{1-x^{2}}}$$ $$f(x)=\frac{1+x}{e^{\cos x}}$$

Check back soon!

Problem 16

Find the domain of each function.

$$g(t)=\sin \left(e^{-t}\right) \quad \text { (b) } g(t)=\sqrt{1-2^{t}}$$

Check back soon!

Problem 19

Suppose the graphs of $f(x)=x^{2}$ and $g(x)=2^{x}$ are drawn

on a coordinate grid where the unit of measurement is

1 inch. Show that, at a distance 2 ft to the right of the ori-

gin, the height of the graph of $f$ is 48 ft but the height

of the graph of $g$ is about 265 mi.

Check back soon!

Problem 20

Compare the rates of growth of the functions $f(x)=x^{5}$

and $g(x)=5^{x}$ by graphing both functions in several view-

ing rectangles. Find all points of intersection of the graphs

correct to one decimal place.

Check back soon!

Problem 21

Compare the functions $f(x)=x^{10}$ and $g(x)=e^{x}$ by graph-

ing both $f$ and $g$ in several viewing rectangles. When does

the graph of $g$ finally surpass the graph of $f ?$

Check back soon!

Problem 22

Use a graph to estimate the values of $x$ such that

$e^{x}>1,000,000,000 .$

Check back soon!

Problem 25

Find the limit.

$$\lim _{x \rightarrow \infty} \frac{e^{3 x}-e^{-3 x}}{e^{3 x}+e^{-3 x}}$$

Check back soon!

Problem 26

Find the limit.

$$\lim _{x \rightarrow \infty} \frac{2+10^{x}}{3-10^{x}}$$

Check back soon!

Problem 29

Find the limit.

$$\lim _{x \rightarrow \infty}\left(e^{-2 x} \cos x\right)$$

Check back soon!

Problem 31

If you graph the function

$f(x)=\frac{1-e^{1 / x}}{1+e^{1 / x}}$

you'll see that $f$ appears to be an odd function. Prove it.

Check back soon!

Problem 32

Graph several members of the family of functions

$f(x)=\frac{1}{1+a e^{b x}}$

where $a>0 .$ How does the graph change when $b$

changes? How does it change when $a$ changes?

Check back soon!