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Calculus and its applications

Bittinger, Marvin L. Ellenbogen, David J. Surgent

Chapter 3

Exponential and Logarithmic Functions - all with Video Answers

Educators

Section 1

Exponential Functions

01:10

Problem 1

Graph.
$$
y=5^{x}
$$

Willis James
Willis James
Numerade Educator
00:35

Problem 2

Graph.
$$
y=4^{x}
$$

AG
Ankit Gupta
Numerade Educator
00:52

Problem 3

Graph.
$$
y=2 \cdot 3^{x}
$$

James Kiss
James Kiss
Numerade Educator
01:31

Problem 4

Graph.
$$
y=3 \cdot 4^{x}
$$

James Kiss
James Kiss
Numerade Educator
01:16

Problem 5

Graph.
$$
y=5\left(\frac{1}{4}\right)^{x}
$$

James Kiss
James Kiss
Numerade Educator
01:31

Problem 6

Graph.
$$
y=4\left(\frac{1}{3}\right)^{x}
$$

James Kiss
James Kiss
Numerade Educator
00:28

Problem 7

Graph.
$$
y=1.3(1.2)^{x}
$$

Linh Vu
Linh Vu
Numerade Educator
01:07

Problem 8

Graph.
$$
y=2.4(1.25)^{x}
$$

James Kiss
James Kiss
Numerade Educator
00:57

Problem 9

Graph.
$$
y=2.6(0.8)^{x}
$$

James Kiss
James Kiss
Numerade Educator
01:24

Problem 10

Graph.
$$
y=1.13(0.81)^{x}
$$

James Kiss
James Kiss
Numerade Educator
01:04

Problem 11

Differentiate.
$$
f(x)=e^{x}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
01:03

Problem 12

Differentiate.
$$
f(x)=e^{-x}
$$

Tyler Moulton
Tyler Moulton
Numerade Educator
00:50

Problem 13

Differentiate.
$$
g(x)=e^{2 x}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
00:52

Problem 14

Differentiate.
$$
g(x)=e^{3 x}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
00:54

Problem 15

Differentiate.
$$
f(x)=6 e^{x}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
00:49

Problem 16

Differentiate.
$$
f(x)=4 e^{x}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
00:35

Problem 17

Differentiate.
$$
F(x)=e^{-7 x}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
00:37

Problem 18

Differentiate.
$$
F(x)=e^{-4 x}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
00:58

Problem 19

Differentiate.
$$
g(x)=3 e^{5 x}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
00:59

Problem 20

Differentiate.
$$
G(x)=-7 e^{-x}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
00:59

Problem 21

Differentiate.
$$
G(x)=-7 e^{-x}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
01:01

Problem 22

Differentiate.
$$
f(x)=-3 e^{-x}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
01:09

Problem 23

Differentiate.
$$
g(x)=\frac{1}{2} e^{-5 x}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
01:07

Problem 24

Differentiate.
$$
f(x)=\frac{1}{3} e^{-4 x}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
01:17

Problem 25

Differentiate.
$$
F(x)=-\frac{2}{3} e^{x^{2}}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
01:23

Problem 26

Differentiate.
$$
g(x)=-\frac{4}{5} e^{x^{3}}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
00:42

Problem 27

Differentiate.
$$
F(x)=4-e^{2 x}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
00:45

Problem 28

Differentiate.
$$
G(x)=7+3 e^{5 x}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
03:11

Problem 29

Differentiate.
$$
G(x)=x^{3}-5 e^{2 x}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
01:02

Problem 30

Differentiate.
$$
f(x)=x^{5}-2 e^{6 x}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
03:11

Problem 31

Differentiate.
$$
g(x)=x^{5} e^{2 x}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
02:37

Problem 32

Differentiate.
$$
f(x)=x^{7} e^{4 x}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
02:06

Problem 33

Differentiate.
$$
F(x)=\frac{e^{2 x}}{x^{4}}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
02:00

Problem 34

Differentiate.
$$
g(x)=\frac{e^{3 x}}{x^{6}}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
02:02

Problem 35

Differentiate.
$$
f(x)=\left(x^{2}-2 x+2\right) e^{x}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
02:30

Problem 36

Differentiate.
$$
f(x)=\left(x^{2}+3 x-9\right) e^{x}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
02:07

Problem 37

Differentiate.
$$
f(x)=\frac{e^{x}}{x^{4}}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
01:50

Problem 38

Differentiate.
$$
f(x)=\frac{e^{x}}{x^{5}}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
01:03

Problem 39

Differentiate.
$$
f(x)=e^{-x^{2}+8 x}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
01:34

Problem 40

Differentiate.
$$
f(x)=e^{-x^{2}+7 x}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
00:55

Problem 41

Differentiate.
$$
f(x)=e^{x^{2} / 2}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
00:39

Problem 42

Differentiate.
$$
f(x)=e^{-x^{2} / 2}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
01:30

Problem 43

Differentiate.
$$
y=e^{\sqrt{x-7}}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
01:21

Problem 44

Differentiate.
$$
y=e^{\sqrt{x-4}}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
00:55

Problem 45

Differentiate.
$$
y=\sqrt{e^{x}-1}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
00:55

Problem 46

Differentiate.
$$
y=\sqrt{e^{x}+1}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
02:00

Problem 47

Differentiate.
$$
y=e^{x}+x^{3}-x e^{x}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
01:51

Problem 48

Differentiate.
$$
y=x e^{-2 x}+e^{-x}+x^{3}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
00:52

Problem 49

Differentiate.
$$
y=1-e^{-3 x}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
01:47

Problem 50

Differentiate.
$$
y=1-e^{-x}
$$

Clarissa Noh
Clarissa Noh
Numerade Educator
00:59

Problem 51

Differentiate.
$$
y=1-e^{-k x}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
01:47

Problem 52

Differentiate.
$$
y=1-e^{-m x}
$$

Clarissa Noh
Clarissa Noh
Numerade Educator
03:25

Problem 53

Differentiate.
$$
g(x)=\left(4 x^{2}+3 x\right) e^{x^{2}-7 x}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
02:49

Problem 54

Differentiate.
$$
g(x)=\left(5 x^{2}-8 x\right) e^{x^{2}-4 x}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
06:15

Problem 55

Graph each function. Then determine any critical values, inflection points, intervals over which the function is increasing or decreasing, and the concavity.
$$
g(x)=e^{-2 x}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
08:00

Problem 56

Graph each function. Then determine any critical values, inflection points, intervals over which the function is increasing or decreasing, and the concavity.
$$
f(x)=e^{2 x}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
06:47

Problem 57

Graph each function. Then determine any critical values, inflection points, intervals over which the function is increasing or decreasing, and the concavity.
$$
f(x)=e^{(1 / 3) x}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
06:18

Problem 58

Graph each function. Then determine any critical values, inflection points, intervals over which the function is increasing or decreasing, and the concavity.
$$
g(x)=e^{(1 / 2) x}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
05:33

Problem 59

Graph each function. Then determine any critical values, inflection points, intervals over which the function is increasing or decreasing, and the concavity.
$$
f(x)=\frac{1}{2} e^{-x}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
05:41

Problem 60

Graph each function. Then determine any critical values, inflection points, intervals over which the function is increasing or decreasing, and the concavity.
$$
g(x)=\frac{1}{3} e^{-x}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
06:50

Problem 61

Graph each function. Then determine any critical values, inflection points, intervals over which the function is increasing or decreasing, and the concavity.
$$
F(x)=-e^{(1 / 3) x}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
06:15

Problem 62

Graph each function. Then determine any critical values, inflection points, intervals over which the function is increasing or decreasing, and the concavity.
$$
G(x)=-e^{(1 / 2) x}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
07:27

Problem 63

Graph each function. Then determine any critical values, inflection points, intervals over which the function is increasing or decreasing, and the concavity.
$$
f(x)=3-e^{-x}, \text { for } x \geq 0
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
08:07

Problem 64

Graph each function. Then determine any critical values, inflection points, intervals over which the function is increasing or decreasing, and the concavity.
$$
g(x)=2\left(1-e^{-x}\right), \text { for } x \geq 0
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
01:29

Problem 65

For each function given in Exercises $55-64,$ graph the function and its first and second derivatives using a graphing utility.

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
01:29

Problem 66

For each function given in Exercises $55-64,$ graph the function and its first and second derivatives using a graphing utility.

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
01:29

Problem 67

For each function given in Exercises $55-64,$ graph the function and its first and second derivatives using a graphing utility.

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
01:29

Problem 68

For each function given in Exercises $55-64,$ graph the function and its first and second derivatives using a graphing utility.

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
01:29

Problem 69

For each function given in Exercises $55-64,$ graph the function and its first and second derivatives using a graphing utility.

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
01:29

Problem 70

For each function given in Exercises $55-64,$ graph the function and its first and second derivatives using a graphing utility.

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
01:29

Problem 71

For each function given in Exercises $55-64,$ graph the function and its first and second derivatives using a graphing utility.

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
01:29

Problem 72

For each function given in Exercises $55-64,$ graph the function and its first and second derivatives using a graphing utility.

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
01:29

Problem 73

For each function given in Exercises $55-64,$ graph the function and its first and second derivatives using a graphing utility.

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
01:29

Problem 74

For each function given in Exercises $55-64,$ graph the function and its first and second derivatives using a graphing utility.

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
01:31

Problem 75

Find the slope of the line tangent to the graph of $f(x)=2 e^{-3 x}$ at the point (0,2)

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
01:18

Problem 76

Find the slope of the line tangent to the graph of $f(x)=e^{x}$ at the point (0,1)

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
03:03

Problem 77

Find an equation of the line tangent to the graph of $f(x)=e^{2 x}$ at the point (0,1)

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
00:41

Problem 78

Find an equation of the line tangent to the graph of $G(x)=e^{-x}$ at the point (0,1)

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
02:29

Problem 79

For each of Exercises 77 and $78,$ graph the function and the tangent line using a graphing utility.

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
02:29

Problem 80

For each of Exercises 77 and $78,$ graph the function and the tangent line using a graphing utility.

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
04:26

Problem 81

U.S. travel exports (goods and services that international travelers buy while visiting the United States) are increasing exponentially. The value of such exports, $t$ years after $2011,$ can be approximated by
$$
V(t)=115.32 e^{0.094 t}
$$
where $V$ is in billions of dollars. (Source: www.census. gov/foreign-trade/data/index.html.)
a) Estimate the value of U.S. travel exports in 2016 and 2018
b) Estimate the growth rate for U.S. travel exports in 2016 and 2018

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
02:46

Problem 82

More Americans are buying organic fruit and vegetables and products made with organic ingredients. The amount $A(t)$, in billions of dollars, spent on organic food and beverages $t$ years after 1995 can be approximated by
$$
A(t)=2.43 e^{0.18 t}
$$
a) Estimate the amount that Americans spent on organic food and beverages in $2009 .$
b) Estimate the rate at which spending on organic food and beverages was growing in $2006 .$

Jonathon Brumley
Jonathon Brumley
Numerade Educator
06:00

Problem 83

The total cost, in millions of dollars, for Cheevers, Inc., is given by
$$
C(t)=100-50 e^{-t}
$$
where $t$ is the time in years since the start-up date.
Find each of the following.
a) The marginal cost, $C^{\prime}(t)$
b) $C^{\prime}(0)$
c) $C^{\prime}(4)$ (Round to the nearest thousand.)
d) Find $\lim _{t \rightarrow \infty} C(t)$ and $\lim _{t \rightarrow \infty} C^{\prime}(t)$.

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
04:20

Problem 84

The total cost, in millions of dollars, for Marcotte Industries is given by
$$
C(t)=200-40 e^{-t}
$$
where $t$ is the time in years since the start-up date.
Find each of the following.
a) The marginal $\operatorname{cost} C^{\prime}(t)$
b) $C^{\prime}(1)$
c) $C^{\prime}(5)$ (Round to the nearest thousand.)
d) Find $\lim _{t \rightarrow \infty} C(t)$ and $\lim _{t \rightarrow \infty} C^{\prime}(t)$.

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
04:42

Problem 85

At a price of $x$ dollars, the demand, in thousands of units, for a certain turntable is given by the demand function
$$
q=240 e^{-0.003 x}
$$
a) How many turntables will be bought at a price of $\$ 250 ?$ Round to the nearest thousand.
b) Graph the demand function for $0 \leq x \leq 400$.
c) Find the marginal demand, $q^{\prime}(x)$.
d) Interpret the meaning of the derivative.

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
04:46

Problem 86

At a price of $x$ dollars, the supply function for the turntable in Exercise 85 is given by
$$
q=75 e^{0.004 x}
$$
where $q$ is in thousands of units.
a) How many turntables will be supplied at a price of $\$ 250 ?$ Round to the nearest thousand.
b) Graph the supply function for $0 \leq x \leq 400$.
c) Find the marginal supply, $q^{\prime}(x)$.
d) Interpret the meaning of the derivative.

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
05:12

Problem 87

Use the Tangent feature from the DRAW menu to find the rate of change in part (b).
Maria deposits $\$ 20,000$ in an IRA whose value increases by $5.6 \%$ every year. The value of the IRA after $t$ years is modeled by
$$
V(t)=20,000(1.056)^{t}
$$
a) Use the model to estimate the value of Maria's IRA after 7 yr.
b) What is the rate of change in the value of the IRA at the end of 7 yr?
c) When will Maria's IRA have a value of $\$ 40,000 ?$

Bobby Barnes
Bobby Barnes
University of North Texas
01:33

Problem 88

Use the Tangent feature from the DRAW menu to find the rate of change in part (b).
Pelican Fabrics purchases a new video surveillance system. The value of the system is modeled by
$$
V(t)=17,500(0.92)^{t}
$$
where $V$ is the value of the system, in dollars, $t$ years after its purchase.
a) Use the model to estimate the value of the system $5 \mathrm{yr}$ after it was purchased.
b) What is the rate of change in the value of the system at the end of 5 yr?
c) When will the system be worth half of its original value?

Kimberly Waterbury
Kimberly Waterbury
Numerade Educator
01:33

Problem 89

Use the Tangent feature from the DRAW menu to find the rate of change in part (b).
Perriot's Restaurant purchased kitchen equipment on January 1,2014 . The value of the equipment decreases by $15 \%$ every year. On January $1,2016,$ the value was $\$ 14,450 .$
a) Find an exponential model for the value, $V,$ of the equipment, in dollars, $t$ years after January 1 $2016 .$
b) What is the rate of change in the value of the equipment on January $1,2016 ?$
c) What was the original value of the equipment on January $1,2014 ?$
d) How many years after January 1,2014 will the value of the equipment have decreased by half?

Kimberly Waterbury
Kimberly Waterbury
Numerade Educator
02:23

Problem 90

Use the Tangent feature from the DRAW menu to find the rate of change in part (b).
The value (price) of a share of stock in Barrington Gold was $\$ 90$ on June $15,2014,$ and was increasing by $3 \%$ every week.
a) Find an exponential model for the value, $V,$ of a share of the stock, in dollars, $t$ weeks after June $15,$ 2014
b) What was the rate of change in the value of a share of the stock 6 weeks prior to June $15,2014 ?$
c) Use the model to estimate the value of a share of the stock 6 weeks prior to June 15,2014 .
d) How many weeks after June $15,2014,$ will the stock's share value have doubled?

Lewis Groves
Lewis Groves
Numerade Educator
07:22

Problem 91

The concentration $C,$ in parts per million, of a medication in the body $t$ hours after ingestion is given by the function $C(t)=10 t^{2} e^{-t}$
a) Find the concentration after $0 \mathrm{hr}, 1 \mathrm{hr}, 2 \mathrm{hr}, 3 \mathrm{hr},$ and $10 \mathrm{hr}$.
b) Sketch a graph of the function for $0 \leq t \leq 10$.
c) Find the rate of change of the concentration, $C^{\prime}(t)$.
d) Find the maximum value of the concentration and the time at which it occurs.
e) Interpret the meaning of the derivative.

Kian Manafi
Kian Manafi
Numerade Educator
09:23

Problem 92

Suppose that you are given the task of learning $100 \%$ of a block of knowledge. Human nature is such that we retain only a percentage $P$ of knowledge $t$ weeks after we have learned it. The $E b b$ inghaus learning model asserts that $P$ is given by
$$
P(t)=Q+(100-Q) e^{-k t}
$$
where $Q$ is the percentage that we would never forget and $k$ is a constant that depends on the knowledge learned. Suppose that $Q=40$ and $k=0.7$.
a) Find the percentage retained after 0 weeks, 1 week, 2 weeks, 6 weeks, and 10 weeks.
b) Find $\lim _{t \rightarrow \infty} P(t)$.
c) Sketch a graph of $P$.
d) Find the rate of change of $P(t)$ with respect to time $t$.
e) Interpret the meaning of the derivative.

Kian Manafi
Kian Manafi
Numerade Educator
10:33

Problem 93

Differentiate.
$$
y=\left(e^{3 x}+1\right)^{5}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
01:13

Problem 94

Differentiate.
$$
y=\frac{e^{3 t}-e^{7 t}}{e^{4 t}}
$$

Christopher Stanley
Christopher Stanley
Numerade Educator
00:56

Problem 95

Differentiate.
$$
y=\frac{e^{x}}{x^{2}+1}
$$

Christopher Stanley
Christopher Stanley
Numerade Educator
01:05

Problem 96

Differentiate.
$$
f(x)=e^{\sqrt{x}}+\sqrt{e^{x}}
$$

Christopher Stanley
Christopher Stanley
Numerade Educator
02:20

Problem 97

Differentiate.
$$
f(x)=e^{x / 2} \cdot \sqrt{x-1}
$$

Kian Manafi
Kian Manafi
Numerade Educator
01:03

Problem 98

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

Christopher Stanley
Christopher Stanley
Numerade Educator
01:43

Problem 99

Differentiate.
$$
f(x)=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}
$$

Christopher Stanley
Christopher Stanley
Numerade Educator
01:04

Problem 100

Differentiate.
$$
f(x)=e^{e^{x}}
$$

Jonathan Mezzenga
Jonathan Mezzenga
Numerade Educator
01:04

Problem 101

Use the results from Exercises 85 and 86 to determine the equilibrium point (the point at which supply equals demand) and the rates at which supply and demand are changing at that point.

Carson Merrill
Carson Merrill
Numerade Educator
01:59

Problem 102

Find the function values that are approximations for e. Round to five decimal places.
$$
\begin{aligned}
&\text { For } f(t)=(1+t)^{1 / t}, \text { we have } e=\lim _{t \rightarrow 0} f(t) . \text { Find } f(1)\\
&f(0.5), f(0.2), f(0.1), \text { and } f(0.001)
\end{aligned}
$$

Christopher Stanley
Christopher Stanley
Numerade Educator
01:59

Problem 103

Find the function values that are approximations for e. Round to five decimal places.
$$
\begin{aligned}
&\text { For } g(t)=t^{1 /(t-1)}, \text { we have } e=\lim _{t \rightarrow 1} g(t) . \text { Find } g(0.5)\\
&g(0.9), g(0.99), g(0.999), \text { and } g(0.9998)
\end{aligned}
$$

Christopher Stanley
Christopher Stanley
Numerade Educator
02:00

Problem 104

Find the minimum value of $f(x)=x e^{x}$ over [-2,0]

Christopher Stanley
Christopher Stanley
Numerade Educator
01:52

Problem 105

A student made the following error on a test:
$\frac{d}{d x} e^{x}=x e^{x-1}$
Identify the error and explain how to correct it.

Kian Manafi
Kian Manafi
Numerade Educator
02:10

Problem 106

Describe the differences in the graphs of $f(x)=3^{x}$ and $g(x)=x^{3}$

Kian Manafi
Kian Manafi
Numerade Educator
00:41

Problem 107

Use a graphing calculator (or Graphicus) to graph each function and find all relative extrema.
$$
f(x)=x^{2} e^{-x}
$$

Lauren Shelton
Lauren Shelton
Numerade Educator
00:30

Problem 108

Use a graphing calculator (or Graphicus) to graph each function and find all relative extrema.
$$
f(x)=e^{-x^{2}}
$$

Lauren Shelton
Lauren Shelton
Numerade Educator
00:07

Problem 109

Graph $f, f^{\prime},$ and $f^{\prime \prime}$
$$
f(x)=e^{x}
$$

Katelyn Chen
Katelyn Chen
Numerade Educator
00:07

Problem 110

Graph $f, f^{\prime},$ and $f^{\prime \prime}$
$$
f(x)=e^{-x}
$$

Katelyn Chen
Katelyn Chen
Numerade Educator
01:37

Problem 111

Graph $f, f^{\prime},$ and $f^{\prime \prime}$
$$
f(x)=2 e^{0.3 x}
$$

Vicki Stebbins
Vicki Stebbins
Numerade Educator
00:07

Problem 112

Graph $f, f^{\prime},$ and $f^{\prime \prime}$
$$
f(x)=1000 e^{-0.08 x}
$$

Katelyn Chen
Katelyn Chen
Numerade Educator
02:28

Problem 113

Graph
$$
f(x)=\left(1+\frac{1}{x}\right)^{x}
$$
Use the TABLE feature and very large values of $x$ to confirm that $e$ is approached as a limit.

Christopher Stanley
Christopher Stanley
Numerade Educator
01:58

Problem 114

Consider the expression $2^{\pi}$, where $\pi=3.1415926 \ldots$ Recall that $\pi$ is an irrational number, that is, a number with an infinite, nonrepeating decimal expansion.
a) Using a calculator, complete the following table, giving answers to eight decimal places.
b) Based on the results from part (a), estimate the value of $2^{\pi}$
c) Using a calculator, find $2^{\pi}$.
d) Use the results of parts (a)-(c) to explain how a base raised to an irrational number can be found as a numerical limit. Does this work for a base raised to any irrational number? Explain.

Foster Wisusik
Foster Wisusik
Numerade Educator