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  • Calculus: Early Transcendentals
  • Applications of Differentiation

Calculus: Early Transcendentals

James Stewart

Chapter 4

Applications of Differentiation - all with Video Answers

Educators

+ 11 more educators

Section 1

Maximum and Minimum Values

14:20

Problem 1

Explain the difference between an absolute minimum and a local minimum.

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
07:01

Problem 2

Suppose $ f $ is a continuous function defined on a closed interval $ [a, b] $.

(a) What theorem guarantees the existence of an absolute maximum value and an absolute minimum value for $ f $?
(b) What steps would you take to find those maximum and minimum values?

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
09:26

Problem 3

For each of the numbers $ a $, $ b $, $ c $, $ d $, $ r $, and $ s $, state whether the function whose graph is shown has an absolute maximum or minimum, a local maximum or minimum, or neither a maximum nor a minimum.

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
View

Problem 4

For each of the numbers $ a $, $ b $, $ c $, $ d $, $ r $, and $ s $, state whether the function whose graph is shown has an absolute maximum or minimum, a local maximum or minimum, or neither a maximum nor a minimum.

DM
David Mccaslin
Numerade Educator
13:13

Problem 5

Use the graph to state the absolute and local maximum and minimum values of the function.

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
01:27

Problem 6

Use the graph to state the absolute and local maximum and minimum values of the function.

Carson Merrill
Carson Merrill
Numerade Educator
03:48

Problem 7

Sketch the graph of a function $ f $ that is continuous on $ [1, 5] $ and has the given properties.

Absolute maximum at $ 5 $, absolute minimum at $ 2 $, local maximum at $ 3 $, local minima at $ 2 $ and $ 4 $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
03:44

Problem 8

Sketch the graph of a function $ f $ that is continuous on $ [1, 5] $ and has the given properties.

Absolute maximum at $ 4 $, absolute minimum at $ 5 $, local maximum at $ 2 $, local minimum at $ 3 $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
02:53

Problem 9

Sketch the graph of a function $ f $ that is continuous on $ [1, 5] $ and has the given properties.

Absolute minimum at $ 3 $, absolute maximum at $ 4 $, local maximum at $ 2 $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
03:12

Problem 10

Sketch the graph of a function $ f $ that is continuous on $ [1, 5] $ and has the given properties.

Absolute maximum at $ 2 $, absolute minimum at $ 5 $, $ 4 $ is a critical number but there is no local maximum and minimum there.

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
05:18

Problem 11

(a) Sketch the graph of a function that has a local maximum of $ 2 $ and is differentiable at $ 2 $.
(b) Sketch the graph of a function that has a local maximum of $ 2 $ and is continuous but not differentiable at $ 2 $.
(c) Sketch the graph of a function that has a local maximum of $ 2 $ and is not continuous at $ 2 $.

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
04:59

Problem 12

(a) Sketch the graph of a function on $ [-1, 2] $ that has an absolute maximum but no local maximum.
(b) Sketch the graph of a function on $ [-1, 2] $ that has a local maximum but no absolute maximum.

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
05:21

Problem 13

(a) Sketch the graph of a function on $ [-1, 2] $ that has an absolute maximum but no absolute minimum.
(b) Sketch the graph of a function on $ [-1, 2] $ that is discontinuous but has both an absolute maximum and an absolute minimum.

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
07:21

Problem 14

(a) Sketch the graph of a function that has two local maximum, one local minimum, and no absolute minimum.
(b) Sketch the graph of a function that has three local minima, two local maxima, and seven critical numbers.

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
05:25

Problem 15

Sketch the graph of $ f $ by hand and use your sketch to find the absolute and local maximum and minimum values of $ f $. (Use the graphs and transformations of Section 1.2 and 1.3).

$ f(x) = \frac{1}{2}(3x-1) $, $ x \leqslant 3 $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
06:26

Problem 16

Sketch the graph of $ f $ by hand and use your sketch to find the absolute and local maximum and minimum values of $ f $. (Use the graphs and transformations of Section 1.2 and 1.3).

$ f(x) = 2 - \frac{1}{3}x $, $ x \geqslant -2 $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
04:13

Problem 17

Sketch the graph of $ f $ by hand and use your sketch to find the absolute and local maximum and minimum values of $ f $. (Use the graphs and transformations of Section 1.2 and 1.3).

$ f(x) = 1/x $, $ x \geqslant 1 $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
02:33

Problem 18

Sketch the graph of $ f $ by hand and use your sketch to find the absolute and local maximum and minimum values of $ f $. (Use the graphs and transformations of Section 1.2 and 1.3).

$ f(x) = 1/x $, $ 1 < x < 3 $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
04:10

Problem 19

Sketch the graph of $ f $ by hand and use your sketch to find the absolute and local maximum and minimum values of $ f $. (Use the graphs and transformations of Section 1.2 and 1.3).

$ f(x) = \sin x $, $ 0 \leqslant x < \pi /2 $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
04:32

Problem 20

Sketch the graph of $ f $ by hand and use your sketch to find the absolute and local maximum and minimum values of $ f $. (Use the graphs and transformations of Section 1.2 and 1.3).

$ f(x) = \sin x $, $ 0 < x \leqslant \pi /2 $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
04:43

Problem 21

Sketch the graph of $ f $ by hand and use your sketch to find the absolute and local maximum and minimum values of $ f $. (Use the graphs and tranformations of Section 1.2 and 1.3).

$ f(x) = \sin x $, $ -\pi /2 \leqslant x \leqslant \pi /2 $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
05:34

Problem 22

Sketch the graph of $ f $ by hand and use your sketch to find the absolute and local maximum and minimum values of $ f $. (Use the graphs and transformations of Section 1.2 and 1.3).

$ f(x) = \cos t $, $ -3 \pi /2 \leqslant t \leqslant 3\pi /2 $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
05:44

Problem 23

Sketch the graph of $ f $ by hand and use your sketch to find the absolute and local maximum and minimum values of $ f $. (Use the graphs and transformations of Section 1.2 and 1.3).

$ f(x) = \ln x $, $ 0 < x \leqslant 2 $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
05:31

Problem 24

Sketch the graph of $ f $ by hand and use your sketch to find the absolute and local maximum and minimum values of $ f $. (Use the graphs and transformations of Section 1.2 and 1.3).

$ f(x) = | x | $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
09:57

Problem 25

Sketch the graph of $ f $ by hand and use your sketch to find the absolute and local maximum and minimum values of $ f $. (Use the graphs and transformations of Section 1.2 and 1.3).

$ f(x) = 1 - \sqrt{x} $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
04:22

Problem 26

Sketch the graph of $ f $ by hand and use your sketch to find the absolute and local maximum and minimum values of $ f $. (Use the graphs and transformations of Section 1.2 and 1.3).

$ f(x) = e^x $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
07:45

Problem 27

Sketch the graph of $ f $ by hand and use your sketch to find the absolute and local maximum and minimum values of $ f $. (Use the graphs and transformations of Section 1.2 and 1.3).

$ f(x) = \left\{
\begin{array}{ll}
x^2 & \mbox{ if}-1 \leqslant x \leqslant 0\\
2 - 3x & \mbox{ if} 0 < x \leqslant 1
\end{array} \right. $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
07:16

Problem 28

Sketch the graph of $ f $ by hand and use your sketch to find the absolute and local maximum and minimum values of $ f $. (Use the graphs and transformations of Section 1.2 and 1.3).

$ f(x) = \left\{
\begin{array}{ll}
2x + 1 & \mbox{ if} 0 \leqslant x < 1\\
4 - 2x & \mbox{ if} 1 \leqslant x \leqslant 3
\end{array} \right. $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
03:15

Problem 29

Find the critical numbers of the function.

$ f(x) = 4 + \frac{1}{3}x - \frac{1}{2}x^2 $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
02:03

Problem 30

Find the critical numbers of the function.

$ f(x) = x^3 + 6x^2 - 15x $

Mary Wakumoto
Mary Wakumoto
Numerade Educator
06:29

Problem 31

Find the critical numbers of the function.

$ f(x) = 2x^3 - 3x^2 - 36x $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
06:02

Problem 32

Find the critical numbers of the function.

$ f(x) = 2x^3 + x^2 + 2x $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
00:44

Problem 33

Find the critical numbers of the function.

$ g(t) = t^4 + t^3 + t^2 + 1 $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:26

Problem 34

Find the critical numbers of the function.

$ g(t) = | 3t - 4 | $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:45

Problem 35

Find the critical numbers of the function.

$ g(y) = \frac{ y - 1}{ y^2 - y + 1} $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
11:16

Problem 36

Find the critical numbers of the function.

$ h(p) = \frac{p - 1}{p^2 + 4} $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
00:49

Problem 37

Find the critical numbers of the function

$ h(t) = t^\frac{3}{4} - 2t^\frac{1}{4} $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:46

Problem 38

Find the critical numbers of the function.

$ g(x) = \sqrt[3]{4 - x^2} $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:02

Problem 39

Find the critical numbers of the function.

$ F(x) = x^\frac{4}{5} (x - 4)^2 $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
09:59

Problem 40

Find the critical numbers of the function.

$ g(\theta) = 4\theta - \tan\theta $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
00:59

Problem 41

Find the critical numbers of the function.

$ f(\theta) = 2 \cos \theta + \sin^2 \theta $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
12:26

Problem 42

Find the critical numbers of the function.

$ h(t) = 3t - \arcsin t $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
06:30

Problem 43

Find the critical numbers of the function.

$ f(x) = x^2 e^{-3x} $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
07:53

Problem 44

Find the critical numbers of the function.

$ f(x) = x^{-2} \ln x $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
04:01

Problem 45

A formula for the derivative of a function $f$ is given. How many critical numbers does $f$ have?

$ f'(x) = 5e^{-0.1 |x|} \sin x - 1 $

Yuki Hotta
Yuki Hotta
Numerade Educator
12:10

Problem 46

A formula for the derivative of a function $f$ is given. How many critical numbers does $f$ have?

$ f'(x) = \frac{100 \cos^2 x}{10 + x^2} - 1 $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
07:40

Problem 47

Find the absolute maximum and absolute minimum values of $f$ on the given interval.

$ f(x) = 12 + 4x - x^2 $, $ [0, 5] $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
09:23

Problem 48

Find the absolute maximum and absolute minimum values of $f$ on the given interval.

$ f(x) = 5 + 54x - 2x^3 $, $ [0, 4] $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
08:41

Problem 49

Find the absolute maximum and absolute minimum values of $f$ on the given interval.

$ f(x) = 2x^3 - 3x^2 - 12x + 1 $, $ [-2, 3] $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
09:09

Problem 50

Find the absolute maximum and absolute minimum values of $f$ on the given interval.

$ f(x) = x^3 - 6x^2 + 5 $, $ [-3, 5] $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
11:12

Problem 51

Find the absolute maximum and absolute minimum values of $f$ on the given interval.

$ f(x) = 3x^4 - 4x^3 - 12x^2 + 1 $, $ [-2, 3] $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
09:45

Problem 52

Find the absolute maximum and absolute minimum values of $f$ on the given interval.

$ f(t) = (t^2 - 4)^3 $, $ [-2, 3] $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
08:05

Problem 53

Find the absolute maximum and absolute minimum values of $f$ on the given interval.

$ f(x) = x + \frac{1}{x} $, $ [0.2, 4] $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
05:03

Problem 54

Find the absolute maximum and absolute minimum values of $f$ on the given interval.

$ f(x) = \frac{x}{x^2 - x + 1} $, $ [0, 3] $

Yuki Hotta
Yuki Hotta
Numerade Educator
15:23

Problem 55

Find the absolute maximum and absolute minimum values of $f$ on the given interval.

$ f(t) = t - \sqrt[3]{x} $, $ [-1, 4] $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
14:22

Problem 56

Find the absolute maximum and absolute minimum values of $f$ on the given interval.

$ f(t) = \frac{\sqrt{t}}{1 + t^2} $, $ [0, 2] $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
16:57

Problem 57

Find the absolute maximum and absolute minimum values of $f$ on the given interval.

$ f(t) = 2 \cos t + \sin 2t $, $ [0, \pi /2] $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
26:55

Problem 58

Find the absolute maximum and absolute minimum values of $f$ on the given interval.

$ f(t) = t + \cot (t/2) $, $ [\pi /4, 7\pi /4] $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
16:40

Problem 59

Find the absolute maximum and absolute minimum values of $f$ on the given interval.

$ f(x) = x^{-2} \ln x $, $ [\frac{1}{2}, 4] $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
11:05

Problem 60

Find the absolute maximum and absolute minimum values of $f$ on the given interval.

$ f(x) = xe^{x/2} $, $ [-3, 1] $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
11:47

Problem 61

Find the absolute maximum and absolute minimum values of $f$ on the given interval.

$ f(x) = \ln (x^2 + x + 1) $, $ [-1, 1] $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
13:51

Problem 62

Find the absolute maximum and absolute minimum values of $f$ on the given interval.

$ f(x) = x - 2 \tan^{-1} x $, $ [0, 4] $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
01:13

Problem 63

If $a$ and $b$ are positive numbers, find the maximum value of $ f(x) = x^a (1 - x)^b $, $ 0 \leqslant x \leqslant 1 $.

Amrita Bhasin
Amrita Bhasin
Numerade Educator
09:25

Problem 64

Use a graph to estimate the critical numbers of $ f(x) = | 1 + 5x - x^3 | $ correct to one decimal place.

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
07:23

Problem 65

(a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places.
(b) Use calculus to find the exact maximum and minimum values.

$ f(x) = x^5 - x^3 + 2 $, $ -1 \leqslant x \leqslant 1 $

Yuki Hotta
Yuki Hotta
Numerade Educator
16:23

Problem 66

(a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places.
(b) Use calculus to find the exact maximum and minimum values.

$ f(x) = e^x + e^{-2x} $, $ 0 \leqslant x \leqslant 1 $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
01:32

Problem 67

(a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places.
(b) Use calculus to find the exact maximum and minimum values.

$ f(x) = x \sqrt{x - x^2} $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
18:59

Problem 68

(a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places.
(b) Use calculus to find the exact maximum and minimum values.

$ f(x) = x - 2\cos x $, $ -2 \leqslant x \leqslant 0 $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
03:16

Problem 69

After the consumption of an alcoholic beverage, the concentration of alcohol in the bloodstream (blood alcohol concentration, or BAC) surges as the alcohol is absorbed, followed by a gradual decline as the alcohol is metabolized. The function

$$ C(t) = 1.35te^{-2.802t} $$

models the average BAC, measured in mg/mL, in a group of eight male subjects $t$ hours after rapid consumption of $15$ ml of ethanol (corresponding to one alcoholic drink). What is the maximum average BAC during the first $3$ hours? When does it occur?

Source: Adapted from P. Wilkinson et. al., "Pharmacokinetics of Ethanol after Oral Administration in the Fasting State," Journal of Pharmacokinetics and Biopharmaceutics 5 (1977): 207-24.

Leon Druch
Leon Druch
Numerade Educator
01:45

Problem 70

After an antibiotic taken is taken, the concentration of the antibiotic in the bloodstream is modeled by the function

$$ C(t) = 8(e^{-0.4t} - e^{-0.6t}) $$

where the time $t$ is measured in hours and $C$ is measured in $ \mu $g/mL. What is the maximum concentration of the antibiotic during the first $12$ hours?

Leon Druch
Leon Druch
Numerade Educator
07:17

Problem 71

Between 0 $ ^\circ C $ and 30 $ ^\circ C $, the volume $V$ (in cubic centimeters) of $1$ kg of water at a temperature $T$ is given approximately by the formula
$$ V = 999.87 - 0.06426T + 0.0085043T^3 - 0.0000679T^3 $$
Find the temperature at which water has its maximum density.

Mutahar Mehkri
Mutahar Mehkri
Numerade Educator
01:10

Problem 72

An object with weight $W$ is dragged along a horizontal plane by force acting along a rope attached to the object. If the rope makes an angle $ \theta $ with the plane, then the magnitude of the force is
$$ F = \frac{\mu W}{\mu \sin \theta + \cos \theta} $$
where $ \mu $ is a positive constant called the coefficient of friction and where $ 0 \leqslant \theta \leqslant \pi /2 $. Show that $F$ is minimized when $ \tan \theta = \mu $.

Amrita Bhasin
Amrita Bhasin
Numerade Educator
05:30

Problem 73

The water level, measured in feet above mean sea level, of Lake Lanier in Georgia, USA, during 2012 can be modeled by the function
$$ L(t) = 0.01441t^3 - 0.4177t^3 + 2.703t + 1060.1 $$
where $t$ is measured in months since January 1, 2012. Estimate when the water level was highest during 2012.

Bobby Barnes
Bobby Barnes
University of North Texas
03:58

Problem 74

On May 7, 1992, the space shuttle Endeavour was launched on mission STS-49, the purpose of which was to install a new perigee kick motor in an Intelsat communications satellite. The table gives the velocity data for the shuttle between liftoff and the jettisoning of the solid rocket boosters.
(a) Use a graphing calculator or computer to find the cubic polynomial that best models the velocity of the shuttle for the time interval $ t \in [0, 125] $. Then graph this polynomial.
(b) Find a model for the acceleration of the shuttle and use it to estimate the maximum and minimum values of the acceleration during the first $125$ seconds.

DM
David Mccaslin
Numerade Educator
09:15

Problem 75

When a foreign object lodged in the trachea (wind pipe) forces a person to cough, the diaphragm thrusts upward causing an increase in pressure in the lungs. This is accompanied by a contraction in the trachea, making a narrower channel for the expelled air to flow through. For a given amount of air to escape in a fixed time, it must move faster through the narrower channel than the wider one. The greater the velocity of the air-stream, the greater the force on the foreign object. X rays show that the radius of the circular tracheal tube contracts to about two-thirds of its normal radius during a cough. According to a mathematical model of coughing, the velocity $v$ of the air-stream is related to the radius $r$ of the trachea by the equation
$$ v(r) = k(r_o - r) r^2 $$ $$ \frac{1}{2}r_o \leqslant r \leqslant r_o $$
where $k$ is constant and $r_o$ is the normal radius of the trachea.
The restriction of $r$ is due to the fact that the tracheal wall stiffens under pressure and a contraction greater than $ \frac{1}{2}r_o $ is prevented (otherwise the person would suffocate.
(a) Determine the value of $r$ in the interval $ [\frac{1}{2}r_o, r_o] $ at which $v$ has an absolute maximum. How does this compare with experimental evidence?
(b) What is the absolute maximum value of $v$ on the interval?
(c) Sketch the graph of $v$ on the interval $ [0, r_o] $.

Mary Wakumoto
Mary Wakumoto
Numerade Educator
02:13

Problem 76

Show that $5$ is a critical number of the function
$$ g(x) = 2 + (x - 5)^3 $$
but $g$ does not have a local extreme value at $5$.

Yuki Hotta
Yuki Hotta
Numerade Educator
00:36

Problem 77

Prove that the function
$$ f(x) = x^101 + x^51 + x + 1 $$
has neither a local maximum nor a local minimum.

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:44

Problem 78

If $f$ has a local minimum value at $c$, show that the function $ g(x) = - f(x) $ has a local maximum value at $c$.

Amrita Bhasin
Amrita Bhasin
Numerade Educator
06:13

Problem 79

Prove Fermat's Theorem for the case in which $f$ has a local minimum at $c$.

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
16:05

Problem 80

A cubic function is a polynomial of degree $3$; that is, it has the form $ f(x) = ax^3 + bx^2 + cx + d$, where $ a \not= 0 $.
(a) Show that a cubic function can have two, one, or no critical number(s). Give examples and sketches to illustrate the three possibilities.
(b) How many local extreme values can a cubic function have?

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator

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