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Chapter 4

Applications of Differentiation

Educators

MP
FP
WZ

Problem 1

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

FP
Fahad P.
Numerade Educator

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?

FP
Fahad P.
Numerade Educator

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.

FP
Fahad P.
Numerade Educator

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.

WZ
Wen Z.
Numerade Educator

Problem 5

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

FP
Fahad P.
Numerade Educator

Problem 6

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

WZ
Wen Z.
Numerade Educator

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 $

FP
Fahad P.
Numerade Educator

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 $

FP
Fahad P.
Numerade Educator

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 $

FP
Fahad P.
Numerade Educator

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.

FP
Fahad P.
Numerade Educator

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 $.

FP
Fahad P.
Numerade Educator

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.

FP
Fahad P.
Numerade Educator

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.

FP
Fahad P.
Numerade Educator

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.

FP
Fahad P.
Numerade Educator

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 $

FP
Fahad P.
Numerade Educator

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 $

FP
Fahad P.
Numerade Educator

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 $

FP
Fahad P.
Numerade Educator

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 $

FP
Fahad P.
Numerade Educator

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 $

FP
Fahad P.
Numerade Educator

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 $

FP
Fahad P.
Numerade Educator

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 $

FP
Fahad P.
Numerade Educator

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 $

FP
Fahad P.
Numerade Educator

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 $

FP
Fahad P.
Numerade Educator

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 | $

FP
Fahad P.
Numerade Educator

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} $

FP
Fahad P.
Numerade Educator

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 $

FP
Fahad P.
Numerade Educator

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. $

FP
Fahad P.
Numerade Educator

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. $

FP
Fahad P.
Numerade Educator

Problem 29

Find the critical numbers of the function.

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

FP
Fahad P.
Numerade Educator

Problem 30

Find the critical numbers of the function.

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

MP
Mathrix P.
Numerade Educator

Problem 31

Find the critical numbers of the function.

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

Amrita B.
Numerade Educator

Problem 32

Find the critical numbers of the function.

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

Amrita B.
Numerade Educator

Problem 33

Find the critical numbers of the function.

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

Amrita B.
Numerade Educator

Problem 34

Find the critical numbers of the function.

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

Amrita B.
Numerade Educator

Problem 35

Find the critical numbers of the function.

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

Amrita B.
Numerade Educator

Problem 36

Find the critical numbers of the function.

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

Amrita B.
Numerade Educator

Problem 37

Find the critical numbers of the function

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

Amrita B.
Numerade Educator

Problem 38

Find the critical numbers of the function.

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

Amrita B.
Numerade Educator

Problem 39

Find the critical numbers of the function.

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

Amrita B.
Numerade Educator

Problem 40

Find the critical numbers of the function.

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

Amrita B.
Numerade Educator

Problem 41

Find the critical numbers of the function.

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

Amrita B.
Numerade Educator

Problem 42

Find the critical numbers of the function.

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

Amrita B.
Numerade Educator

Problem 43

Find the critical numbers of the function.

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

Amrita B.
Numerade Educator

Problem 44

Find the critical numbers of the function.

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

Amrita B.
Numerade Educator

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 $

Amrita B.
Numerade Educator

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 $

Amrita B.
Numerade Educator

Problem 47

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

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

Amrita B.
Numerade Educator

Problem 48

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

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

Amrita B.
Numerade Educator

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] $

Amrita B.
Numerade Educator

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] $

Amrita B.
Numerade Educator

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] $

Amrita B.
Numerade Educator

Problem 52

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

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

Amrita B.
Numerade Educator

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] $

Amrita B.
Numerade Educator

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] $

Amrita B.
Numerade Educator

Problem 55

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

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

Amrita B.
Numerade Educator

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] $

Amrita B.
Numerade Educator

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] $

Amrita B.
Numerade Educator

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] $

Amrita B.
Numerade Educator

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] $

Amrita B.
Numerade Educator

Problem 60

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

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

Amrita B.
Numerade Educator

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] $

Amrita B.
Numerade Educator

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] $

Amrita B.
Numerade Educator

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 B.
Numerade Educator

Problem 64

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

Amrita B.
Numerade Educator

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 $

Amrita B.
Numerade Educator

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 $

Amrita B.
Numerade Educator

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 B.
Numerade Educator

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 $

Amrita B.
Numerade Educator

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.

Amrita B.
Numerade Educator

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?

Amrita B.
Numerade Educator

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.

Amrita B.
Numerade Educator

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 B.
Numerade Educator

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.

Amrita B.
Numerade Educator

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.

Amrita B.
Numerade Educator

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] $.

WZ
Wen Z.
Numerade Educator

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 $5$.

Amrita B.
Numerade Educator

Problem 77

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

Amrita B.
Numerade Educator

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 B.
Numerade Educator

Problem 79

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

Amrita B.
Numerade Educator

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?

Amrita B.
Numerade Educator