Deck 9: Applications of the Derivative

Question 1

Find the rate of change of x with respect to p. $$p = \sqrt { \frac { 200 - x } { 2 x } } , \quad 0 < x \leq 200$$

A) $- \frac { 4 x p } { 2 p ^ { 2 } + 1 }$
B) $\frac { 4 x p } { 2 p ^ { 2 } + 1 }$
C) $- \frac { 4 x } { 2 p ^ { 2 } + 1 }$
D) $\frac { 4 x } { 2 p ^ { 2 } + 1 }$
E) $- \frac { 4 x p } { 2 p + 1 }$

Answer 1

$- \frac { 4 x p } { 2 p ^ { 2 } + 1 }$

Question 2

Find the slope of the graph at the given point. $x ^ { 2 } + y ^ { 2 } = 4$

A)0
B)3
C)5
D)4
E)7

Answer 2

0

Question 3

Find the $f ^ { ( 6 ) } ( x )$ of $f ^ { ( 4 ) } ( x ) = \left( x ^ { 2 } + 1 \right) ^ { 2 }$ .

A) $12 x ^ { 2 } + 4$
B) $12 x ^ { 2 } + 2$
C) $6 x ^ { 2 } + 4$
D) $6 x ^ { 2 } + 2$
E) $12 x ^ { 2 } + 1$

Answer 3

$12 x ^ { 2 } + 4$

Question 4

A brick becomes dislodged from the Empire State Building (at a height of 1175 feet) and falls to the sidewalk below. Write the position s(t), velocity v(t), and acceleration a(t) as functions of time.

A) $s ( t ) = 16 t ^ { 2 } + 1175$ ; $v ( t ) = 32 t$ ; $a ( t ) = 32$
B) $s ( t ) = - 16 t ^ { 2 } - 1175$ ; $v ( t ) = - 32 t$ ; $a ( t ) = - 32$
C) $s ( t ) = - 16 t ^ { 2 } + 1175$ ; $v ( t ) = - 32 t$ ; $a ( t ) = - 32$
D) $s ( t ) = 16 t ^ { 2 } - 1175$ ; $v ( t ) = - 32 t$ ; $a ( t ) = - 32$
E) $s ( t ) = - 16 t ^ { 2 } + 1175$ ; $v ( t ) = - 32$ ; $a ( t ) = - 32 t$

Question 5

Find $y ^ { \prime }$ implicitly for $8 x ^ { 9 } - y ^ { 9 } = 5$

A) $y ^ { \prime } = \frac { 8 x ^ { 9 } } { y ^ { 9 } }$
B) $y ^ { \prime } = \frac { y ^ { 9 } } { 8 x ^ { 9 } }$
C) $y ^ { \prime } = \frac { 8 x ^ { 8 } } { y ^ { 8 } }$
D) $y ^ { \prime } = \frac { y ^ { 8 } } { 8 x ^ { 8 } }$
E) $y ^ { \prime } = \frac { x ^ { 8 } } { 8 y ^ { 8 } }$

Question 6

Find the second derivative of the function. $f ( x ) = 3 x ^ { \frac { 4 } { 7 } }$

A) $f ^ { \prime \prime } ( x ) = \frac { - 36 } { 49 } x ^ { \frac { 3 } { 7 } }$
B) $f ^ { \prime \prime } ( x ) = \frac { 4 } { 49 } x ^ { \frac { - 10 } { 7 } }$
C) $f ^ { \prime \prime } ( x ) = \frac { 147 } { 49 } x ^ { \frac { - 10 } { 7 } }$
D) $f ^ { \prime \prime } ( x ) = \frac { - 36 } { 49 } x ^ { \frac { - 10 } { 7 } }$
E)None of the above

Question 7

Find the slope of the graph at the given point. $( 4 - x ) y ^ { 2 } = x ^ { 3 }$

A)2
B)0
C)1
D)3
E)5

Question 8

Find $\frac { d y } { d x }$ for the equation $\sqrt { x y } = x - 20 y$ by implicit differentiation and evaluate the derivative at the point $( 100,4 )$ .

A) $- \frac { 1 } { 25 }$
B) $\frac { 1 } { 25 }$
C) $\frac { 3 } { 25 }$
D) $- \frac { 3 } { 25 }$
E)0

Question 9

Find the third derivative of the function $f ( x ) = x ^ { 5 } - 3 x ^ { 4 }$ .

A) $60 x ^ { 2 } - 72 x$
B) $30 x ^ { 2 } - 36 x$
C) $60 x ^ { 2 } - 72 x ^ { 2 }$
D) $60 x ^ { 2 } - 36 x$
E) $30 x ^ { 2 } - 36 x$ .

Question 10

Find the second derivative for the function $f ( x ) = \frac { 7 x } { 7 x + 3 }$ and solve the equation $f ^ { \prime \prime } ( x ) = 0$ .

A)0
B)3
C)no solution
D)-3
E) $- \frac { 1 } { 3 }$

Question 11

Find the third derivative. $y = \frac { 4 } { x ^ { 9 } }$

A) $\frac { - 3960 } { x ^ { 11 } }$
B) $\frac { 3960 } { x ^ { 12 } }$
C) $0$
D) $\frac { 440 } { x ^ { 11 } }$
E) $\frac { - 3960 } { x ^ { 12 } }$

Question 12

Let x represent the units of labor and y the capital invested in a manufacturing process. When 135,540 units are produced, the relationship between labor and capital can be modeled by $100 x ^ { 0.75 } y ^ { 0.25 } = 135,540$ . Find the rate of change of y with respect to x when $x = 1500 \text { and } y = 135,540$ .

A)-2
B)0
C)3
D)-7
E)5

Question 13

Find $\frac { d y } { d x }$ for the equation $\frac { x + 4 y } { 5 x - 6 y } = 6$ .

A) $\frac { d ^ { 2 } y } { d x } = - \frac { 29 } { 40 }$
B) $\frac { d y } { d x } = \frac { 31 } { 40 }$
C) $\frac { d y } { d x } = \frac { 29 } { 40 }$
D) $\frac { d ^ { 2 } y } { d x } = - \frac { 31 } { 40 }$
E) $\frac { d y } { d x } = 6$

Question 14

Find the second derivative for the function $f ( x ) = 3 x ^ { 3 } + 27 x ^ { 2 } - 12 x - 32$ and solve the equation $f ^ { \prime \prime } ( x ) = 0$ .

A)-3
B)3
C)0
D)32
E)12

Question 15

Find the indicated derivative. Find $y ^ { ( 4 ) } \text { if } y = x ^ { 7 } - 3 x ^ { 3 } \text {. }$

A) $210 x ^ { 4 }$
B) $210 x ^ { 3 }$
C) $210 x ^ { 3 } - 18 x$
D) $840 x ^ { 4 } - 18 x$
E) $840 x ^ { 3 }$

Question 16

Find the rate of change of x with respect to p. $$p = \frac { 2 } { 0.00001 x ^ { 3 } + 0.1 x } x \geq 0$$

A) $- \frac { 2 } { p ^ { 2 } \left( 0.00003 x ^ { 2 } + 0.1 \right) }$
B) $- \frac { 2 } { p \left( 0.00003 x ^ { 2 } + 0.1 \right) }$
C) $- \frac { 2 } { p ^ { 2 } x \left( 0.00003 x ^ { 2 } + 0.1 \right) }$
D) $- \frac { 2 } { p x \left( 0.00003 x ^ { 2 } + 0.1 \right) }$
E) $- \frac { 2 x } { p ^ { 2 } \left( 0.00003 x ^ { 2 } + 0.1 \right) }$

Question 17

Find dy/dx for the following equation: $8 x + y ^ { 2 } - 7 y + 9 = 0$

A) $\frac { d y } { d x } = \frac { 7 } { 8 - 2 y }$
B) $\frac { d y } { d x } = \frac { 8 } { 7 - 2 y }$
C) $\frac { d y } { d x } = \frac { 4 } { 7 - y }$
D) $\frac { d y } { d x } = \frac { 7 } { 7 - y }$
E) $\frac { d y } { d x } = \frac { 4 } { 8 - y }$

Question 18

Find the value $g ^ { \prime \prime } ( 9 )$ for the function $g ( t ) = 5 t ^ { 6 } + 5 t ^ { 4 } + 6$ .

A)989,010
B)1,786,050
C)2,690,016
D)1,786,056
E)10,701,720

Question 19

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If $y = f ( x ) g ( x ) \text {, then } y ^ { \prime } = f ^ { \prime } ( x ) g ^ { \prime } ( x )$

A)True
B)False. The product rule is $[ f ( x ) g ( x ) ]^\prime = f ( x ) g ^ { \prime } ( x ) + g ( x ) f ^ { \prime } ( x )$

Question 20

Find $d y / d x$ implicitly and explicitly(the explicit functions are shown on the graph) and show that the results are equivalent. Use the graph to estimate the slope of the tangent line at the labeled point. Then verify your result analytically by evaluating $d y / d x$ at the point.

A) $\frac { 1 } { 2 y } , - \frac { 1 } { 2 }$
B) $- \frac { 1 } { 2 y } , \frac { 1 } { 2 }$
C) $- \frac { 1 } { 2 y } , - \frac { 1 } { 2 }$
D) $\frac { 1 } { 2 y } , \frac { 1 } { 2 }$
E) $\frac { 1 } { 2 } , - \frac { 1 } { 2 }$

Question 21

An airplane flying at an altitude of 5 miles passes directly over a radar antenna. When the airplane is 50 miles away (s = 50), the radar detects that the distance s is changing at a rate of 280 miles per hour. What is the speed of the airplane? Round your answer to the nearest integer.

A)281 mi/hr
B)271 mi/hr
C)563 mi/hr
D)141 mi/hr
E)135 mi/hr

Question 22

A baseball diamond has the shape of a square with sides 90 feet long (see figure). A player running from second base to third base at a speed of 26 feet per second is 80 feet from third base. At what rate is the player's distance s from home plate changing? Round your answer to one decimal place.

A)-50.4 feet/second
B)-0.2 feet/second
C)-0.7 feet/second
D)-17.3 feet/second
E)-1.9 feet/second

Question 23

Volume and radius. Suppose that air is being pumped into a spherical balloon at a rate of $8 \mathrm { in. } ^ { 3 } / \mathrm { min }$ At what rate is the radius of the balloon increasing when the radius is 3 in.?

A) $\frac { d r } { d t } = \frac { 8 } { 9 \pi }$
B) $\frac { d r } { d t } = \frac { 2 } { 3 \pi }$
C) $\frac { d r } { d t } = \frac { 9 } { 8 \pi }$
D) $\frac { d r } { d t } = \frac { 3 } { 8 \pi }$
E) $\frac { d r } { d t } = \frac { 2 } { 9 \pi }$

Question 24

Use the graph of $y = f ( x )$ to identify at which of the indicated points the derivative $f ^ { \prime } ( x )$ changes from positive to negative.

A)(5,6)
B)(-1,2), (2,4)
C)(-1,2), (5,6)
D)(-1,2)
E)(2,4), (5,6)

Question 25

Use the graph of $y = f ( x )$ to identify at which of the indicated points the derivative $f ^ { \prime } ( x )$ changes from negative to positive.

A)(2,4), (5,6)
B)(-1,2)
C)(2,4)
D)(-1,2), (2,4)
E)(-1,2), (5,6)

Question 26

Given $x y = 6,$ find $\frac { d y } { d t }$ when x = 4 and $\frac { d x } { d t } = - 2$

A) $\frac { d y } { d t } = \frac { 27 } { 4 }$
B) $\frac { d y } { d t } = - \frac { 3 } { 4 }$
C) $\frac { d y } { d t } = \frac { 3 } { 4 }$
D) $\frac { d y } { d t } = \frac { 4 } { 3 }$
E) $\frac { d y } { d t } = - \frac { 4 } { 27 }$

Question 27

Profit. Suppose that the monthly revenue and cost (in dollars) for x units of a product are $R = 200 x - \frac { x ^ { 2 } } { 10 } \text { and } C = 6000 + 10 x$ At what rate per month is the profit changing if the number of units produced and sold is 100 and is increasing at a rate of 10 units per month?

A) $\$ 18,800$ per month
B) $\$ 1700$ per month
C) $\$ 1800$ per month
D) $\$ - 100$ per month
E) $\$ 19,800$ per month

Question 28

The lengths of the edges of a cube are increasing at a rate of 7 ft/min. At what rate is the surface area changing when the edges are 22 ft long?

A)294 ft2/min
B)1848 ft2/min
C)924 ft2/min
D)6468 ft2/min
E)154 ft2/min

Question 29

The radius r of a sphere is increasing at a rate of 3 inches per minute. Find the rate of change of volume when r = 17 inches. Round your answer to one decimal place.

A)3631.7 cubic inches per minute
B)20,579.5 cubic inches per minute
C)61,738.6 cubic inches per minute
D)10,895.0 cubic inches per minute
E)32,685.1 cubic inches per minute

Question 30

Identify the open intervals where the function $f ( x ) = 4 x ^ { 2 } + 6 x - 1$ is increasing or decreasing.

A)decreasing: $\left( - \infty , - \frac { 3 } { 4 } \right)$ ; increasing: $\left( - \frac { 3 } { 4 } , \infty \right)$
B)increasing: $\left( - \infty , - \frac { 3 } { 4 } \right)$ ; decreasing: $\left( - \frac { 3 } { 4 } , \infty \right)$
C)increasing on $( - \infty , \infty )$
D)decreasing on $( - \infty , \infty )$
E)none of the above

Question 31

Both a function and its derivative are given. Use them to find all critical numbers. $f ( x ) = x - 9 x ^ { 2 / 3 } + 7 \quad f ^ { \prime } ( x ) = \frac { x ^ { 1 / 3 } - 6 } { x ^ { 1 / 3 } }$

A) $x = 0$
B) $x = 216$
C) $x = 0 , x = - 101$
D) $x = 0 , x = 216$
E) $x = - 101 , x = 216$

Question 32

Assume that x and y are differentiable functions of t. Find dx/dt given that $x = - 2$ , $y = - 8$ , and $d y / d t = 6$ $y ^ { 2 } - x ^ { 2 } = 60$

A)-3.00
B)2.67
C)1.50
D)-48.00
E)24.00

Question 33

Assume that x and y are differentiable functions of t. Find dy/dt using the given values. $y = 4 x ^ { 3 } + 7 x ^ { 2 } - x$ for $x = 2 , d x / d t = 2$

A)152
B)58
C)116
D)150
E)75

Question 34

Area. The radius, r, of a circle is decreasing at a rate of 2 centimeters per minute. Find the rate of change of area, A, when the radius is $5$ .

A) $\frac { d A } { d t } = - 10 \pi$
B) $\frac { d A } { d t } = - 100 \pi$
C) $\frac { d A } { d t } = 100 \pi$
D) $\frac { d A } { d t } = - 20 \pi$
E) $\frac { d A } { d t } = 20 \pi$

Question 35

Identify the open intervals where the function $f ( x ) = 4 x ^ { 2 } - 5 x - 5$ is increasing or decreasing.

A)decreasing: $\left( - \infty , \frac { 5 } { 8 } \right)$ ; increasing: $\left( \frac { 5 } { 8 } , \infty \right)$
B)increasing: $\left( - \infty , \frac { 5 } { 8 } \right)$ ; decreasing: $\left( \frac { 5 } { 8 } , \infty \right)$
C)increasing on $( - \infty , \infty )$
D)decreasing on $( - \infty , \infty )$
E)none of the above

Question 36

For the given function, find all critical numbers. $y = x ^ { 3 } - 3 x ^ { 2 } - 105 x + 5$

A) $x = 0$
B) $x = - 7$ and $x = - 5$
C) $x = - 7$ and $x = 5$
D) $x = - 5$ and $x = 7$
E) $x = 5$ and $x = 7$

Question 37

A retail sporting goods store estimates that weekly sales and weekly advertising costs are related by the equation $S = 2290 + 90 x + 0.35 x ^ { 2 }$ . The current weekly advertising costs are $1500, and these costs are increasing at a rate of $140 per week. Find the current rate of change of weekly sales.

A)159,600 dollars per week
B)161,890 dollars per week
C)88,390 dollars per week
D)86,100 dollars per week
E)802,390 dollars per week

Question 38

A point is moving along the graph of the function $y = 9 x ^ { 2 } + 4$ such that $\frac { d x } { d t } = 3$ centimeters per second. Find dy/dt for the given values of x. (a) $x = 4~~~~~~~~~~~~~~~~~~~~~~~~~~$ (b) $x = 7$

A) $\frac { d y } { d t } = 4$ $\frac { d y } { d t } = 378$
B) $\frac { d y } { d t } = 216$ $\frac { d y } { d t } = 378$
C) $\frac { d y } { d t } = 378$ $\frac { d y } { d t } = 216$
D) $\frac { d y } { d t } = 7$ $\frac { d y } { d t } = - 216$
E) $\frac { d y } { d t } = 7$ $\frac { d y } { d t } = 378$

Question 39

A point is moving along the graph of the function $y = \frac { 1 } { 8 x ^ { 2 } + 3 }$ such that $\frac { d x } { d t } = 5$ centimeters per second. Find dy/dt when $x = 1$ .

A) $\frac { d y } { d t } = - \frac { 16 } { 121 }$
B) $\frac { d y } { d t } = - \frac { 80 } { 121 }$
C) $\frac { d y } { d t } = \frac { 80 } { 121 }$
D) $\frac { d y } { d t } = \frac { 16 } { 121 }$
E) $\frac { d y } { d t } = - \frac { 80 } { 11 }$

Question 40

Boat docking. Suppose that a boat is being pulled toward a dock by a winch that is 24 ft above the level of the boat deck. If the winch is pulling the cable at a rate of 23 ft/min, at what rate is the boat approaching the dock when it is 32 ft from the dock? Use the figure below.

A) $28.75 ~\mathrm { ft } / \mathrm { min }$
B) $23.00 ~\mathrm { ft } / \mathrm { min }$
C) $38.33~ \mathrm { ft } / \mathrm { min }$
D) $17.25~ \mathrm { ft } / \mathrm { min }$
E) $13.80 ~\mathrm { ft } / \mathrm { min }$

Question 41

Suppose the number y of medical degrees conferred in the United States can be modeled by $y = 0.813 t ^ { 3 } - 55.70 t ^ { 2 } + 1185.2 t + 7752,$ for $0 \leq t \leq 32$ , where t is the time in years, with $t = 0$ corresponding to 1972. Use the test for increasing and decreasing functions to estimate the years during which the number of medical degrees is increasing and the years during which it is decreasing.

A)The number of medical degrees is increasing from 1972 to 1989 and 1997 to 2002, and decreasing during 1989 to 1997.
B)The number of medical degrees is increasing from 1972 to 1988 and 1996 to 2002, and decreasing during 1988 to 1996.
C)The number of medical degrees is increasing from 1972 to 1989 and 1996 to 2002, and decreasing during 1989 to 1996.
D)The number of medical degrees is increasing from 1972 to 1990 and 1996 to 2002, and decreasing during 1990 to 1996.
E)The number of medical degrees is increasing from 1972 to 1989 and 1995 to 2002, and decreasing during 1989 to 1995.

Question 42

Locate the absolute extrema of the function $f ( x ) = - 4 x ^ { 2 } + 8 x - 2$ on the closed interval $[ - 2,2 ]$ .

A)no absolute max; absolute min: f(1)= 2
B)absolute max: f(-2)= -34 ; absolute min: f(1)= 2
C)absolute max: f(1)= 2 ; no absolute min
D)absolute max: f(1)= 2 ; absolute min: f(-2)= -34
E)no absolute max or min

Question 43

Locate the absolute extrema of the function $f ( x ) = x ^ { 3 } - 3 x$ on the closed interval [0,2].

A)absolute max: f(2)= 2 ; absolute min: f(1)= -2
B)absolute max: f(1)= -2 ; absolute min: f(2)= 2
C)absolute max: f(2)= 2 ; no absolute min
D)no absolute max; absolute min: f(2)= 2
E)no absolute max or min

Question 44

Approximate the critical numbers of the function shown in the graph and determine whether the function has a relative maximum, a relative minimum, an absolute maximum, an absolute minimum, or none of these at each critical number on the interval shown.

A)The critical number $x = - 1$ yields an absolute maximum and the critical number $x = 1$ yields an absolute minimum..
B)Both the critical numbers $x = - 1$ & $x = 1$ yield an absolute maximum.
C)The critical number $x = - 1$ yields an absolute minimum and the critical number $x = 1$ yields an absolute maximum.
D)Both the critical numbers $x = - 1$ and $x = 1$ yield an absolute minimum.
E)The critical number $x = - 1$ yields a relative minimum and the critical number $x = 1$ yields a relative maximum.

Question 45

Find the open intervals on which the function $y = \left\{ \begin{array} { l } 2 x + 7 , x \leq 7 \\4 - x ^ { 2 } , x > 7\end{array} \right.$ is increasing or decreasing.

A)The function is increasing on the interval $- \infty < x < 0$ and decreasing on the interval $0 < x < \infty$ .
B)The function is increasing on the interval $7 < x < \infty$ and decreasing on the interval $- \infty < x < 7$ .
C)The function is increasing on the interval $- \infty < x \leq 7$ and decreasing on the interval $7 < x < \infty$ .
D)The function is increasing on the interval $0 < x < \infty$ and decreasing on the interval $- \infty < x < 0$ .
E)The function is increasing on the interval $- \infty < x < 7$ and decreasing on the interval $7 < x < \infty$ .

Question 46

Find the open intervals on which the function $f ( x ) = \frac { x } { x ^ { 2 } + 36 }$ is increasing or decreasing.

A)The function is increasing on the interval $- 6 < x < 6$ , and decreasing on the intervals $- \infty < x < - 6$ and $6 < x < \infty$ .
B)The function is increasing on the interval $- \infty < x < - 6$ , and decreasing on the intervals $- 6 < x < 6$ and $6 < x < \infty$ .
C)The function is increasing on the interval $6 < x < \infty$ , and decreasing on the intervals $- \infty < x < - 6$ and $- 6 < x < 6$ .
D)The function is decreasing on the interval $- 6 < x < 6$ , and increasing on the intervals $- \infty < x < - 6$ and $6 < x < \infty$ .
E)The function is decreasing on the interval $- \infty < x < - 6$ , and increasing on the intervals $- 6 < x < 6$ and $6 < x < \infty$ .

Question 47

Find all relative minima of the given function. $y = x ^ { 4 } - 8 x ^ { 3 } + 16 x ^ { 2 } + 16$

A) $( 0,16 )$
B) $( 2,32 )$
C) $( 4,16 )$
D) $( 0,16 )$ , $( 4,16 )$
E)no relative minima

Question 48

Find the absolute extrema of the function $f ( x ) = \frac { 4 x } { x ^ { 2 } + 1 }$ on the interval $[ 0 , \infty )$ .

A)The maximum of the function is 1 and the minimum of the function is 0.
B)The maximum of the function is 0 and the minimum of the function is -2.
C)The maximum of the function is -2 and the minimum of the function is 0.
D)The maximum of the function is 2 and the minimum of the function is 0.
E)The maximum of the function is 0 and the minimum of the function is 2.

Question 49

A fast-food restaurant determines the cost model, $C = 0.6 x + 5500,0 \leq x \leq 40000$ and revenue model, $R = \frac { 1 } { 30000 } \left( 55000 x - x ^ { 2 } \right)$ for $0 \leq x \leq 40000$ where x is the number of hamburgers sold. Determine the intervals on which the profit function is increasing and on which it is decreasing.

A)The profit function is increasing on the interval $( 18500,40000 )$ and decreasing on the interval $( 0,18500 )$ .
B)The profit function is increasing on the interval $( 0,12500 )$ and decreasing on the interval $( 12500,40000 )$ .
C)The profit function is increasing on the interval $( 0,18500 )$ and decreasing on the interval $( 18500,40000 )$ .
D)The profit function is increasing on the interval $( 12500,40000 )$ and decreasing on the interval $( 0,12500 )$ .
E)The profit function is increasing on the interval $( 0,5500 )$ and decreasing on the interval $( 5500,40000 )$ .

Question 50

Find the x-values of all relative maxima of the given function. $y = \frac { 1 } { 3 } x ^ { 3 } - 4 x ^ { 2 } + 12 x + 9$

A) $x = 0$
B) $x = 6$
C) $x = 4$
D) $x = 2$
E)no relative maxima

Question 51

For the function $f ( x ) = 2 x ^ { 3 } - 12 x ^ { 2 } + 2$ : (a) Find the critical numbers of f (if any);
(b) Find the open intervals where the function is increasing or decreasing; and
(c) Apply the First Derivative Test to identify all relative extrema.
Then use a graphing utility to confirm your results.

A) (a) $x = 0,4$
(b) increasing: $( - \infty , 0 ) \cup ( 4 , \infty )$ ; decreasing: $( 0,4 )$
(c) relative max: $f ( 0 ) = 2$ ; relative min: $f ( 4 ) = - 62$
B) (a) $x = 0,4$
(b) decreasing: $( - \infty , 0 ) \cup ( 4 , \infty )$ ; increasing: $( 0,4 )$
(c) relative min: $f ( 0 ) = 2$ ; relative max: $f ( 4 ) = - 62$
C) (a) $x = 0,1$
(b) increasing: $( - \infty , 0 ) \cup ( 1 , \infty )$ ; decreasing: $( 0,1 )$
(c) relative max: $f ( 0 ) = 2$ ; relative min: $f ( 1 ) = - 8$
D) (a) $x = 0,1$
(b) decreasing: $( - \infty , 0 ) \cup ( 1 , \infty )$ ; increasing: $( 0,1 )$
(c) relative min: $f ( 0 ) = 2$ ; relative max: $f ( 1 ) = - 8$
E) (a) $x = 0,1$
(b) increasing: $( - \infty , 0 ) \cup ( 1 , \infty )$ ; decreasing: $( 0,1 )$
(c) relative $\max : f ( 0 ) = 2$ ; no relative min.

Question 52

For the given function, find the relative minima. $y = x ^ { 3 } - 9 x ^ { 2 } - 48 x + 10$

A) $( - 2,62 )$
B) $( 8 , - 438 )$
C) $( - 2 , - 114 )$
D) $( - 8 , - 694 )$
E)no relative minima

Question 53

Identify the open intervals where the function $f ( x ) = x \sqrt { 24 - x ^ { 2 } }$ is increasing or decreasing.

A)decreasing: $( - \infty , \sqrt { 12 } )$ ; increasing: $( \sqrt { 12 } , \infty )$
B)increasing: $( - \sqrt { 12 } , \sqrt { 12 } )$ ; decreasing: $( - \sqrt { 24 } , - \sqrt { 12 } ) \cup ( \sqrt { 12 } , \sqrt { 24 } )$
C)increasing: $( - \infty , \sqrt { 24 } )$ ; decreasing: $( \sqrt { 24 } , \infty )$
D)increasing: $( - \sqrt { 24 } , - \sqrt { 12 } ) \cup ( \sqrt { 12 } , \sqrt { 24 } )$ ; decreasing: $( - \sqrt { 12 } , \sqrt { 12 } )$
E)decreasing for all x

Question 54

For the given function, find the critical numbers. $y = \frac { x ^ { 4 } } { 4 } - \frac { x ^ { 3 } } { 3 } - 4$

A) $x = 0 \text { and } x = 1$
B) $x = 0 \text { and } x = 4$
C) $x = 0 \text { and } x = - 4$
D) $x = 0 \text { and } x = - 1$
E) $x = - 1 \text { and } x = 1$

Question 55

Find any critical numbers of the function $g ( t ) = t \sqrt { 14 - t }$ , t < 14.

A)0
B) $- \frac { 28 } { 3 }$
C) $\frac { 28 } { 3 }$
D)both A and B
E)both A and C

Question 56

Locate the absolute extrema of the given function on the closed interval [-48,48]. $f ( x ) = \frac { 48 x } { x ^ { 2 } + 16 }$

A)absolute max: f(4)= 6
B)absolute min: f(-4)= -6
C)no absolute max
D)no absolute min
E)both A and D
F)both A and B

Question 57

Find the x-value at which the absolute minimum of f (x) occurs on the interval [a, b]. $f ( x ) = x ^ { 3 } - 75 x + 6 , [ - 15,6 ]$

A) $x = - 10$
B) $x = - 5$
C) $x = 0$
D) $x = 5$
E) $x = 6$

Question 58

Graph a function on the interval $[ - 1,5 ]$ having the following characteristics. Absolute maximum at $x = 4.75$ Absolute minimum at $x = - 1$ Relative maximum at $x = 0.3$ Relative minimum at $x = 3.1$

A)
B)
C)
D)
E)

Question 59

Find all relative maxima of the given function. $y = x ^ { 4 } - 8 x ^ { 3 } + 16 x ^ { 2 } + 5$

A) $( 0,5 )$
B) $( 2,21 )$
C) $( 4,5 )$
D) $( 0,5 )$ , $( 4,5 )$
E)no relative maxima

Question 60

Find the absolute extrema of the function $h ( t ) = ( t - 3 ) ^ { 2 / 3 }$ on the closed interval $[ - 1,6 ]$ . Round your answer to two decimal places.

A)The maximum of the function is 1 and the minimum of the function is 0.
B)The maximum of the function is 2.52 and the minimum of the function is 1.
C)The maximum of the function is 2.52 and the minimum of the function is 0.
D)The maximum of the function is1 and the minimum of the function is 2.08.
E)The maximum of the function is 0 and the minimum of the function is 2.08.

Question 61

The graph of f is shown. Graph f, f' and f'' on the same set of coordinate axes.

A)
B)
C)
D)
E)none of the above

Question 62

The graph of f is shown in the figure. Sketch a graph of the derivative of f.

A)
B)The derivative of f does not exist.
C)
D)
E)

Question 63

State the signs of $f ^ { \prime } ( x )$ and $f ^ { \prime \prime } ( x )$ on the interval (0, 2).

A) $\begin{array} { l } f ^ { \prime } = 0 \\f ^ { \prime \prime } < 0\end{array}$
B) $\begin{array} { l } f ^ { \prime } > 0 \\f ^ { \prime \prime } < 0\end{array}$
C) $\begin{array} { l } f ^ { \prime } < 0 \\f ^ { \prime \prime } < 0\end{array}$
D) $\begin{array} { l } f ^ { \prime } < 0 \\f ^ { \prime \prime } > 0\end{array}$
E) $\begin{array} { l } f ^ { \prime } > 0 \\f ^ { \prime \prime } > 0\end{array}$

Question 64

Find all relative extrema of the function $f ( x ) = \sqrt { 36 - x ^ { 2 } }$ . Use the Second-Derivative Test when applicable.

A)The relative minimum is $( 0,6 )$ and the relative maximum is $( - 6,0 )$ .
B)The relative maximum is $( 0,6 )$ .
C)The relative minimum is $( 0,6 )$ .
D)The relative maximum is $( 0,6 )$ and the relative minima are $( 6,0 )$ and $( - 6,0 )$ .
E)The relative minimum is $( 0,6 )$ and the relative maximum is $( 6,0 )$ .

Question 65

Determine the open intervals on which the graph of $y = - 3 x ^ { 3 } + 8 x ^ { 2 } + 8 x - 5$ is concave downward or concave upward.

A)concave downward on $( - \infty , \infty )$
B)concave downward on $\left( - \infty , \frac { 8 } { 9 } \right)$ ; concave upward on $\left( \frac { 8 } { 9 } , \infty \right)$
C)concave upward on $\left( - \infty , - \frac { 8 } { 9 } \right)$ ; concave downward on $\left( - \frac { 8 } { 9 } , \infty \right)$
D)concave downward on $\left( - \infty , - \frac { 8 } { 9 } \right)$ ; concave upward on $\left( - \frac { 8 } { 9 } , \infty \right)$
E)concave upward on $\left( - \infty , \frac { 8 } { 9 } \right)$ ; concave downward on $\left( \frac { 8 } { 9 } , \infty \right)$

Question 66

Find all relative extrema of the function $f ( x ) = \frac { 4 } { x ^ { 2 } + 2 }$ . Use the Second-Derivative Test when applicable.

A)The relative maximum is $( 2,0 )$ .
B)The relative minimum is $( 0,2 )$ .
C)The relative maximum is $( 0,2 )$ .
D)The relative minimum is $( 0,2 )$
E)The relative maximum is $( 4,0 )$ .

Question 67

The graph of f is shown in the figure. Sketch a graph of the derivative of f.

A)
B)
C)
D)
E)

Question 68

Medication. The number of milligrams x of a medication in the bloodstream t hours after a dose is taken can be modeled by $x ( t ) = \frac { 5000 t } { t ^ { 2 } + 13 }$ $t > 0$ . Find the t-value at which x is maximum. Round your answer to two decimal places.

A)0 hours
B)3.61 hours
C)693.38 hours
D)5.61 hours
E)8.80 hours

Question 69

Sketch a graph of a function f having the following characteristics. $$\begin{array} { l } f ( - 1 ) = f ( - 3 ) = 0 \\f ^ { \prime }( x ) < 0 \text { if } x < - 2 \\f ^ { \prime } ( - 2 ) = 0 \\f ^ { \prime } ( x ) > 0 \text { if } x > - 2 \\f ^ { \prime } ( x ) > 0\end{array}$$

A)
B)
C)
D)
E)

Question 70

Find all relative extrema of the function $f ( x ) = x ^ { 8 / 9 } - 8$ . Use the Second Derivative Test where applicable.

A)relative max: f(1)= -7
B)relative min: f(0)= -8
C)no relative max or min
D)both A and B
E)none of the above

Question 71

The graph of f is shown in the figure. Sketch a graph of the derivative of f.

A)
B)
C)
D)
E)

Question 72

Find all relative extrema of the function $f ( x ) = - 2 x ^ { 2 } - 8 x - 12$ . Use the Second Derivative Test where applicable.

A)relative max: $f ( - 2 ) = - 4$
B)relative min: $f ( 0 ) = - 12$
C)no relative min
D)no relative max
E)both A and C
F) both B and D

Question 73

The graph of f is shown in the figure. Sketch a graph of the derivative of f.

A)
B)
C)
D)
E)

Question 74

Find the x-value at which the given function has a point of inflection. $y = \frac { 1 } { 3 } x ^ { 3 } - 4 x ^ { 2 } + 15 x + 9$

A) $x = 0$
B) $x = 5$
C) $x = 4$
D) $x = 3$
E)no point of inflection

Question 75

Find the points of inflection and discuss the concavity of the function. $f ( x ) = 7 x ^ { 3 } + 8 x ^ { 2 } + 5 x - 8$

A)inflection point at $x = - \frac { 8 } { 21 }$ ; concave downward on $\left( - \infty , - \frac { 8 } { 21 } \right)$ ; concave upward on $\left( - \frac { 8 } { 21 } , \infty \right)$
B)inflection point at $x = - \frac { 8 } { 21 }$ ; concave upward on $\left( - \infty , - \frac { 8 } { 21 } \right)$ ; concave downward on $\left( - \frac { 8 } { 21 } , \infty \right)$
C)inflection point at $x = \frac { 8 } { 21 }$ ; concave downward on $\left( - \infty , \frac { 8 } { 21 } \right)$ ; concave upward on $\left( \frac { 8 } { 21 } , \infty \right)$
D)inflection point at $x = \frac { 8 } { 21 }$ ; concave upward on $\left( - \infty , \frac { 8 } { 21 } \right)$ ; concave downward on $\left( \frac { 8 } { 21 } , \infty \right)$
E)none of the above

Question 76

Find all relative extrema of the function $f ( x ) = 4 x ^ { 4 } - 16 x ^ { 3 } + 5$ Use the Second Derivative Test where applicable.

A)relative max: $f ( 6 ) = 1733$ ; no relative min
B)relative max: $f ( 3 ) = 103$ ; no relative min
C)no relative max or min
D)relative min: $f ( 6 ) = 1733$ ; no relative max
E)relative min: $f ( 3 ) = - 103$ ; no relative max

Question 77

A function and its graph are given. Use the second derivative to locate all x-values of points of inflection on the graph of $y = f ( x )$ . Check these results against the graph shown. $$y = 18 x ^ { 4 } - 54 x ^ { 2 } + 36$$

A) $x = - \frac { \sqrt { 2 } } { 2 }$
B) $x = 0$
C) $x = \frac { \sqrt { 2 } } { 2 }$
D) $x = \frac { \sqrt { 2 } } { 2 }$ , $x = - \frac { \sqrt { 2 } } { 2 }$
E) $x = - \frac { \sqrt { 2 } } { 2 }$ , $x = 0$ , $x = - \frac { \sqrt { 2 } } { 2 }$

Question 78

Determine the open intervals on which the graph of $f ( x ) = 2 x ^ { 2 } + 5 x - 4$ is concave downward or concave upward.

A)concave upward on $( - \infty , 0 )$ ; concave downward on $( 0 , \infty )$
B)concave downward on $( - \infty , \infty )$
C)concave upward on $( - \infty , \infty )$
D)concave downward on $( - \infty , 0 )$ ; concave upward on $( 0 , \infty )$
E)concave upward on $( - \infty , 1 )$ ; concave downward on $( 1 , \infty )$

Question 79

Medication. The number of milligrams x of a medication in the bloodstream t hours after a dose is taken can be modeled by $x ( t ) = \frac { 3000 t } { t ^ { 2 } + 12 }$ $t > 0$ . Find the maximum value of x. Round your answer to two decimal places.

A)3.46 mg
B)433.01 mg
C)952.63 mg
D)10.74 mg
E)1125.30 mg

Question 80

Suppose the resident population P(in millions) of the United States can be modeled by $P = 0.00000583 t ^ { 3 } + 0.005003 t ^ { 2 } + 0.13776 t + 4.658 , - 6 \leq t \leq 193$ , where $t = 0$ corresponds to 1800. Analytically find the minimum and maximum populations in the U.S. for $- 6 \leq t \leq 193$ .

A)The population is minimum at $t = - 6$ and maximum at $t = 0$ .
B)The population is minimum at $t = 0$ and maximum at $t = 193$ .
C)The population is minimum at $t = 193$ and maximum at $t = - 6$ .
D)The population is minimum at $t = - 6$ and maximum at $t = 193$ .
E)The population is minimum at $t = 0$ and maximum at $t = - 6$ .