Deck 3: Additional Applications of the Derivative

Question 1

Determine where the graph of $f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 8 x + 4$ is concave down.

A)For x > 2
B)For x < -2
C)For x > -2
D)For x < 2

Answer 1

For x < 2

Question 2

Locate all inflection points of $f ( x ) = \frac { x } { x ^ { 2 } + 1 }$ .

A) $( 0,0 ) , \left( 1 , \frac { 1 } { 2 } \right) , \left( - 1 , - \frac { 1 } { 2 } \right)$
B)(0,0)
C) $( 0,0 ) , \left( - 1 , - \frac { 1 } { 2 } \right)$
D) $\left( - \sqrt { 3 } , - \frac { \sqrt { 3 } } { 4 } \right) , ( 0,0 ) , \left( \sqrt { 3 } , \frac { \sqrt { 3 } } { 4 } \right)$

Answer 2

$\left( - \sqrt { 3 } , - \frac { \sqrt { 3 } } { 4 } \right) , ( 0,0 ) , \left( \sqrt { 3 } , \frac { \sqrt { 3 } } { 4 } \right)$

Question 3

Sketch the graph of the given function. $f ( x ) = \frac { x + 5 } { x + 2 }$

A)
B)
C)
D)

Answer 3

Question 4

The owner of a novelty store can obtain joy buzzers from the manufacturer for 50 cents apiece.He estimates he can sell 60 buzzers when he charges $1.00 apiece for them and that he will be able to sell 10 more buzzers for every 8 cent decrease in price.What price should he charge in order to maximize profit?

A)$0.79
B)$0.89
C)$1.09
D)$0.99

Question 5

Name the vertical and horizontal asymptotes of the given graph.

A)Vertical asymptotes: x = -2,x = 3,x = 0; horizontal asymptotes: y = 5,y = 0
B)Vertical asymptotes: x = -2,x = 3,x = 0; horizontal asymptote: y = 5
C)Vertical asymptotes: x = -2,x = 3; horizontal asymptote: y = 5
D)Vertical asymptotes: x = -2,x = 3; horizontal asymptotes: y = 5,y = 0

Question 6

Use the second derivative test to find the relative maxima and minima of the function $f ( x ) = 2 x ^ { 3 } + 6 x ^ { 2 } - 18 x + 1$ .

A)Relative maximum at (-3,-8); relative minimum at (1,55)
B)Relative maximum at (-3,55); relative minimum at (1,-9)
C)Relative maximum at (-3,-9); relative minimum at (-1,23)
D)Relative maximum at (1,55); relative minimum at (-3,-9)

Question 7

Determine the critical points of the given function and classify each critical point as a relative maximum,a relative minimum,or neither. $f ( x ) = 3 x ^ { 4 } - 8 x ^ { 3 } + 6 x ^ { 2 } + 3$

A)(0,2)neither; (1,3)relative minimum
B)(0,2)relative minimum; (1,3)relative maximum
C)(0,3)relative minimum; (1,4)neither
D)(0,2)relative minimum; (1,4)neither

Question 8

A 5-year projection of population trends suggests that t years from now,the population of a certain community will be $P ( t ) = - t ^ { 3 } + 15 t ^ { 2 } + 72 t + 55$ thousand.
a.At what time during the 5-year period will the population be growing most rapidly?
b. At what time during the 5-year period will the population be growing least rapidly?
c. At what time is the rate of population growth changing most rapidly?

A)a.t = 2 yearsb.t = 0 yearsc.t = 2 years
B)a.t = 2 yearsb.t = 1 yearc.t = 3 years
C)a.t = 5 yearsb.t = 0 yearsc.t = 0 years
D)a.t = 0 yearsb.t = 0 yearsc.t = 2 years

Question 9

Find two non-negative numbers x and y whose sum is 14 and are such that the xy² is as large as possible.

A)7 and 7
B) $\frac { 14 } { 3 }$ and $\frac { 28 } { 3 }$
C)1 and 13
D)14 and 28

Question 10

Find the absolute maximum of the function $f ( x ) = x ^ { 7 } - x ^ { 6 }$ on the interval -1 $\leq$ x $\leq$ 1.

A)-2
B)0
C)1
D)-1

Question 11

Determine the critical numbers of the given function and classify each critical point as a relative maximum,a relative minimum,or neither.Round numbers to two decimal places,if necessary. $f ( x ) = \frac { 1 } { x ^ { 2 } - 8 x + 7 }$

A)x = 1,4,7; (1,1)relative maximum; (4,-0.11)relative minimum;(7,1)relative maximum
B)x = 1,7; (1,1)relative maximum; (7,1)relative maximum
C)x = 1,7; (1,1)relative maximum; (7,1)relative minimum
D)x = 1,4,7; (4,-0.11)relative maximum

Question 12

The cost of producing x units of a certain commodity is $C ( x ) = 3 x ^ { 2 } + 5 x + 5$ dollars.If the price is p(x)= (41 - x)dollars per unit,determine the level of production that maximizes profit.

A)x = 3
B)x = 1
C)x = 5
D)x = 2

Question 13

Find all vertical and horizontal asymptotes of the graph of the given function. $f ( x ) = \frac { x ^ { 2 } - 3 x } { x ^ { 2 } + 5 x + 6 }$

A)Vertical asymptotes: x = -3,x = -2; horizontal asymptote: y = 1
B)Vertical asymptotes: x = -3,x = -2; horizontal asymptote: y = 3
C)Vertical asymptotes: x = -3,x = -2,x = 3; horizontal asymptote: y = 0
D)Vertical asymptotes: x = -3,x = -2,x = 3; horizontal asymptote: y = 1

Question 14

Find the intervals of increase and decrease for $f ( x ) = 6 x ^ { 3 } + 9 x ^ { 2 } - 108 x - 3$ .

A)Increasing on x < -3 and x > 2,decreasing on -3 < x < 2
B)Increasing on -3 < x < 2,decreasing on x < -3 and x > 2
C)Increasing on x < -2,decreasing on x > 3
D)Increasing on x $\leq$ -2 and x $\geq$ 3 ,decreasing on - 2 $\leq$ x $\leq$ 3

Question 15

A cable is to be run from a power plant on one side of a river 900 meters wide to a factory on the other side,6,000 meters downstream.The cost of running cable under the water is $5 per meter,while over land is $4 per meter.What is the most economical route over which to run the cable?

A)The cable reaches the opposite bank 6,000 meters downstream.
B)The cable reaches the opposite bank 3,000 meters downstream.
C)The cable reaches the opposite bank 1,200 meters downstream.
D)The cable reaches the opposite bank directly across stream.

Question 16

Find constants a,b,and c so that the graph of the function $f ( x ) = c x ^ { 2 } + b x + c$ has a relative maximum at (7,34)and crosses the y-axis at (0,6).

A) $a = \frac { 4 } { 7 }$ ,b = 8,c = 6
B) $a = - \frac { 4 } { 7 }$ ,b = -8,c = -6
C) $a = \frac { 4 } { 7 }$ ,b = -8,c = 6
D) $a = - \frac { 4 } { 7 }$ ,b = 8,c = 6

Question 17

The demand function for a certain commodity is $D ( p ) = \frac { 30 } { p + 5 }$ .For what values of p is the demand inelastic?

A)p > -5
B)p > 0
C)p < -5
D)p < 0

Question 18

Determine where the graph of $f ( x ) = x ^ { 4 } - 54 x ^ { 2 } + 6$ is concave up.

A)Everywhere
B)For x < -3 and x > 3
C)Nowhere
D)For -3 < x < 3

Question 19

The revenue derived from the production of x units of a particular commodity is $R ( x ) = \frac { 80 x - x ^ { 2 } } { x ^ { 2 } + 80 }$ million dollars.What level of production results in maximum revenue? What is the maximum revenue? Round numbers to two decimal places,if necessary.

A)Maximum at x = 8 and maximum revenue is R(8)= 4 (million dollars)
B)Maximum at x = 8 and maximum revenue is R(8)= 32 (million dollars)
C)Maximum at x = 9 and maximum revenue is R(9)= 2.67 (million dollars)
D)Maximum at x = 8 and maximum revenue is R(8)= 2.67 (thousand dollars)

Question 20

Find the absolute minimum of the function $f ( x ) = x ^ { 3 } - 3 x ^ { 2 }$ on the interval 1 $\leq$ x $\leq$ 5.

A)5
B)-4
C)-1
D)0

Question 21

An open box is to be made from a square piece of cardboard,36 inches by 36 inches,by removing a small square from each corner and folding up the flaps to form the sides.What are the dimensions of the box of greatest volume that can be constructed in this way?

A)24 inches long by 24 inches wide by 6 inches deep
B)6 inches long by 6 inches wide by 24 inches deep
C)12 inches long by 12 inches wide by 6 inches deep
D)18 inches long by 18 inches wide by 6 inches deep