Deck 12: Calculus Practice Questions

Question 1

Differentiate: $f ( x ) = x ^ { 6 } + 7$

A) $6 x ^ { 7 } + 7 x$
B) $5 x ^ { 5 }$
C) $6 x ^ { 5 } + 7$
D) $6 x ^ { 5 }$

Answer 1

$6 x ^ { 5 }$

Question 2

The owner of an appliance store expects to sell 500 toasters this year.Each toaster costs her $6 dollars to purchase,and each time she orders a shipment of toasters,it costs $30.In addition,it costs $4 a year to store each toaster.Assuming the toasters sell out at a uniform rate and that the owner never allows herself to run out of toasters,how many toasters should be ordered in each shipment to minimize the annual cost? (Round any fractional amounts.)

A)87
B)22
C)44
D)33

Answer 2

87

Question 3

What is the rate of change of $f ( t ) = \frac { 2 t - 7 } { t + 6 }$ with respect to t when t = 13?

A)-19
B)19
C) $\frac { 1 } { 19 }$
D) $- \frac { 1 } { 19 }$

Answer 3

$\frac { 1 } { 19 }$

Question 4

Find the equation of the tangent line to the graph of $f ( x ) = \frac { 3 } { x }$ at the point $\left( 4 , \frac { 3 } { 4 } \right)$ .

A) $y = \frac { 3 } { 4 } x + \frac { 3 } { 2 }$
B) $y = - \frac { 3 } { 16 } x + \frac { 3 } { 2 }$
C) $y = - \frac { 3 } { 4 } x + \frac { 3 } { 2 }$
D) $y = \frac { 3 } { 16 } x$

Question 5

The fraction of television sets manufactured by a certain company that are still in working condition after t years of use is approximately $f ( t ) = e ^ { - 0.2 t }$ .What fraction can be expected to fail before 2 years of use? Round your answer to two decimal places,if necessary.

A)1.49
B)0.91
C)0.67
D)0.33

Question 6

The second derivative test reveals that $f ( x ) = x ^ { 4 } - 4 x ^ { 2 } + 1$ has

A)neither a relative maximum nor a relative minimum.
B)a relative maximum and two relative minima.
C)a relative maximum only.
D)a relative minimum only.

Question 7

Find the equation of the tangent line to $f ( x ) = 4 \ln x ^ { 3 }$ at x = e.

A) $y = \frac { 12 } { e } x + 24$
B)y = 12x
C) $y = \frac { 12 } { e } x$
D) $y = \frac { 12 } { e } x - 24$

Question 8

Solve the given equation for x. $6 = 5 + 9 e ^ { 3 x }$

A) $\frac { e ^ { 9 } } { 3 }$
B) $\frac { \ln 9 } { 3 }$
C) $- \frac { \ln 9 } { 3 }$
D) $3 e ^ { 9 }$

Question 9

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

A)1
B)-1
C)0
D) $\frac { 1 } { 8 }$

Question 10

A company makes a certain product for $4 each and sells it for $8.If the company has overhead expenses of $10,000 per year,how many of its products must be made and sold to break even?

A)20,000
B)40,000
C)2,500
D)10,000

Question 11

Find the intervals of increase and decrease for $f ( x ) = \frac { 8 x - 1 } { - 2 x + 10 }$ .Round numbers to two decimal places,if necessary.

A)Increasing on x 0.12 and on x > 5,decreasing on 0.12 < x 5
B)Increasing on 0.12 < x 5,decreasing on x 0.12 and on x > 5
C)Increasing on x < 5 and x > 5
D)Increasing on x < 5,decreasing on x > 5

Question 12

If $f ( x ) = \left( x + 7 e ^ { - x } \right) ^ { 2 }$ ,then $f ^ { \prime } ( x ) = x + 7 e ^ { - x }$ .

Question 13

The function y = ln 6x is concave downward everywhere.

Question 14

Find the points of intersection (if any)of the given pair of curves. y = x + 8 and y = 2x + 4

A)(12,20)
B)(4,12)
C)(1,-4)
D)(0,6)

Question 15

Find the indicated limit if it exists. $\lim _ { x \rightarrow 4 } \frac { \sqrt { x } - 2 } { x - 4 }$

A) $- \frac { 1 } { 4 }$
B)Does not exist
C) $\frac { 1 } { 4 }$
D)4

Question 16

An equation for the tangent line to the curve $y = \left( x ^ { 5 } + x - 1 \right) ^ { 5 }$ at the point where x = 1 is

A)y = 30x
B)y = 5x + 1
C)y = 30x - 1
D)y = 30x - 29

Question 17

Determine where the graph of $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 9 x + 1$ is concave down.

A)x > -1
B)x < -1
C)x < 1
D)x > 1

Question 18

An efficiency study of the morning shift at a certain factory indicates that an average worker who arrives on the job at 8:00 A.M.will have assembled $f ( x ) = - x ^ { 3 } + 5 x ^ { 2 } + 16 x$ transistor radios x hours later.How many radios will such a worker assemble between 10:00 and 11:00 A.M.?

A)18
B)20
C)22
D)15

Question 19

The output at a certain plant is $Q = 0.06 x ^ { 2 } + 0.15 x y + 0.05 y ^ { 2 }$ units per day,where x is the number of hours of skilled labor used and y is the number of hours of unskilled labor used.Currently 60 hours of skilled labor and 150 hours of unskilled labor are used each day.Use calculus to estimate the change in unskilled labor that should be made to offset a 1 hour increase in skilled labor so that output will remain the same.Round your answer to two decimal places,if necessary.

A)No change
B)An increase of 1.24 hours
C)A decrease of 1.24 hours
D)It cannot be determined

Question 20

Find the slope (if possible)of the line that passes through the given pair of points. (7,0)and (17,19)

A) $\frac { 10 } { 19 }$
B) $- \frac { 19 } { 10 }$
C) $\frac { 19 } { 10 }$
D) $- \frac { 10 } { 19 }$

Question 21

Evaluate the following integral: $\int 3 e ^ { 7 x } d x$

A) $\frac { 3 } { 7 } e ^ { 7 x } + C$
B) $3 e ^ { 7 x } + C$
C) $21 e ^ { 7 x } + C$
D) $\frac { 3 } { 8 } e ^ { 8 x } + C$

Question 22

Money is transferred continuously into an account at the constant rate of $1,000 per year.Assume the account earns interest at the annual rate of 8% compounded continuously.Compute the future value of the income stream over a 5 year period.

A)$12,295.62
B)$6,147.81
C)$491.82
D)$24,591.23

Question 23

Sketch the region R and then use calculus to find the area of R.R is the region between the curve $y = x ^ { 3 }$ and the line y = 18x for x $\geq$ 0.

A)20.25
B)4.5
C)0
D)81

Question 24

Evaluate $\int x \sqrt { x ^ { 2 } + 2 } d x$ .

A) $\left( x ^ { 2 } + 2 \right) ^ { 3 / 2 } + C$
B) $\frac { x ^ { 3 } } { 3 } + 2 x + C$
C) $\frac { 3 \left( x ^ { 2 } + 2 \right) ^ { 3 / 2 } } { 4 } + C$
D) $\frac { \left( x ^ { 2 } + 2 \right) ^ { 3 / 2 } } { 3 } + C$

Question 25

$\int \frac { x ^ { 3 } - 3 x ^ { 2 } + 2 } { x } d x = \frac { x ^ { 3 } } { 3 } - \frac { 3 x ^ { 2 } } { 2 } + 2 \ln | x | + C$