Deck 5: Itegration

Question 1

Suppose the marginal cost is $C ^ { \prime } ( x ) = e ^ { - 0.4 x }$ ,where x is measured in units of 300 items and the cost is measured in units of $10,000.Find the cost corresponding to the production interval [300,1,500].Round to the nearest dollar.

A)$13,375
B)$5,350
C)$9,029
D)$2,140

Answer 1

$13,375

Question 2

Evaluate $\int 5 x ^ { 4 } e ^ { x ^ { 5 } } d x$ .

A) $e ^ { x ^ { n } } + C$
B) $x ^ { 5 } e ^ { x ^ { 5 } } + C$
C) $25 x ^ { 5 } e ^ { x ^ { 5 } } + C$
D) $e ^ { x ^ { 3 } } + C$

Answer 2

$e ^ { x ^ { 3 } } + C$

Question 3

The velocity v(t)(t)= x'(t)at time t of an object moving along the x axis is given,along with the initial position x(0)of the object. $x ^ { t } ( t ) = - 3 ( 3 t + 1 ) ^ { 1 / 2 }$ ; x(0)= 4 Find:
(a)The position x(t)at time t.
(b)The position of the object at time t = 1.
(c)The time when the object is at x = 2.
Round answers for parts (b)and (c)to one decimal place.

A)(a) $x ( t ) = - \frac { 2 } { 3 } ( 3 t + 1 ) ^ { 3 / 2 }$ (b)x(1)= -5.3(c)t = -0.3
B)(a) $x ( t ) = - \frac { 2 } { 3 } ( 3 t + 1 ) ^ { 3 / 2 } + \frac { 14 } { 3 }$ (b)x(1)= 0.7(c)t = -7.7
C)(a) $x ( t ) = - \frac { 2 } { 3 } ( 3 t + 1 ) ^ { 3 / 2 }$ (b)x(1)= 0.1(c)t = -12.3
D)(a) $x ( t ) = - \frac { 2 } { 3 } ( 3 t + 1 ) ^ { 3 / 2 } + \frac { 14 } { 3 }$ (b)x(1)= -0.7(c)t = 0.5

Answer 3

(a) $x ( t ) = - \frac { 2 } { 3 } ( 3 t + 1 ) ^ { 3 / 2 } + \frac { 14 } { 3 }$ (b)x(1)= -0.7(c)t = 0.5

Question 4

A study indicates that x months from now the population of a certain city will be increasing at the rate of $( 3 + 3 x ) x ^ { - 1 / 2 }$ people per month.By how much will the population increase over the next 9 months?

A)54 people
B)62 people
C)72 people
D)82 people

Question 5

Evaluate $\int \left( 5 x ^ { 6 } - 9 x + 7 \right) d x$ .

A) $\frac { 5 x ^ { 7 } } { 7 } - \frac { 9 x ^ { 2 } } { 2 } + C$
B) $\frac { 5 x ^ { 7 } } { 7 } - \frac { 9 x ^ { 2 } } { 2 } + 7 x + C$
C) $30 x ^ { 5 } - 9 + C$
D)5x⁷ - 9x² + 7x + C

Question 6

Compute the area of the finite region bounded by the graphs of the functions $y = 5 x ^ { 2 }$ ,and $y = - 5 x ^ { 2 } - 20 x + 80$ .

A)360
B)320
C)280
D)240

Question 7

It is estimated that t days from now a farmer's crop will be increasing at the rate of $0.3 t ^ { 2 } + 0.6 t + 1$ bushels per day.By how much will the value of the crop increase during the next 10 days if the market price remains fixed at $6 per bushel?

A)$870.00
B)$138.00
C)$840.00
D)$780.00

Question 8

The slope f '(x)at each point (x,y)on a curve y = f (x)is given,along with a particular point (a,b)on the curve.Use this information to find f (x). $f ^ { \prime } ( x ) = x e ^ { 36 - x ^ { 2 } }$ ; (-6,1)

A) $y = - 1 - \frac { 1 } { 2 } e ^ { 36 - x ^ { 2 } }$
B) $y = - \frac { 1 } { 2 } e ^ { 3 - x ^ { 2 } }$
C) $y = \frac { 3 } { 2 } - \frac { 1 } { 2 } e ^ { 36 - x ^ { 2 } }$
D) $y = 1 - \frac { 1 } { 2 } e ^ { 36 - x ^ { 2 } }$

Question 9

Evaluate $\int x ^ { 6 } \sqrt { x ^ { 7 } + 4 } d x$ .

A) $x ^ { 7 } + 4 + C$
B) $C \left( x ^ { 7 } + 4 \right) ^ { 3 / 2 }$
C) $\frac { 2 \left( x ^ { 7 } + 4 \right) ^ { 3 / 2 } } { 21 } + C$
D) $\left( x ^ { 7 } + 4 \right) ^ { 3 / 2 } + C$

Question 10

Water flows into a tank at the rate of $\sqrt { 2 t + 7 }$ ft³/min.If the tank is empty when t = 0,how much water does it contain 8 minutes later? Express the answer to two decimal places.

A)49.45 ft³
B)38.37 ft³
C)30.59 ft³
D)0.72 ft³

Question 11

Evaluate $\int \left( \frac { x } { 9 } - 9 \right) ^ { 2 / 9 } d x$ .

A) $\left( \frac { x } { 9 } - 9 \right) ^ { 11 / 9 } + C$
B) $\frac { 9 } { 11 } \left( \frac { x } { 9 } - 9 \right) ^ { 11 / } + C$
C) $\frac { 81 } { 11 } \left( \frac { x } { 9 } - 9 \right) ^ { 11 / 9 } + C$
D) $\frac { x ^ { 2 } } { 18 } - 9 x + C$

Question 12

Find the Gini index for the given Lorentz curve.Round your answer to four decimal places,if necessary. $$L ( x ) = 0.6 x ^ { 2 } + 0.3 x$$

A)0.3600
B)1.7000
C)0.3000
D)0.2571

Question 13

An object is moving so that its velocity after t minutes is $v ( t ) = 1 + 8 t + 9 t ^ { 2 }$ meters per minute.How far does the object travel from the end of minute 2 to the end of minute 3?

A)37 meters
B)78 meters
C)157 meters
D)77 meters

Question 14

Solve the given initial value problem for y = f (x). $\frac { d y } { d x } = \frac { x + 5 } { x ^ { 2 } + 10 x + 3 }$ where y = 6 when x = 1

A) $y = \frac { 1 } { 2 } \ln \left| x ^ { 2 } + 10 x + 3 \right| - \frac { 1 } { 2 } \ln 14 + 6$
B) $y = \frac { 1 } { 2 } \ln \left| x ^ { 2 } + 10 x + 3 \right| - \frac { 1 } { 2 } \ln 14$
C) $y = \frac { 1 } { 2 } \ln \left| x ^ { 2 } + 10 x + 3 \right| + 6$
D) $y = \frac { 1 } { 2 } \ln \left| x ^ { 2 } + 10 x + 3 \right|$

Question 15

Evaluate $\int _ { - 1 } ^ { 1 } x ^ { 11 } d x$ .

A) $- \frac { 1 } { 22 }$
B)0
C) $\frac { 1 } { 11 }$
D) $\frac { 1 } { 22 }$

Question 16

In a certain section of the country,the price of chicken is currently $3 per kilogram.It is estimated that x weeks from now,the price will be increasing at a rate of $3 \sqrt { x + 1 }$ cents per kilogram.How much will chicken cost 4 weeks from now?

A)$4.20 per kilogram
B)$3.20 per kilogram
C)$2.21 per kilogram
D)$0.21 per kilogram

Question 17

Find the average value of the function $f ( x ) = \frac { x } { 4 x + 1 }$ over the interval [1,3].Round your answer to two decimal places.

A)0.28
B)1.22
C)0.44
D)0.22

Question 18

Find the area of the region under the line y = 4x + 7 over the interval 1 $\leq$ x $\leq$ 8 .

A)189
B)173
C)193
D)175

Question 19

Evaluate the indicated integral: $\int \frac { 7 x ^ { 4 } + 56 x ^ { 3 } + 56 } { x ^ { 5 } + 10 x ^ { 4 } + 40 x + 4 } d x$

A) $\frac { 7 } { 5 } \ln \left| x ^ { 5 } + 10 x ^ { 4 } + 40 x + 4 \right| + C$
B) $\frac { 5 } { 7 } \ln \left| x ^ { 5 } + 10 x ^ { 4 } + 40 x + 4 \right| + C$
C) $\frac { 7 } { 5 } \ln \left| x ^ { 4 } + 8 x ^ { 3 } + 8 \right| + C$
D) $\frac { 7 } { 5 } \ln \left| x ^ { 5 } + 10 x ^ { 4 } + 32 x \right| + C$

Question 20

The chief economist for a meat-packing plant estimates that if L worker-hours of labor are employed each week,the number of pounds of meat made ready for sale will be given by $Q ( L ) = 650 L ^ { 3 / 4 }$ .Find the average weekly output as labor varies from 480 to 790 hours.Round to the nearest pound.

A)82,069 pounds
B)73,954 pounds
C)34,248 pounds
D)86,461 pounds

Question 21

Find the volume of the solid of revolution formed by rotating the region R about the x-axis. R is the region under the curve $y = \sqrt { 49 - x ^ { 2 } }$ from x = -7 to x = 7.

A) $343 \pi$
B) $\frac { 1,372 } { 3 } \pi$
C) $49 \pi$
D) $\frac { 196 } { 3 } \pi$

Question 22

Money is transferred continuously into an account at the constant rate of $1,100 per year.The account earns interest at the annual rate of 8% compounded continuously.How much will be in the account at the end of 3 years?

A)$3,300.00
B)$1,746.97
C)$151,567.68
D)$3,729.68

Question 23

Given a consumer's demand function, $D ( q ) = \frac { 300 } { 0.4 q + 8 }$ dollars per unit,find the total amount of money consumers are willing to spend to get 15 units of the commodity.

A)$419.71
B)$0.56
C)$167.88
D)$209.86

Question 24

Given an initial population, $P _ { 0 } = 140,000$ ,a renewal rate,R = 355,and a survival function, $S ( t ) = e ^ { - 000 t }$ ,with time t measured in years,determine the population at the end of 19 years.

A)34,087
B)5,408
C)11,009,981
D)136,533

Question 25

A certain nuclear power plant produces nuclear waste in the form of strontium-90 at the constant rate of 200 pounds per year.The waste decays exponentially with a half-life of 28 years.How much of the radioactive waste from the nuclear plant will be present after 130 years? [Hint: Think of this as a survival and renewal problem.] Round to two decimal places.

A)4,033.07 pounds
B)7,755.70 pounds
C)8,402.48 pounds
D)3,877.85 pounds