Question 1

Compute the area of the finite region bounded by the graphs of the functions $y = 2 x ^ { 2 }$ , and $y = - 2 x ^ { 2 } - 24 x + 288$ .

Accepted Answer

A)  2,592 
B)  3,024 
C)  3,888 
D)  3,456 
A)  2,592 
B)  3,024 
C)  3,888 
D)  3,456 C

Question 2

Determine the area of the region bounded by the curve $y = x ^ { 3 }$ and the line y = 3x + 2.

Accepted Answer

A)  $\frac { 2.5 } { 4 }$ 
B)  $\frac { 2.7 } { 4 }$ 
C)  $\frac { 2.9 } { 4 }$ 
D)  $\frac { 31 } { 4 }$ 
A)  $\frac { 2.5 } { 4 }$ 
B)  $\frac { 2.7 } { 4 }$ 
C)  $\frac { 2.9 } { 4 }$ 
D)  $\frac { 31 } { 4 }$ B

Question 3

Money is transferred continuously into an account at the constant rate of $1,500 per year. Assume the account earns interest at the annual rate of 5% compounded continuously. Compute the future value of the income stream over a 13 year period.

Accepted Answer

A)  $1,373.31 
B)  $54,932.45 
C)  $109,864.90 
D)  $27,466.22 
A)  $1,373.31 
B)  $54,932.45 
C)  $109,864.90 
D)  $27,466.22 D

Question 4

$\int 2 x \sqrt { x ^ { 2 } - 2 } d x = \frac { 2 \left( x ^ { 2 } - 2 \right) ^ { 3 / 2 } } { 3 } + C$

Accepted Answer

A) True 
 B)False

Question 5

Evaluate $\int x \sqrt { x ^ { 2 } + 4 } d x$ .

Accepted Answer

A)  $\frac { 1 } { 3 } \left( x ^ { 2 } + 4 \right) ^ { 3 / 2 } + C$ 
B)  $\left( x ^ { 2 } + 4 \right) ^ { 3 / 2 } + C$ 
C)  $\frac { x ^ { 3 } } { 3 } + 4 x + C$ 
D)  $\left( \frac { x ^ { 3 } } { 3 } + 4 \right) ^ { 3 / 2 } + C$ 
A)  $\frac { 1 } { 3 } \left( x ^ { 2 } + 4 \right) ^ { 3 / 2 } + C$ 
B)  $\left( x ^ { 2 } + 4 \right) ^ { 3 / 2 } + C$ 
C)  $\frac { x ^ { 3 } } { 3 } + 4 x + C$ 
D)  $\left( \frac { x ^ { 3 } } { 3 } + 4 \right) ^ { 3 / 2 } + C$

Question 6

Evaluate $\int _ { 0 } ^ { 1 } \left( x ^ { 3 } - 3 x ^ { 2 } + 5 \right) d x$ .

Accepted Answer

The answer of Evaluate \(\int _ { 0 } ^...

Question 7

The population density r miles from the center of a city is $D ( r ) = k e ^ { - 0.5 r ^ { 2 } }$ people per square mile. If 300,000 people live within 5 miles from the center of the city, then k is 15,000.

Accepted Answer

A) True 
 B)False

Question 8

An object is moving so that its velocity after t minutes is $v ( t ) = 4 + 6 t + 9 t ^ { 2 }$ meters per minute. How far does the object travel from the end of minute 4 to the end of minute 5?

Accepted Answer

A)  214 meters 
B)  690 meters 
C)  210 meters 
D)  156 meters 
A)  214 meters 
B)  690 meters 
C)  210 meters 
D)  156 meters

Question 9

Suppose the marginal cost is $C ^ { \prime } ( x ) = e ^ { - 0.9 x }$ , where x is measured in units of 400 items and the cost is measured in units of $1,000. Find the cost corresponding to the production interval [1,200, 1,600]. Round to the nearest dollar.

Accepted Answer

The answer of Suppose the marginal cost is \(C ^...

Question 10

Given a consumer's demand function, $D ( q ) = \frac { 500 } { 0.8 q + 8 }$ dollars per unit, find the total amount of money consumers are willing to spend to get 20 units of the commodity.

Accepted Answer

The answer of Given a consumer's demand function, \(D (...

Question 11

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 = 6x for x $\ge$ 0.

Accepted Answer

A)  0 
B)  9 
C)  1.5 
D)  2.25 
A)  0 
B)  9 
C)  1.5 
D)  2.25

Question 12

Specify the substitution you would choose to evaluate the integrals. $$\int \sqrt { 4 - 2 t } d t$$

Accepted Answer

A)  u = t 
B)  $u = 4 - 2 t$ 
C)  u = 2t 
D)  $u = \sqrt { 4 - 2 t }$ 
A)  u = t 
B)  $u = 4 - 2 t$ 
C)  u = 2t 
D)  $u = \sqrt { 4 - 2 t }$

Question 13

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 $Q ( L ) = 600 L ^ { 3 / 4 }$ . Find the average weekly output as labor varies from 500 to 760 hours. Round to the nearest pound.

Accepted Answer

A)  28,570 pounds 
B)  79,741 pounds 
C)  67,234 pounds 
D)  75,349 pounds 
A)  28,570 pounds 
B)  79,741 pounds 
C)  67,234 pounds 
D)  75,349 pounds

Question 14

Evaluate $\int \left( 5 x ^ { 3 } - 5 x + 6 \right) d x$ .

Accepted Answer

A)  $\frac { 5 x ^ { 9 } } { 9 } - \frac { 5 x ^ { 2 } } { 2 } + 6 x + C$ 
B)  $40 x ^ { 7 } - 5 + C$ 
C)  5x9 -5x2 +6x + C 
D)  $\frac { 5 x ^ { 9 } } { 9 } - \frac { 5 x ^ { 2 } } { 2 } + C$ 
A)  $\frac { 5 x ^ { 9 } } { 9 } - \frac { 5 x ^ { 2 } } { 2 } + 6 x + C$ 
B)  $40 x ^ { 7 } - 5 + C$ 
C)  5x9 -5x2 +6x + C 
D)  $\frac { 5 x ^ { 9 } } { 9 } - \frac { 5 x ^ { 2 } } { 2 } + C$

Question 15

Evaluate $\int _ { - 5 } ^ { 5 } \left( 5 x ^ { 5 } + 2 x ^ { 3 } - 6 x \right) d x$ .

Accepted Answer

The answer of Evaluate \(\int _ { - 5 }...

Question 16

Evaluate $\int \left( e ^ { 5 t } + e ^ { - 2 t } \right) d t$ .

Accepted Answer

The answer of Evaluate \(\int \left( e ^ { 5...

Question 17

Given an initial population, $P _ { 0 } = 100,000$ , a renewal rate, R = 2,000, and a survival function, $S ( t ) = e ^ { - .05 t }$ , with time t measured in years, determine the population at the end of 10 years.

Accepted Answer

A)  $4,000 + 100,000 e ^ { - 0 .5 }$ 
B)  $96,000 + 4,000 e ^ { - 0.5 }$ 
C)  $100,000 \left( 1 + 20 e ^ { - 0.5 } \right)$ 
D)  $40,000 + 60,000 e ^ { - 0.5 }$ 
A)  $4,000 + 100,000 e ^ { - 0 .5 }$ 
B)  $96,000 + 4,000 e ^ { - 0.5 }$ 
C)  $100,000 \left( 1 + 20 e ^ { - 0.5 } \right)$ 
D)  $40,000 + 60,000 e ^ { - 0.5 }$

Question 18

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). $f ^ { \prime } ( x ) = x e ^ { 16 - x ^ { 2 } }$ ; (4, -5)

Accepted Answer

The answer of The slope f '(x) at each point...

Question 19

Suppose the marginal cost is $C ^ { \prime } ( x ) = e ^ { - 0.9 x }$ , where x is measured in units of 500 items and the cost is measured in units of $10,000. Find the cost corresponding to the production interval [1,000, 3,000]. Round to the nearest dollar.

Accepted Answer

A)  $1,608 
B)  $1,447 
C)  $1,786 
D)  $1,476 
A)  $1,608 
B)  $1,447 
C)  $1,786 
D)  $1,476

Question 20

Find the consumer's surplus for a commodity whose demand function is $D ( q ) = \frac { 500 } { 0.4 q + 3 }$ dollars per unit if the market price is $p _ { 0 } = \$ 100$ per unit. (Hint: Find the quantity q0 that corresponds to the given price p0 = D(q0).)

Accepted Answer

A)  $130.52 
B)  $138.53 
C)  $142.01 
D)  $145.43 
A)  $130.52 
B)  $138.53 
C)  $142.01 
D)  $145.43

Compute the area of the finite region bounded by the graphs of the functions $y = 2 x ^ { 2 }$ , and $y = - 2 x ^ { 2 } - 24 x + 288$ .

Determine the area of the region bounded by the curve $y = x ^ { 3 }$ and the line y = 3x + 2.

Money is transferred continuously into an account at the constant rate of $1,500 per year. Assume the account earns interest at the annual rate of 5% compounded continuously. Compute the future value of the income stream over a 13 year period.

$\int 2 x \sqrt { x ^ { 2 } - 2 } d x = \frac { 2 \left( x ^ { 2 } - 2 \right) ^ { 3 / 2 } } { 3 } + C$

Evaluate $\int x \sqrt { x ^ { 2 } + 4 } d x$ .

Evaluate $\int _ { 0 } ^ { 1 } \left( x ^ { 3 } - 3 x ^ { 2 } + 5 \right) d x$ .

The population density r miles from the center of a city is $D ( r ) = k e ^ { - 0.5 r ^ { 2 } }$ people per square mile. If 300,000 people live within 5 miles from the center of the city, then k is 15,000.

An object is moving so that its velocity after t minutes is $v ( t ) = 4 + 6 t + 9 t ^ { 2 }$ meters per minute. How far does the object travel from the end of minute 4 to the end of minute 5?

Suppose the marginal cost is $C ^ { \prime } ( x ) = e ^ { - 0.9 x }$ , where x is measured in units of 400 items and the cost is measured in units of $1,000. Find the cost corresponding to the production interval [1,200, 1,600]. Round to the nearest dollar.

Given a consumer's demand function, $D ( q ) = \frac { 500 } { 0.8 q + 8 }$ dollars per unit, find the total amount of money consumers are willing to spend to get 20 units of the commodity.

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 = 6x for x $\ge$ 0.

Specify the substitution you would choose to evaluate the integrals. $\int \sqrt { 4 - 2 t } d t$

Evaluate $\int \left( 5 x ^ { 3 } - 5 x + 6 \right) d x$ .

Evaluate $\int _ { - 5 } ^ { 5 } \left( 5 x ^ { 5 } + 2 x ^ { 3 } - 6 x \right) d x$ .

Evaluate $\int \left( e ^ { 5 t } + e ^ { - 2 t } \right) d t$ .

Given an initial population, $P _ { 0 } = 100,000$ , a renewal rate, R = 2,000, and a survival function, $S ( t ) = e ^ { - .05 t }$ , with time t measured in years, determine the population at the end of 10 years.

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). $f ^ { \prime } ( x ) = x e ^ { 16 - x ^ { 2 } }$ ; (4, -5)

Suppose the marginal cost is $C ^ { \prime } ( x ) = e ^ { - 0.9 x }$ , where x is measured in units of 500 items and the cost is measured in units of $10,000. Find the cost corresponding to the production interval [1,000, 3,000]. Round to the nearest dollar.

Find the consumer's surplus for a commodity whose demand function is $D ( q ) = \frac { 500 } { 0.4 q + 3 }$ dollars per unit if the market price is $p _ { 0 } = \$ 100$ per unit. (Hint: Find the quantity q₀ that corresponds to the given price p₀ = D(q₀).)

Functions, Graphs, and Limits

Differentiation: Basic Concepts

Additional Applications of the Derivative

Exponential and Logarithmic Functions

Additional Topics in Integration

Calculus of Several Variables

Appendix: Algebra Review

Filters

Exam 5: Integration

Compute the area of the finite region bounded by the graphs of the functions y=2x2y = 2 x ^ { 2 }y=2x2 , and y=−2x2−24x+288y = - 2 x ^ { 2 } - 24 x + 288y=−2x2−24x+288 .

Determine the area of the region bounded by the curve y=x3y = x ^ { 3 }y=x3 and the line y = 3x + 2.

Money is transferred continuously into an account at the constant rate of $1,500 per year. Assume the account earns interest at the annual rate of 5% compounded continuously. Compute the future value of the income stream over a 13 year period.

∫2xx2−2dx=2(x2−2)3/23+C\int 2 x \sqrt { x ^ { 2 } - 2 } d x = \frac { 2 \left( x ^ { 2 } - 2 \right) ^ { 3 / 2 } } { 3 } + C∫2xx2−2​dx=32(x2−2)3/2​+C

Evaluate ∫xx2+4dx\int x \sqrt { x ^ { 2 } + 4 } d x∫xx2+4​dx .

Evaluate ∫01(x3−3x2+5)dx\int _ { 0 } ^ { 1 } \left( x ^ { 3 } - 3 x ^ { 2 } + 5 \right) d x∫01​(x3−3x2+5)dx .

The population density r miles from the center of a city is D(r)=ke−0.5r2D ( r ) = k e ^ { - 0.5 r ^ { 2 } }D(r)=ke−0.5r2 people per square mile. If 300,000 people live within 5 miles from the center of the city, then k is 15,000.

An object is moving so that its velocity after t minutes is v(t)=4+6t+9t2v ( t ) = 4 + 6 t + 9 t ^ { 2 }v(t)=4+6t+9t2 meters per minute. How far does the object travel from the end of minute 4 to the end of minute 5?

Suppose the marginal cost is C′(x)=e−0.9xC ^ { \prime } ( x ) = e ^ { - 0.9 x }C′(x)=e−0.9x , where x is measured in units of 400 items and the cost is measured in units of $1,000. Find the cost corresponding to the production interval [1,200, 1,600]. Round to the nearest dollar.

Given a consumer's demand function, D(q)=5000.8q+8D ( q ) = \frac { 500 } { 0.8 q + 8 }D(q)=0.8q+8500​ dollars per unit, find the total amount of money consumers are willing to spend to get 20 units of the commodity.

Sketch the region R and then use calculus to find the area of R. R is the region between the curve y=x3y = x ^ { 3 }y=x3 and the line y = 6x for x ≥\ge≥ 0.

Specify the substitution you would choose to evaluate the integrals. ∫4−2tdt\int \sqrt { 4 - 2 t } d t∫4−2t​dt

Evaluate ∫(5x3−5x+6)dx\int \left( 5 x ^ { 3 } - 5 x + 6 \right) d x∫(5x3−5x+6)dx .

Evaluate ∫−55(5x5+2x3−6x)dx\int _ { - 5 } ^ { 5 } \left( 5 x ^ { 5 } + 2 x ^ { 3 } - 6 x \right) d x∫−55​(5x5+2x3−6x)dx .

Evaluate ∫(e5t+e−2t)dt\int \left( e ^ { 5 t } + e ^ { - 2 t } \right) d t∫(e5t+e−2t)dt .

Given an initial population, P0=100,000P _ { 0 } = 100,000P0​=100,000 , a renewal rate, R = 2,000, and a survival function, S(t)=e−.05tS ( t ) = e ^ { - .05 t }S(t)=e−.05t , with time t measured in years, determine the population at the end of 10 years.

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′(x)=xe16−x2f ^ { \prime } ( x ) = x e ^ { 16 - x ^ { 2 } }f′(x)=xe16−x2 ; (4, -5)

Suppose the marginal cost is C′(x)=e−0.9xC ^ { \prime } ( x ) = e ^ { - 0.9 x }C′(x)=e−0.9x , where x is measured in units of 500 items and the cost is measured in units of $10,000. Find the cost corresponding to the production interval [1,000, 3,000]. Round to the nearest dollar.

Functions, Graphs, and Limits

Differentiation: Basic Concepts

Additional Applications of the Derivative

Exponential and Logarithmic Functions

Additional Topics in Integration

Calculus of Several Variables

Appendix: Algebra Review

Filters

Compute the area of the finite region bounded by the graphs of the functions $y = 2 x ^ { 2 }$ , and $y = - 2 x ^ { 2 } - 24 x + 288$ .

Determine the area of the region bounded by the curve $y = x ^ { 3 }$ and the line y = 3x + 2.

$\int 2 x \sqrt { x ^ { 2 } - 2 } d x = \frac { 2 \left( x ^ { 2 } - 2 \right) ^ { 3 / 2 } } { 3 } + C$

Evaluate $\int x \sqrt { x ^ { 2 } + 4 } d x$ .

Evaluate $\int _ { 0 } ^ { 1 } \left( x ^ { 3 } - 3 x ^ { 2 } + 5 \right) d x$ .

The population density r miles from the center of a city is $D ( r ) = k e ^ { - 0.5 r ^ { 2 } }$ people per square mile. If 300,000 people live within 5 miles from the center of the city, then k is 15,000.

An object is moving so that its velocity after t minutes is $v ( t ) = 4 + 6 t + 9 t ^ { 2 }$ meters per minute. How far does the object travel from the end of minute 4 to the end of minute 5?

Suppose the marginal cost is $C ^ { \prime } ( x ) = e ^ { - 0.9 x }$ , where x is measured in units of 400 items and the cost is measured in units of $1,000. Find the cost corresponding to the production interval [1,200, 1,600]. Round to the nearest dollar.

Given a consumer's demand function, $D ( q ) = \frac { 500 } { 0.8 q + 8 }$ dollars per unit, find the total amount of money consumers are willing to spend to get 20 units of the commodity.

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 = 6x for x $\ge$ 0.

Specify the substitution you would choose to evaluate the integrals. $\int \sqrt { 4 - 2 t } d t$

Evaluate $\int \left( 5 x ^ { 3 } - 5 x + 6 \right) d x$ .

Evaluate $\int _ { - 5 } ^ { 5 } \left( 5 x ^ { 5 } + 2 x ^ { 3 } - 6 x \right) d x$ .

Evaluate $\int \left( e ^ { 5 t } + e ^ { - 2 t } \right) d t$ .

Given an initial population, $P _ { 0 } = 100,000$ , a renewal rate, R = 2,000, and a survival function, $S ( t ) = e ^ { - .05 t }$ , with time t measured in years, determine the population at the end of 10 years.

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). $f ^ { \prime } ( x ) = x e ^ { 16 - x ^ { 2 } }$ ; (4, -5)

Suppose the marginal cost is $C ^ { \prime } ( x ) = e ^ { - 0.9 x }$ , where x is measured in units of 500 items and the cost is measured in units of $10,000. Find the cost corresponding to the production interval [1,000, 3,000]. Round to the nearest dollar.