Question 1

$\int _ { 0 } ^ { 2 } \left( x ^ { 3 } + 3 x ^ { 2 } + x + 1 \right) d x$ Enter just an integer.

Accepted Answer

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

Question 2

Set up an integral or sum of integrals and then find the area bounded by y = $x ^ { 2 }$ + 3 and y = -5x - 3.

Accepted Answer

A)  $\int _ { - 3 } ^ { - 2 } \left[ \left( x ^ { 2 } + 3 \right) - ( - 5 x - 3 ) \right] d x$ = - $\frac { 1 } { 6 }$ 
B)  $\int _ { 2 } ^ { 3 } \left[ \left( x ^ { 2 } + 3 \right) - ( - 5 x - 3 ) \right] d x$ = $\frac { 149 } { 6 }$ 
C)  $\int _ { - 3 } ^ { - 2 } \left[ ( - 5 x - 3 ) - \left( x ^ { 2 } + 3 \right) \right] d x$ = $\frac { 1 } { 6 }$ 
D)  $\int _ { 2 } ^ { 3 } \left[ ( - 5 x - 3 ) - \left( x ^ { 2 } + 3 \right) \right] d x$ = - $\frac { 149 } { 6 }$ 
E)  none of these 
A)  $\int _ { - 3 } ^ { - 2 } \left[ \left( x ^ { 2 } + 3 \right) - ( - 5 x - 3 ) \right] d x$ = - $\frac { 1 } { 6 }$ 
B)  $\int _ { 2 } ^ { 3 } \left[ \left( x ^ { 2 } + 3 \right) - ( - 5 x - 3 ) \right] d x$ = $\frac { 149 } { 6 }$ 
C)  $\int _ { - 3 } ^ { - 2 } \left[ ( - 5 x - 3 ) - \left( x ^ { 2 } + 3 \right) \right] d x$ = $\frac { 1 } { 6 }$ 
D)  $\int _ { 2 } ^ { 3 } \left[ ( - 5 x - 3 ) - \left( x ^ { 2 } + 3 \right) \right] d x$ = - $\frac { 149 } { 6 }$ 
E)  none of these

Question 3

Find: $\int$ $\left( \frac { x ^ { 2 } } { 4 } - 4 \right)$ dx

Accepted Answer

A)  $\frac { 3 x ^ { 3 } } { 4 }$ - 4 + C 
B)  $\frac { x ^ { 3 } } { 12 }$ - 4x + C 
C)  $\frac { x ^ { 2 } } { 4 }$ - 4 + C 
D)  $\frac { x } { 2 }$ + C 
E)  none of these 
A)  $\frac { 3 x ^ { 3 } } { 4 }$ - 4 + C 
B)  $\frac { x ^ { 3 } } { 12 }$ - 4x + C 
C)  $\frac { x ^ { 2 } } { 4 }$ - 4 + C 
D)  $\frac { x } { 2 }$ + C 
E)  none of these

Question 4

Use a Riemann sum to approximate the area under the graph of $f ( x ) = c , - 2 \leq x \leq 3 , n = 6$ Use the left endpoints of the interval.
Enter an unlabeled answer.

Accepted Answer

The answer of Use a Riemann sum to approximate the...

Question 5

Find all functions f(x) with the following property: f'(x) = 3 $x ^ { 2 }$ + 2x + 1.
Enter your answer as a polynomial in x in standard form with any fractional coefficients or powers reduced of form $\frac { a } { b }$ .

Accepted Answer

The answer of Find all functions f(x) with the following...

Question 6

Suppose that a colony of fruit flies is growing exponentially with growth constant 0.04 . If there are currently 30,000 flies present, what will be the average population over the next 6 months?

Accepted Answer

A)  1,500,000( $e ^ { 0.02 }$ - 1) 
B)  30,000( $e ^ { 0.02 }$ - 1) 
C)  75,000( $e ^ { 0.24 }$ - 1) 
D)  60,000( $e ^ { 0.24 }$ - 1) 
E)  none of these 
A)  1,500,000( $e ^ { 0.02 }$ - 1) 
B)  30,000( $e ^ { 0.02 }$ - 1) 
C)  75,000( $e ^ { 0.24 }$ - 1) 
D)  60,000( $e ^ { 0.24 }$ - 1) 
E)  none of these

Question 7

A company estimates that its marginal profit for producing a product is given by $\mathrm { MP } ( q ) = 60 - 0.12 q$ where $\operatorname { MP } ( q )$ is in dollars per unit. Given that $P = - 500$ when $\mathrm { q } = 0$ find the total profit realized from selling 300 units of product.

Accepted Answer

A)  $12,100 
B)  $7700 
C)  $6700 
D)  $4100 
A)  $12,100 
B)  $7700 
C)  $6700 
D)  $4100

Question 8

Suppose that during a controlled experiment, the temperature in a test tube at time t is rising at a rate of $6 t ^ { 2 } + 2$ degrees centigrade per minute. If the initial temperature is 0°C, what is the temperature in the test tube after 10 minutes?

Accepted Answer

A)  524° 
B)  120° 
C)  602° 
D)  2020° 
E)  none of these 
A)  524° 
B)  120° 
C)  602° 
D)  2020° 
E)  none of these

Question 9

$\int _ { 0 } ^ { 1 } \left( 2 x ^ { 4 } + 5 x + 1 \right) d x$ Enter a reduced fraction of form $\frac { a } { b }$ .

Accepted Answer

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

Question 10

Does $\int _ { 1 } ^ { 3 } \left( e ^ { x } - 1 \right) d x$ express the area between the curve y = $e ^ { x }$ - 1 and the x-axis between x = 1 and $x = 3 ?$ Give a reason for your answer.

Accepted Answer

A)  Yes, because y = $e ^ { x }$ - 1 does not cross the x-axis between 1 and 3 . 
B)  Yes, because y = $e ^ { x }$ - 1 is nonnegative for all x between 1 and 3 . 
C)  No, because $\int _ { 1 } ^ { 3 } \left( e ^ { x } - 1 \right) d x$ is negative. 
D)  No, because y = $e ^ { x }$ - 1 crosses the x-axis between 1 and 3 . 
E)  none of these 
A)  Yes, because y = $e ^ { x }$ - 1 does not cross the x-axis between 1 and 3 . 
B)  Yes, because y = $e ^ { x }$ - 1 is nonnegative for all x between 1 and 3 . 
C)  No, because $\int _ { 1 } ^ { 3 } \left( e ^ { x } - 1 \right) d x$ is negative. 
D)  No, because y = $e ^ { x }$ - 1 crosses the x-axis between 1 and 3 . 
E)  none of these

Question 11

Find the producers' surplus for the supply curve p = 4 + $\frac { 1 } { 2 }$ $\sqrt { x }$ at x = 144.

Accepted Answer

A)  $392 
B)  $1152 
C)  $724 
D)  $288 
A)  $392 
B)  $1152 
C)  $724 
D)  $288

Question 12

Use a Riemann sum to approximate the area under the graph of f(x) on the given interval. Use the left endpoints. f(x) = 3x - 4; 1 ≤ x ≤ 4, n = 6

Accepted Answer

A)  $\frac { 51 } { 4 }$ = 12.75 
B)  $\frac { 63 } { 4 }$ = 15.75 
C)  $\frac { 33 } { 4 }$ = 8.25 
D)  $\frac { 87 } { 4 }$ = 21.75 
A)  $\frac { 51 } { 4 }$ = 12.75 
B)  $\frac { 63 } { 4 }$ = 15.75 
C)  $\frac { 33 } { 4 }$ = 8.25 
D)  $\frac { 87 } { 4 }$ = 21.75

Question 13

Suppose f(x) is an antiderivative of $\frac { 2 } { \sqrt { x } }$ and f(0) = 1. What is f(9)?

Accepted Answer

A)  13 
B)  - $\frac { 4 } { 3 }$ 
C)  1 
D)  -4 
E)  none of these 
A)  13 
B)  - $\frac { 4 } { 3 }$ 
C)  1 
D)  -4 
E)  none of these

Question 14

Find the producers' surplus for the supply curve p = 0.02x + 7 at x = 100.

Accepted Answer

A)  $100 
B)  $200 
C)  $900 
D)  $800 
A)  $100 
B)  $200 
C)  $900 
D)  $800

Question 15

Determine the average value of $f ( x ) = e ^ { 3 x }$ over the interval from x = 1 to x = 3.
Enter your answer in the form a( $\mathrm { e } ^ { \mathrm {b} }$ ± $\mathrm { e } ^ { \mathrm { C } }$ )

Accepted Answer

The answer of Determine the average value of \(f (...

Question 16

Find the area of the region bounded by y = 6x - $x ^ { 2 }$ and y = $x ^ { 2 }$ - 2x.
Enter a reduced fraction of form $\frac { a } { b }$ .

Accepted Answer

The answer of Find the area of the region bounded...

Question 17

Draw the region whose area is given by the definite integral. $\int _ { 4 } ^ { 5 } \frac { 2 } { x ^ { 3 } }$ dx

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 18

Draw the region whose area is given by the definite integral. $\int _ { 2 } ^ { 9 } \frac { 2 } { \sqrt { x } }$ dx

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 19

Find: $\int \frac { x - 7 } { x }$ dx
Enter terms in the same order in which they appear in the integral.

Accepted Answer

The answer of Find: \(\int \frac { x - 7...

Question 20

Use the graph below to determine the area of the shaded region.
Enter your answer exactly in the form a ± $\mathrm { e } ^ { \mathrm {b } }$ .

Accepted Answer

The answer of Use the graph below to determine the...

$\int _ { 0 } ^ { 2 } \left( x ^ { 3 } + 3 x ^ { 2 } + x + 1 \right) d x$ Enter just an integer.

Set up an integral or sum of integrals and then find the area bounded by y = $x ^ { 2 }$ + 3 and y = -5x - 3.

Find: $\int$ $\left( \frac { x ^ { 2 } } { 4 } - 4 \right)$ dx

Use a Riemann sum to approximate the area under the graph of $f ( x ) = c , - 2 \leq x \leq 3 , n = 6$ Use the left endpoints of the interval. Enter an unlabeled answer.

Find all functions f(x) with the following property: f'(x) = 3 $x ^ { 2 }$ + 2x + 1. Enter your answer as a polynomial in x in standard form with any fractional coefficients or powers reduced of form $\frac { a } { b }$ .

Suppose that a colony of fruit flies is growing exponentially with growth constant 0.04 . If there are currently 30,000 flies present, what will be the average population over the next 6 months?

A company estimates that its marginal profit for producing a product is given by $\mathrm { MP } ( q ) = 60 - 0.12 q$ where $\operatorname { MP } ( q )$ is in dollars per unit. Given that $P = - 500$ when $\mathrm { q } = 0$ find the total profit realized from selling 300 units of product.

Suppose that during a controlled experiment, the temperature in a test tube at time t is rising at a rate of $6 t ^ { 2 } + 2$ degrees centigrade per minute. If the initial temperature is 0°C, what is the temperature in the test tube after 10 minutes?

$\int _ { 0 } ^ { 1 } \left( 2 x ^ { 4 } + 5 x + 1 \right) d x$ Enter a reduced fraction of form $\frac { a } { b }$ .

Does $\int _ { 1 } ^ { 3 } \left( e ^ { x } - 1 \right) d x$ express the area between the curve y = $e ^ { x }$ - 1 and the x-axis between x = 1 and $x = 3 ?$ Give a reason for your answer.

Find the producers' surplus for the supply curve p = 4 + $\frac { 1 } { 2 }$ $\sqrt { x }$ at x = 144.

Use a Riemann sum to approximate the area under the graph of f(x) on the given interval. Use the left endpoints. f(x) = 3x - 4; 1 ≤ x ≤ 4, n = 6

Suppose f(x) is an antiderivative of $\frac { 2 } { \sqrt { x } }$ and f(0) = 1. What is f(9)?

Find the producers' surplus for the supply curve p = 0.02x + 7 at x = 100.

Determine the average value of $f ( x ) = e ^ { 3 x }$ over the interval from x = 1 to x = 3. Enter your answer in the form a( $\mathrm { e } ^ { \mathrm {b} }$ ± $\mathrm { e } ^ { \mathrm { C } }$ )

Find the area of the region bounded by y = 6x - $x ^ { 2 }$ and y = $x ^ { 2 }$ - 2x. Enter a reduced fraction of form $\frac { a } { b }$ .

Draw the region whose area is given by the definite integral. $\int _ { 4 } ^ { 5 } \frac { 2 } { x ^ { 3 } }$ dx $Draw the region whose area is given by the definite integral. \int _ { 4 } ^ { 5 } \frac { 2 } { x ^ { 3 } } dx$

Draw the region whose area is given by the definite integral. $\int _ { 2 } ^ { 9 } \frac { 2 } { \sqrt { x } }$ dx $Draw the region whose area is given by the definite integral. \int _ { 2 } ^ { 9 } \frac { 2 } { \sqrt { x } } dx$

Find: $\int \frac { x - 7 } { x }$ dx Enter terms in the same order in which they appear in the integral.

Use the graph below to determine the area of the shaded region. Enter your answer exactly in the form a ± $\mathrm { e } ^ { \mathrm {b } }$ . $Use the graph below to determine the area of the shaded region. Enter your answer exactly in the form a ± \mathrm { e } ^ { \mathrm {b } } .$

The Derivative

Applications of the Derivative

Techniques of Differentiation

Logarithm Functions

Applications of the Exponential and Natural Logarithm Functions

Functions of Several Variables

The Trigonometric Functions

Techniques of Integration

Differential Equations

Taylor Polynomials and Infinite Series

Probability and Calculus

Filters

Exam 6: The Definite Integral

∫02(x3+3x2+x+1)dx\int _ { 0 } ^ { 2 } \left( x ^ { 3 } + 3 x ^ { 2 } + x + 1 \right) d x∫02​(x3+3x2+x+1)dx Enter just an integer.

Set up an integral or sum of integrals and then find the area bounded by y = x2x ^ { 2 }x2 + 3 and y = -5x - 3.

Find: ∫\int∫ (x24−4)\left( \frac { x ^ { 2 } } { 4 } - 4 \right)(4x2​−4) dx

Use a Riemann sum to approximate the area under the graph of f(x)=c,−2≤x≤3,n=6f ( x ) = c , - 2 \leq x \leq 3 , n = 6f(x)=c,−2≤x≤3,n=6 Use the left endpoints of the interval. Enter an unlabeled answer.

Find all functions f(x) with the following property: f'(x) = 3 x2x ^ { 2 }x2 + 2x + 1. Enter your answer as a polynomial in x in standard form with any fractional coefficients or powers reduced of form ab\frac { a } { b }ba​ .

Suppose that a colony of fruit flies is growing exponentially with growth constant 0.04 . If there are currently 30,000 flies present, what will be the average population over the next 6 months?

Suppose that during a controlled experiment, the temperature in a test tube at time t is rising at a rate of 6t2+26 t ^ { 2 } + 26t2+2 degrees centigrade per minute. If the initial temperature is 0°C, what is the temperature in the test tube after 10 minutes?

∫01(2x4+5x+1)dx\int _ { 0 } ^ { 1 } \left( 2 x ^ { 4 } + 5 x + 1 \right) d x∫01​(2x4+5x+1)dx Enter a reduced fraction of form ab\frac { a } { b }ba​ .

Does ∫13(ex−1)dx\int _ { 1 } ^ { 3 } \left( e ^ { x } - 1 \right) d x∫13​(ex−1)dx express the area between the curve y = exe ^ { x }ex - 1 and the x-axis between x = 1 and x=3?x = 3 ?x=3? Give a reason for your answer.

Find the producers' surplus for the supply curve p = 4 + 12\frac { 1 } { 2 }21​ x\sqrt { x }x​ at x = 144.

Use a Riemann sum to approximate the area under the graph of f(x) on the given interval. Use the left endpoints. f(x) = 3x - 4; 1 ≤ x ≤ 4, n = 6

Suppose f(x) is an antiderivative of 2x\frac { 2 } { \sqrt { x } }x​2​ and f(0) = 1. What is f(9)?

Find the producers' surplus for the supply curve p = 0.02x + 7 at x = 100.

Determine the average value of f(x)=e3xf ( x ) = e ^ { 3 x }f(x)=e3x over the interval from x = 1 to x = 3. Enter your answer in the form a( eb\mathrm { e } ^ { \mathrm {b} }eb ± eC\mathrm { e } ^ { \mathrm { C } }eC )

Find the area of the region bounded by y = 6x - x2x ^ { 2 }x2 and y = x2x ^ { 2 }x2 - 2x. Enter a reduced fraction of form ab\frac { a } { b }ba​ .

Draw the region whose area is given by the definite integral. ∫452x3\int _ { 4 } ^ { 5 } \frac { 2 } { x ^ { 3 } }∫45​x32​ dx

Draw the region whose area is given by the definite integral. ∫292x\int _ { 2 } ^ { 9 } \frac { 2 } { \sqrt { x } }∫29​x​2​ dx

Find: ∫x−7x\int \frac { x - 7 } { x }∫xx−7​ dx Enter terms in the same order in which they appear in the integral.

Use the graph below to determine the area of the shaded region. Enter your answer exactly in the form a ± eb\mathrm { e } ^ { \mathrm {b } }eb .

The Derivative

Applications of the Derivative

Techniques of Differentiation

Logarithm Functions

Applications of the Exponential and Natural Logarithm Functions

Functions of Several Variables

The Trigonometric Functions

Techniques of Integration

Differential Equations

Taylor Polynomials and Infinite Series

Probability and Calculus

Filters

$\int _ { 0 } ^ { 2 } \left( x ^ { 3 } + 3 x ^ { 2 } + x + 1 \right) d x$ Enter just an integer.

Set up an integral or sum of integrals and then find the area bounded by y = $x ^ { 2 }$ + 3 and y = -5x - 3.

Find: $\int$ $\left( \frac { x ^ { 2 } } { 4 } - 4 \right)$ dx

Use a Riemann sum to approximate the area under the graph of $f ( x ) = c , - 2 \leq x \leq 3 , n = 6$ Use the left endpoints of the interval. Enter an unlabeled answer.

Find all functions f(x) with the following property: f'(x) = 3 $x ^ { 2 }$ + 2x + 1. Enter your answer as a polynomial in x in standard form with any fractional coefficients or powers reduced of form $\frac { a } { b }$ .

Suppose that during a controlled experiment, the temperature in a test tube at time t is rising at a rate of $6 t ^ { 2 } + 2$ degrees centigrade per minute. If the initial temperature is 0°C, what is the temperature in the test tube after 10 minutes?

$\int _ { 0 } ^ { 1 } \left( 2 x ^ { 4 } + 5 x + 1 \right) d x$ Enter a reduced fraction of form $\frac { a } { b }$ .

Does $\int _ { 1 } ^ { 3 } \left( e ^ { x } - 1 \right) d x$ express the area between the curve y = $e ^ { x }$ - 1 and the x-axis between x = 1 and $x = 3 ?$ Give a reason for your answer.

Find the producers' surplus for the supply curve p = 4 + $\frac { 1 } { 2 }$ $\sqrt { x }$ at x = 144.

Suppose f(x) is an antiderivative of $\frac { 2 } { \sqrt { x } }$ and f(0) = 1. What is f(9)?

Determine the average value of $f ( x ) = e ^ { 3 x }$ over the interval from x = 1 to x = 3. Enter your answer in the form a( $\mathrm { e } ^ { \mathrm {b} }$ ± $\mathrm { e } ^ { \mathrm { C } }$ )

Find the area of the region bounded by y = 6x - $x ^ { 2 }$ and y = $x ^ { 2 }$ - 2x. Enter a reduced fraction of form $\frac { a } { b }$ .

Draw the region whose area is given by the definite integral. $\int _ { 4 } ^ { 5 } \frac { 2 } { x ^ { 3 } }$ dx $Draw the region whose area is given by the definite integral. \int _ { 4 } ^ { 5 } \frac { 2 } { x ^ { 3 } } dx$

Draw the region whose area is given by the definite integral. $\int _ { 2 } ^ { 9 } \frac { 2 } { \sqrt { x } }$ dx $Draw the region whose area is given by the definite integral. \int _ { 2 } ^ { 9 } \frac { 2 } { \sqrt { x } } dx$

Find: $\int \frac { x - 7 } { x }$ dx Enter terms in the same order in which they appear in the integral.

Use the graph below to determine the area of the shaded region. Enter your answer exactly in the form a ± $\mathrm { e } ^ { \mathrm {b } }$ . $Use the graph below to determine the area of the shaded region. Enter your answer exactly in the form a ± \mathrm { e } ^ { \mathrm {b } } .$