Question 1

Determine the average value of g(x) = $\sqrt { x }$ over the interval from x = 0 to x = 4.
Enter a reduced fraction of form $\frac { a } { b }$ .

Accepted Answer

The answer of Determine the average value of g(x) =...

Question 2

Find a function f(x) with the following property: f'(x) = - $\frac { 1 } { 4 - x }$ , f(2) = 0.

Accepted Answer

A)  f(x) = $- ( 4 - x ) ^ { - 2 }$ + $\frac { 1 } { \sqrt { 2 } }$ 
B)  f(x) = ln $\frac { 1 } { 4 - x }$ + $\frac { 1 } { 2 }$ 
C)  f(x) = ln $| 4 - x |$ - 2 
D)  f(x) = -ln $| 4 - x |$ + 2 
E)  none of these 
A)  f(x) = $- ( 4 - x ) ^ { - 2 }$ + $\frac { 1 } { \sqrt { 2 } }$ 
B)  f(x) = ln $\frac { 1 } { 4 - x }$ + $\frac { 1 } { 2 }$ 
C)  f(x) = ln $| 4 - x |$ - 2 
D)  f(x) = -ln $| 4 - x |$ + 2 
E)  none of these

Question 3

Find the area of the region bounded by the curve f(x) = $x ^ { 2 }$ and the line y = x.
Enter just a reduced fraction of form $\frac { a } { b }$ .

Accepted Answer

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

Question 4

$\int _ { 2 } ^ { 5 } \left( e ^ { 4 x } - \frac { 1 } { x } \right) d x$ Enter your answer as a( $\mathrm { e } ^ { \mathrm { b } }$ - $\mathrm { e } ^ { \mathrm { C } }$ ) + d.

Accepted Answer

The answer of \(\int _ { 2 } ^ {...

Question 5

Find: $\int \left( x ^ { 3 } + 1 \right) d x$ 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: \(\int \left( x ^ { 3...

Question 6

Suppose that at time t (0 ≤ t ≤ 2 , t in months), the sales of a certain commodity are decreasing at a rate of $1000 \mathrm { e } ^ { - 0.05t }$ units per month. Calculate the total change in sales from t = 0 to t = 2.
Enter your answer as just a(1 - $\mathrm { e } ^ { \mathrm { b } }$ ) where b is a real number to one decimal place.

Accepted Answer

The answer of Suppose that at time t (0 ≤...

Question 7

For the Riemann sum, [ $( 3.5 ) ^ { 4 }$ + 7(3.5) + $( 4 ) ^ { 4 }$ + 7(4) + $( 4.5 ) ^ { 4 }$ + 7(4.5) + $( 5 ) ^ { 4 }$ + 7(5)](0.5); a = 3, find n, b, and f(x).
Enter your answer as just n, b, f(x) (2 integers in that order separated by commas and followed by a polynomial in x).

Accepted Answer

The answer of For the Riemann sum, [ \(( 3.5...

Question 8

Find: $\int \left( x ^ { 2 } - \frac { 1 } { 4 x } \right) d x$ Enter your terms in standard forms in the order in which they appear in the integral.

Accepted Answer

The answer of Find: \(\int \left( x ^ { 2...

Question 9

Set up a definite integral or sum of definite integrals that represent the indicated shaded areas over the interval [a, c]. y = g(x)

Accepted Answer

A)  $\int _ { b } ^ { a } [ f ( x ) - g ( x ) ] d x$ + $\int _ { b } ^ { c } [ g ( x ) - f ( x ) ] d x$ 
B)  $\int _ { a } ^ { b } [ g ( x ) - f ( x ) ] d x$ + $\int _ { b } ^ { c } [ f ( x ) - g ( x ) ] d x$ 
C)  $\int _ { a } ^ { c } [ f ( x ) - g ( x ) ] d x$ 
D)  $\int _ { a } ^ { b } [ f ( x ) - g ( x ) ] d x$ + $\int _ { b } ^ { c } [ g ( x ) - f ( x ) ] d x$ 
A)  $\int _ { b } ^ { a } [ f ( x ) - g ( x ) ] d x$ + $\int _ { b } ^ { c } [ g ( x ) - f ( x ) ] d x$ 
B)  $\int _ { a } ^ { b } [ g ( x ) - f ( x ) ] d x$ + $\int _ { b } ^ { c } [ f ( x ) - g ( x ) ] d x$ 
C)  $\int _ { a } ^ { c } [ f ( x ) - g ( x ) ] d x$ 
D)  $\int _ { a } ^ { b } [ f ( x ) - g ( x ) ] d x$ + $\int _ { b } ^ { c } [ g ( x ) - f ( x ) ] d x$

Question 10

What is the area under the curve y = $\frac { 2 } { x }$ between x = 1 and x = 3?

Accepted Answer

A)  ln 9 
B)  $\frac { 1 } { 2 }$ ln 3 
C)  2 ln 3 - e 
D)  2 ln 3 - 2 
E)  none of these 
A)  ln 9 
B)  $\frac { 1 } { 2 }$ ln 3 
C)  2 ln 3 - e 
D)  2 ln 3 - 2 
E)  none of these

Question 11

Find: $\int ( 2 x + 1 ) ^ { 2 } d x$ 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: \(\int ( 2 x + 1...

Question 12

Find a function f(x) with the following property: f'(x) = $x ^ { 6 }$ + $\frac { x ^ { 4 } } { 3 }$ - $x ^ { 3 }$ + $\frac { 5 } { 3 }$ $x ^ { 2 }$ , f(0) = e.

Accepted Answer

A)  f(x) = 6 $x ^ { 7 }$ + $\frac { 4 } { 3 }$ $x ^ { 5 }$ - $3 x ^ { 4 }$ + $\frac { 10 } { 3 }$ $x ^ { 3 }$ 
B)  f(x) = $\frac { x ^ { 7 } } { 6 }$ + $\frac { x ^ { 5 } } { 12 }$ - $\frac { x ^ { 4 } } { 3 }$ + $\frac { 5 } { 6 }$ $x ^ { 3 }$ + ex 
C)  f(x) = $\frac { x ^ { 7 } } { 7 }$ + $\frac { x ^ { 5 } } { 15 }$ - $\frac { x ^ { 4 } } { 4 }$ + $\frac { 5 } { 9 }$ $x ^ { 3 }$ + e 
D)  $f(x)=7^{e^{7}}+\frac{5}{3 \mathrm{e}^{5}}-4^{e^{4}}+5^{e^{3}}$ 
A)  f(x) = 6 $x ^ { 7 }$ + $\frac { 4 } { 3 }$ $x ^ { 5 }$ - $3 x ^ { 4 }$ + $\frac { 10 } { 3 }$ $x ^ { 3 }$ 
B)  f(x) = $\frac { x ^ { 7 } } { 6 }$ + $\frac { x ^ { 5 } } { 12 }$ - $\frac { x ^ { 4 } } { 3 }$ + $\frac { 5 } { 6 }$ $x ^ { 3 }$ + ex 
C)  f(x) = $\frac { x ^ { 7 } } { 7 }$ + $\frac { x ^ { 5 } } { 15 }$ - $\frac { x ^ { 4 } } { 4 }$ + $\frac { 5 } { 9 }$ $x ^ { 3 }$ + e 
D)  $f(x)=7^{e^{7}}+\frac{5}{3 \mathrm{e}^{5}}-4^{e^{4}}+5^{e^{3}}$

Question 13

$\int _ { a } ^ { b } ( f ( x ) - g ( x ) ) d x$ expresses the area between the curves y = f(x) and y = g(x) from $x = a \operatorname { to } x = b$ only if:

Accepted Answer

A)  f(x) is greater than or equal to g(x) for all x between a and b , and neither f(x) nor g(x) crosses the x-axis . 
B)  f(x) is greater than or equal to g(x) for all x between a and b . 
C)  f(x) and g(x) do not cross each other between x = a and x = b . 
D)  neither f(x) and g(x) cross the x-axis. 
E)  none of these 
A)  f(x) is greater than or equal to g(x) for all x between a and b , and neither f(x) nor g(x) crosses the x-axis . 
B)  f(x) is greater than or equal to g(x) for all x between a and b . 
C)  f(x) and g(x) do not cross each other between x = a and x = b . 
D)  neither f(x) and g(x) cross the x-axis. 
E)  none of these

Question 14

Determine the average value of f(x) = $x ^ { 3 }$ + 3 $x ^ { 2 }$ + 3x + 1 over the interval from x = -1 to x = 3.
Enter an integer.

Accepted Answer

The answer of Determine the average value of f(x) =...

Question 15

Determine the average value of f(x) = $e ^ { x }$ over the interval from x = 1 to x = 4.
Enter your answer in the form a( $\mathrm { e } ^ { \mathrm { b} }$ ± e)

Accepted Answer

The answer of Determine the average value of f(x) =...

Question 16

Find the area bounded by y = $\frac { 1 } { x }$ , the x-axis, and the lines $x = 1 \text { and } x = a > 1$ Enter a real number (no approximations).

Accepted Answer

The answer of Find the area bounded by y =...

Question 17

Find the value of k that makes the antidifferentiation formula true. 
-$\int x ^ { - 7 } d x$ = k $x ^ { - 6 }$ + C

Accepted Answer

A)  6 
B)  -6 
C)  $\frac { 1 } { 6 }$ 
D)  - $\frac { 1 } { 6 }$ 
A)  6 
B)  -6 
C)  $\frac { 1 } { 6 }$ 
D)  - $\frac { 1 } { 6 }$

Question 18

Find all antiderivatives of the function. 
-f(x) = $x ^ { 15 }$ Enter your answer as a polynomial in x in standard form.

Accepted Answer

The answer of Find all antiderivatives of the function. 
-f(x)...

Question 19

Determine the average value of f(x) = $\frac { 1 } { x ^ { 2 } }$ over the interval from x = 1 to x = 100.
Enter a reduced fraction of form $\frac { a } { b }$ .

Accepted Answer

The answer of Determine the average value of f(x) =...

Question 20

Find the area of the region bounded by f(x) = { $x ^ { 2 }$ (0 ≤ x ≤1); 2x - $x ^ { 2 }$ (1 ≤ x ≤ 2)} and the line $y = 0$ Enter just an integer.

Accepted Answer

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

Determine the average value of g(x) = $\sqrt { x }$ over the interval from x = 0 to x = 4. Enter a reduced fraction of form $\frac { a } { b }$ .

Find a function f(x) with the following property: f'(x) = - $\frac { 1 } { 4 - x }$ , f(2) = 0.

Find the area of the region bounded by the curve f(x) = $x ^ { 2 }$ and the line y = x. Enter just a reduced fraction of form $\frac { a } { b }$ .

$\int _ { 2 } ^ { 5 } \left( e ^ { 4 x } - \frac { 1 } { x } \right) d x$ Enter your answer as a( $\mathrm { e } ^ { \mathrm { b } }$ - $\mathrm { e } ^ { \mathrm { C } }$ ) + d.

Find: $\int \left( x ^ { 3 } + 1 \right) d x$ Enter your answer as a polynomial in x in standard form with any fractional coefficients or powers reduced of form $\frac { a } { b }$ .

For the Riemann sum, [ $( 3.5 ) ^ { 4 }$ + 7(3.5) + $( 4 ) ^ { 4 }$ + 7(4) + $( 4.5 ) ^ { 4 }$ + 7(4.5) + $( 5 ) ^ { 4 }$ + 7(5)](0.5); a = 3, find n, b, and f(x). Enter your answer as just n, b, f(x) (2 integers in that order separated by commas and followed by a polynomial in x).

Find: $\int \left( x ^ { 2 } - \frac { 1 } { 4 x } \right) d x$ Enter your terms in standard forms in the order in which they appear in the integral.

Set up a definite integral or sum of definite integrals that represent the indicated shaded areas over the interval [a, c]. y = g(x)

What is the area under the curve y = $\frac { 2 } { x }$ between x = 1 and x = 3?

Find: $\int ( 2 x + 1 ) ^ { 2 } d x$ Enter your answer as a polynomial in x in standard form with any fractional coefficients or powers reduced of form $\frac { a } { b }$ .

Find a function f(x) with the following property: f'(x) = $x ^ { 6 }$ + $\frac { x ^ { 4 } } { 3 }$ - $x ^ { 3 }$ + $\frac { 5 } { 3 }$ $x ^ { 2 }$ , f(0) = e.

$\int _ { a } ^ { b } ( f ( x ) - g ( x ) ) d x$ expresses the area between the curves y = f(x) and y = g(x) from $x = a \operatorname { to } x = b$ only if:

Determine the average value of f(x) = $x ^ { 3 }$ + 3 $x ^ { 2 }$ + 3x + 1 over the interval from x = -1 to x = 3. Enter an integer.

Determine the average value of f(x) = $e ^ { x }$ over the interval from x = 1 to x = 4. Enter your answer in the form a( $\mathrm { e } ^ { \mathrm { b} }$ ± e)

Find the area bounded by y = $\frac { 1 } { x }$ , the x-axis, and the lines $x = 1 \text { and } x = a > 1$ Enter a real number (no approximations).

Find the value of k that makes the antidifferentiation formula true. - $\int x ^ { - 7 } d x$ = k $x ^ { - 6 }$ + C

Find all antiderivatives of the function. -f(x) = $x ^ { 15 }$ Enter your answer as a polynomial in x in standard form.

Determine the average value of f(x) = $\frac { 1 } { x ^ { 2 } }$ over the interval from x = 1 to x = 100. Enter a reduced fraction of form $\frac { a } { b }$ .

Find the area of the region bounded by f(x) = { $x ^ { 2 }$ (0 ≤ x ≤1); 2x - $x ^ { 2 }$ (1 ≤ x ≤ 2)} and the line $y = 0$ Enter just an integer.

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

Determine the average value of g(x) = x\sqrt { x }x​ over the interval from x = 0 to x = 4. Enter a reduced fraction of form ab\frac { a } { b }ba​ .

Find a function f(x) with the following property: f'(x) = - 14−x\frac { 1 } { 4 - x }4−x1​ , f(2) = 0.

Find the area of the region bounded by the curve f(x) = x2x ^ { 2 }x2 and the line y = x. Enter just a reduced fraction of form ab\frac { a } { b }ba​ .

∫25(e4x−1x)dx\int _ { 2 } ^ { 5 } \left( e ^ { 4 x } - \frac { 1 } { x } \right) d x∫25​(e4x−x1​)dx Enter your answer as a( eb\mathrm { e } ^ { \mathrm { b } }eb - eC\mathrm { e } ^ { \mathrm { C } }eC ) + d.

Find: ∫(x3+1)dx\int \left( x ^ { 3 } + 1 \right) d x∫(x3+1)dx 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​ .

Find: ∫(x2−14x)dx\int \left( x ^ { 2 } - \frac { 1 } { 4 x } \right) d x∫(x2−4x1​)dx Enter your terms in standard forms in the order in which they appear in the integral.

Set up a definite integral or sum of definite integrals that represent the indicated shaded areas over the interval [a, c]. y = g(x)

What is the area under the curve y = 2x\frac { 2 } { x }x2​ between x = 1 and x = 3?

Find: ∫(2x+1)2dx\int ( 2 x + 1 ) ^ { 2 } d x∫(2x+1)2dx 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​ .

Find a function f(x) with the following property: f'(x) = x6x ^ { 6 }x6 + x43\frac { x ^ { 4 } } { 3 }3x4​ - x3x ^ { 3 }x3 + 53\frac { 5 } { 3 }35​ x2x ^ { 2 }x2 , f(0) = e.

∫ab(f(x)−g(x))dx\int _ { a } ^ { b } ( f ( x ) - g ( x ) ) d x∫ab​(f(x)−g(x))dx expresses the area between the curves y = f(x) and y = g(x) from x=ato⁡x=bx = a \operatorname { to } x = bx=atox=b only if:

Determine the average value of f(x) = x3x ^ { 3 }x3 + 3 x2x ^ { 2 }x2 + 3x + 1 over the interval from x = -1 to x = 3. Enter an integer.

Determine the average value of f(x) = exe ^ { x }ex over the interval from x = 1 to x = 4. Enter your answer in the form a( eb\mathrm { e } ^ { \mathrm { b} }eb ± e)

Find the area bounded by y = 1x\frac { 1 } { x }x1​ , the x-axis, and the lines x=1 and x=a>1x = 1 \text { and } x = a > 1x=1 and x=a>1 Enter a real number (no approximations).

Find the value of k that makes the antidifferentiation formula true. - ∫x−7dx\int x ^ { - 7 } d x∫x−7dx = k x−6x ^ { - 6 }x−6 + C

Find all antiderivatives of the function. -f(x) = x15x ^ { 15 }x15 Enter your answer as a polynomial in x in standard form.

Determine the average value of f(x) = 1x2\frac { 1 } { x ^ { 2 } }x21​ over the interval from x = 1 to x = 100. Enter a reduced fraction of form ab\frac { a } { b }ba​ .

Find the area of the region bounded by f(x) = { x2x ^ { 2 }x2 (0 ≤ x ≤1); 2x - x2x ^ { 2 }x2 (1 ≤ x ≤ 2)} and the line y=0y = 0y=0 Enter just an integer.

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

Determine the average value of g(x) = $\sqrt { x }$ over the interval from x = 0 to x = 4. Enter a reduced fraction of form $\frac { a } { b }$ .

Find a function f(x) with the following property: f'(x) = - $\frac { 1 } { 4 - x }$ , f(2) = 0.

Find the area of the region bounded by the curve f(x) = $x ^ { 2 }$ and the line y = x. Enter just a reduced fraction of form $\frac { a } { b }$ .

$\int _ { 2 } ^ { 5 } \left( e ^ { 4 x } - \frac { 1 } { x } \right) d x$ Enter your answer as a( $\mathrm { e } ^ { \mathrm { b } }$ - $\mathrm { e } ^ { \mathrm { C } }$ ) + d.

Find: $\int \left( x ^ { 3 } + 1 \right) d x$ Enter your answer as a polynomial in x in standard form with any fractional coefficients or powers reduced of form $\frac { a } { b }$ .

Find: $\int \left( x ^ { 2 } - \frac { 1 } { 4 x } \right) d x$ Enter your terms in standard forms in the order in which they appear in the integral.

What is the area under the curve y = $\frac { 2 } { x }$ between x = 1 and x = 3?

Find: $\int ( 2 x + 1 ) ^ { 2 } d x$ Enter your answer as a polynomial in x in standard form with any fractional coefficients or powers reduced of form $\frac { a } { b }$ .

Find a function f(x) with the following property: f'(x) = $x ^ { 6 }$ + $\frac { x ^ { 4 } } { 3 }$ - $x ^ { 3 }$ + $\frac { 5 } { 3 }$ $x ^ { 2 }$ , f(0) = e.

$\int _ { a } ^ { b } ( f ( x ) - g ( x ) ) d x$ expresses the area between the curves y = f(x) and y = g(x) from $x = a \operatorname { to } x = b$ only if:

Determine the average value of f(x) = $x ^ { 3 }$ + 3 $x ^ { 2 }$ + 3x + 1 over the interval from x = -1 to x = 3. Enter an integer.

Determine the average value of f(x) = $e ^ { x }$ over the interval from x = 1 to x = 4. Enter your answer in the form a( $\mathrm { e } ^ { \mathrm { b} }$ ± e)

Find the area bounded by y = $\frac { 1 } { x }$ , the x-axis, and the lines $x = 1 \text { and } x = a > 1$ Enter a real number (no approximations).

Find the value of k that makes the antidifferentiation formula true. - $\int x ^ { - 7 } d x$ = k $x ^ { - 6 }$ + C

Find all antiderivatives of the function. -f(x) = $x ^ { 15 }$ Enter your answer as a polynomial in x in standard form.

Determine the average value of f(x) = $\frac { 1 } { x ^ { 2 } }$ over the interval from x = 1 to x = 100. Enter a reduced fraction of form $\frac { a } { b }$ .

Find the area of the region bounded by f(x) = { $x ^ { 2 }$ (0 ≤ x ≤1); 2x - $x ^ { 2 }$ (1 ≤ x ≤ 2)} and the line $y = 0$ Enter just an integer.