Question 1

Find the exact average value of $f ( x ) = \sin x + \cos x$ from x = - $\pi$ to x = $\pi$

Accepted Answer

A)  $1 / 2$ 
B)  0 
C)  3/2 
D)  1 
A)  $1 / 2$ 
B)  0 
C)  3/2 
D)  1

Question 2

Find $\sum _ { k = 1 } ^ { n } \frac { k ^ { 2 } - 2 k + 3 } { n ^ { 3 } }$

Accepted Answer

A)  $\frac { 1 } { 5 }$ 
B)  $\frac { 2 } { 3 }$ 
C)  $\frac { 2 } { 5 }$ 
D)  $\frac { 1 } { 3 }$ 
A)  $\frac { 1 } { 5 }$ 
B)  $\frac { 2 } { 3 }$ 
C)  $\frac { 2 } { 5 }$ 
D)  $\frac { 1 } { 3 }$

Question 3

Find $\frac { d } { d x } \int _ { 1 } ^ { x ^ { 2 } } \ln t d t$

Accepted Answer

The answer of Find \(\frac { d } { d...

Question 4

Given $f ( x ) = 4 - x ^ { 2 }$ over the interval [0, 4], find: (a) the signed area, (b) the absolute area, between the graph of f and the x-axis from x = 0 to x = 4.

Accepted Answer

The answer of Given \(f ( x ) = 4...

Question 5

Find the exact average value of $f ( x ) = 3 + \sqrt { x }$ from x = 1 to x = 4.

Accepted Answer

The answer of Find the exact average value of \(f...

Question 6

Use geometry to find the exact value of $\int _ { - 1 } ^ { 4 } ( 2 - | x - 2 | ) d x$

Accepted Answer

A)  9/2 
B)  5/2 
C)  7/2 
D)  11/2 
A)  9/2 
B)  5/2 
C)  7/2 
D)  11/2

Question 7

Find the exact average value of $f ( x ) = x - 2$ from x = -1 to x = 3.

Accepted Answer

A)  -4 
B)  -2 
C)  -1 
D)  2 
A)  -4 
B)  -2 
C)  -1 
D)  2

Question 8

Find the exact average value of $f ( x ) = ( x + 1 ) ^ { 2 } - 2$ from x = -3 to x = 0.

Accepted Answer

A)  3 
B)  0 
C)  1 
D)  2 
A)  3 
B)  0 
C)  1 
D)  2

Question 9

Evaluate $\int _ { - 1 } ^ { 4 } | x - 3 | d x$

Accepted Answer

A)  15/2 
B)  9/2 
C)  17/2 
D)  19/2 
A)  15/2 
B)  9/2 
C)  17/2 
D)  19/2

Question 10

Given $f ( x ) = 2 x + 1$ , find a number c on the interval (1, 3) such that f(c) is the average value of f on [1, 3].

Accepted Answer

The answer of Given \(f ( x ) = 2...

Question 11

Write out each sum in expanded form, and then calculate the value of the sum. $$\sum _ { k = 1 } ^ { 5 } ( 3 + k ) ^ { 2 } + 1$$

Accepted Answer

17 + 26 +

Question 12

Evaluate $\int _ { 1 } ^ { 4 } \frac { 3 \sqrt { x } - 2 } { \sqrt { x } } d x$

Accepted Answer

A)  4 
B)  5 
C)  3 
D)  -3 
A)  4 
B)  5 
C)  3 
D)  -3

Question 13

Approximate the area between the graph of $f ( x ) = \sqrt { x + 1 }$ and the x-axis on the interval [1, 3], for n = 4, using (a) left sums (b) right sums.

Accepted Answer

The answer of Approximate the area between the graph of...

Question 14

Use definite integrals and Fundamental Theorem of Calculus to find the exact area of the region between the graphs of $f ( x ) = x ^ { 2 }$ and $g ( x ) = x + 6$ from x = -3 to x = 0.

Accepted Answer

A)  10 
B)  62/6 
C)  10.5 
D)  61/6 
A)  10 
B)  62/6 
C)  10.5 
D)  61/6

Question 15

Given $f ( x ) = 16 - x ^ { 2 }$ , find a number c on the interval (0, 4) such that f(c) is the average value of f on [0, 4].

Accepted Answer

The answer of Given \(f ( x ) = 16...

Question 16

Given $f ( x ) = 4 - x$ over the interval [1, 6], find: (a) the signed area, (b) the absolute area, between the graph of f and the x-axis from x = 1 to x = 6.

Accepted Answer

The answer of Given \(f ( x ) = 4...

Question 17

Evaluate $\int _ { 0 } ^ { 2 } \left( x ^ { 2 } - 2 \right) ^ { 2 } d x$

Accepted Answer

A)  56/13 
B)  53/15 
C)  56/15 
D)  55/16 
A)  56/13 
B)  53/15 
C)  56/15 
D)  55/16

Question 18

Approximate the area between the graph of $f ( x ) = ( x - 2 ) ^ { 2 } + 1$ and the x-axis on the interval [1, 5], using a left sum with (a) n = 2, (b) n = 4.

Accepted Answer

The answer of Approximate the area between the graph of...

Question 19

Given $f ( x ) = 3 + \sqrt { x }$ , find a number c on the interval (1, 4) such that f(c) is the average value of f on [1, 4].

Accepted Answer

The answer of Given \(f ( x ) = 3...

Question 20

Write the following sum in sigma notation.11 + 18 + 27 + 38 + 51 + 66

Accepted Answer

A)  $\sum _ { k = 2 } ^ { 7 } k ^ { 2 } + 2$ 
B)  $\sum _ { k = 3 } ^ { 8 } k ^ { 2 } + 2$ 
C)  $\sum _ { k = 2 } ^ { 8 } k ^ { 2 } + 2$ 
D)  $\sum _ { k = 3 } ^ { 8 } k ^ { 2 } + 3$ 
A)  $\sum _ { k = 2 } ^ { 7 } k ^ { 2 } + 2$ 
B)  $\sum _ { k = 3 } ^ { 8 } k ^ { 2 } + 2$ 
C)  $\sum _ { k = 2 } ^ { 8 } k ^ { 2 } + 2$ 
D)  $\sum _ { k = 3 } ^ { 8 } k ^ { 2 } + 3$

Find the exact average value of $f ( x ) = \sin x + \cos x$ from x = - $\pi$ to x = $\pi$

Find $\sum _ { k = 1 } ^ { n } \frac { k ^ { 2 } - 2 k + 3 } { n ^ { 3 } }$

Find $\frac { d } { d x } \int _ { 1 } ^ { x ^ { 2 } } \ln t d t$

Given $f ( x ) = 4 - x ^ { 2 }$ over the interval [0, 4], find: (a) the signed area, (b) the absolute area, between the graph of f and the x-axis from x = 0 to x = 4.

Find the exact average value of $f ( x ) = 3 + \sqrt { x }$ from x = 1 to x = 4.

Use geometry to find the exact value of $\int _ { - 1 } ^ { 4 } ( 2 - | x - 2 | ) d x$

Find the exact average value of $f ( x ) = x - 2$ from x = -1 to x = 3.

Find the exact average value of $f ( x ) = ( x + 1 ) ^ { 2 } - 2$ from x = -3 to x = 0.

Evaluate $\int _ { - 1 } ^ { 4 } | x - 3 | d x$

Given $f ( x ) = 2 x + 1$ , find a number c on the interval (1, 3) such that f(c) is the average value of f on [1, 3].

Write out each sum in expanded form, and then calculate the value of the sum. $\sum _ { k = 1 } ^ { 5 } ( 3 + k ) ^ { 2 } + 1$

Evaluate $\int _ { 1 } ^ { 4 } \frac { 3 \sqrt { x } - 2 } { \sqrt { x } } d x$

Approximate the area between the graph of $f ( x ) = \sqrt { x + 1 }$ and the x-axis on the interval [1, 3], for n = 4, using (a) left sums (b) right sums.

Use definite integrals and Fundamental Theorem of Calculus to find the exact area of the region between the graphs of $f ( x ) = x ^ { 2 }$ and $g ( x ) = x + 6$ from x = -3 to x = 0.

Given $f ( x ) = 16 - x ^ { 2 }$ , find a number c on the interval (0, 4) such that f(c) is the average value of f on [0, 4].

Given $f ( x ) = 4 - x$ over the interval [1, 6], find: (a) the signed area, (b) the absolute area, between the graph of f and the x-axis from x = 1 to x = 6.

Evaluate $\int _ { 0 } ^ { 2 } \left( x ^ { 2 } - 2 \right) ^ { 2 } d x$

Approximate the area between the graph of $f ( x ) = ( x - 2 ) ^ { 2 } + 1$ and the x-axis on the interval [1, 5], using a left sum with (a) n = 2, (b) n = 4.

Given $f ( x ) = 3 + \sqrt { x }$ , find a number c on the interval (1, 4) such that f(c) is the average value of f on [1, 4].

Write the following sum in sigma notation.11 + 18 + 27 + 38 + 51 + 66

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Derivatives

Applications of the Derivative

Techniques of Integration

Applications of Integration

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Power Series

Parametric Equations Polar Coordinates and Conic Sections

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Multivariable Functions

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Exam 4: Definite Integrals

Find the exact average value of f(x)=sin⁡x+cos⁡xf ( x ) = \sin x + \cos xf(x)=sinx+cosx from x = - π\piπ to x = π\piπ

Find ∑k=1nk2−2k+3n3\sum _ { k = 1 } ^ { n } \frac { k ^ { 2 } - 2 k + 3 } { n ^ { 3 } }∑k=1n​n3k2−2k+3​

Find ddx∫1x2ln⁡tdt\frac { d } { d x } \int _ { 1 } ^ { x ^ { 2 } } \ln t d tdxd​∫1x2​lntdt

Given f(x)=4−x2f ( x ) = 4 - x ^ { 2 }f(x)=4−x2 over the interval [0, 4], find: (a) the signed area, (b) the absolute area, between the graph of f and the x-axis from x = 0 to x = 4.

Find the exact average value of f(x)=3+xf ( x ) = 3 + \sqrt { x }f(x)=3+x​ from x = 1 to x = 4.

Use geometry to find the exact value of ∫−14(2−∣x−2∣)dx\int _ { - 1 } ^ { 4 } ( 2 - | x - 2 | ) d x∫−14​(2−∣x−2∣)dx

Find the exact average value of f(x)=x−2f ( x ) = x - 2f(x)=x−2 from x = -1 to x = 3.

Find the exact average value of f(x)=(x+1)2−2f ( x ) = ( x + 1 ) ^ { 2 } - 2f(x)=(x+1)2−2 from x = -3 to x = 0.

Evaluate ∫−14∣x−3∣dx\int _ { - 1 } ^ { 4 } | x - 3 | d x∫−14​∣x−3∣dx

Given f(x)=2x+1f ( x ) = 2 x + 1f(x)=2x+1 , find a number c on the interval (1, 3) such that f(c) is the average value of f on [1, 3].

Write out each sum in expanded form, and then calculate the value of the sum. ∑k=15(3+k)2+1\sum _ { k = 1 } ^ { 5 } ( 3 + k ) ^ { 2 } + 1∑k=15​(3+k)2+1

Evaluate ∫143x−2xdx\int _ { 1 } ^ { 4 } \frac { 3 \sqrt { x } - 2 } { \sqrt { x } } d x∫14​x​3x​−2​dx

Approximate the area between the graph of f(x)=x+1f ( x ) = \sqrt { x + 1 }f(x)=x+1​ and the x-axis on the interval [1, 3], for n = 4, using (a) left sums (b) right sums.

Use definite integrals and Fundamental Theorem of Calculus to find the exact area of the region between the graphs of f(x)=x2f ( x ) = x ^ { 2 }f(x)=x2 and g(x)=x+6g ( x ) = x + 6g(x)=x+6 from x = -3 to x = 0.

Given f(x)=16−x2f ( x ) = 16 - x ^ { 2 }f(x)=16−x2 , find a number c on the interval (0, 4) such that f(c) is the average value of f on [0, 4].

Given f(x)=4−xf ( x ) = 4 - xf(x)=4−x over the interval [1, 6], find: (a) the signed area, (b) the absolute area, between the graph of f and the x-axis from x = 1 to x = 6.

Evaluate ∫02(x2−2)2dx\int _ { 0 } ^ { 2 } \left( x ^ { 2 } - 2 \right) ^ { 2 } d x∫02​(x2−2)2dx

Approximate the area between the graph of f(x)=(x−2)2+1f ( x ) = ( x - 2 ) ^ { 2 } + 1f(x)=(x−2)2+1 and the x-axis on the interval [1, 5], using a left sum with (a) n = 2, (b) n = 4.

Given f(x)=3+xf ( x ) = 3 + \sqrt { x }f(x)=3+x​ , find a number c on the interval (1, 4) such that f(c) is the average value of f on [1, 4].

Write the following sum in sigma notation.11 + 18 + 27 + 38 + 51 + 66

Limits

Derivatives

Applications of the Derivative

Techniques of Integration

Applications of Integration

Sequences and Series

Power Series

Parametric Equations Polar Coordinates and Conic Sections

Vectors

Vector Functions

Multivariable Functions

Double and Triple Integrals

Vector Analysis

Functions and Precalculus

Filters

Find the exact average value of $f ( x ) = \sin x + \cos x$ from x = - $\pi$ to x = $\pi$

Find $\sum _ { k = 1 } ^ { n } \frac { k ^ { 2 } - 2 k + 3 } { n ^ { 3 } }$

Find $\frac { d } { d x } \int _ { 1 } ^ { x ^ { 2 } } \ln t d t$

Given $f ( x ) = 4 - x ^ { 2 }$ over the interval [0, 4], find: (a) the signed area, (b) the absolute area, between the graph of f and the x-axis from x = 0 to x = 4.

Find the exact average value of $f ( x ) = 3 + \sqrt { x }$ from x = 1 to x = 4.

Use geometry to find the exact value of $\int _ { - 1 } ^ { 4 } ( 2 - | x - 2 | ) d x$

Find the exact average value of $f ( x ) = x - 2$ from x = -1 to x = 3.

Find the exact average value of $f ( x ) = ( x + 1 ) ^ { 2 } - 2$ from x = -3 to x = 0.

Evaluate $\int _ { - 1 } ^ { 4 } | x - 3 | d x$

Given $f ( x ) = 2 x + 1$ , find a number c on the interval (1, 3) such that f(c) is the average value of f on [1, 3].

Write out each sum in expanded form, and then calculate the value of the sum. $\sum _ { k = 1 } ^ { 5 } ( 3 + k ) ^ { 2 } + 1$

Evaluate $\int _ { 1 } ^ { 4 } \frac { 3 \sqrt { x } - 2 } { \sqrt { x } } d x$

Approximate the area between the graph of $f ( x ) = \sqrt { x + 1 }$ and the x-axis on the interval [1, 3], for n = 4, using (a) left sums (b) right sums.

Use definite integrals and Fundamental Theorem of Calculus to find the exact area of the region between the graphs of $f ( x ) = x ^ { 2 }$ and $g ( x ) = x + 6$ from x = -3 to x = 0.

Given $f ( x ) = 16 - x ^ { 2 }$ , find a number c on the interval (0, 4) such that f(c) is the average value of f on [0, 4].

Given $f ( x ) = 4 - x$ over the interval [1, 6], find: (a) the signed area, (b) the absolute area, between the graph of f and the x-axis from x = 1 to x = 6.

Evaluate $\int _ { 0 } ^ { 2 } \left( x ^ { 2 } - 2 \right) ^ { 2 } d x$

Approximate the area between the graph of $f ( x ) = ( x - 2 ) ^ { 2 } + 1$ and the x-axis on the interval [1, 5], using a left sum with (a) n = 2, (b) n = 4.

Given $f ( x ) = 3 + \sqrt { x }$ , find a number c on the interval (1, 4) such that f(c) is the average value of f on [1, 4].