Question 1

Use a &#64257;nite approximation to estimate the area under the graph of the given function on the stated interval as&#10;instructed.&#10; &#10;A) 174&#10;B) 149&#10;C) 165&#10;D) 126

Accepted Answer

The interval from 4 to 8 is divided into 4 equal parts, each with a width of 1 (since $8 - 4 = 4$ and we are using 4 rectangles). The midpoints for these rectangles will be at $x = 4.5, 5.5, 6.5, $ and $7.5$. Using the midpoint rule, we calculate the area of each rectangle by evaluating the function $f(x) = x^2$ at these midpoints and multiplying by the width (1 in this case). Thus, the total area is approximately:$$f(4.5) \times 1 + f(5.5) \times 1 + f(6.5) \times 1 + f(7.5) \times 1 = 20.25 + 30.25 + 42.25 + 56.25 = 149.$$

Question 2

Use a &#64257;nite approximation to estimate the area under the graph of the given function on the stated interval as&#10;instructed.&#10; &#10;A) 93.75&#10;B) 62.5&#10;C) 10&#10;D) 187.5

Accepted Answer

Using the midpoint rule with two rectangles of equal width, we have
\begin{align*}
\Delta x &= \frac{b-a}{n} = \frac{5-(-5)}{2} = 5 \
x_0 &= -5 + \frac{\Delta x}{2} = -2.5\
x_1 &= x_0 + \Delta x = 2.5\
f(x_0) &= 25 - (-2.5)^2 = 18.75\
f(x_1) &= 25 - 2.5^2 = 18.75\
	ext{Area} &\approx \Delta x \cdot \left[f(x_0) + f(x_1)ight] \
&= 5 \cdot (18.75+18.75) \
&= 187.5.
\end{align*}
Therefore, the best choice is D.

Question 3

Estimate the value of the quantity.&#10;

Accepted Answer

B

Question 4

Use a &#64257;nite sum to estimate the average value of the function on the given interval by partitioning the interval and&#10;evaluating the function at the midpoints of the subintervals.&#10;

Accepted Answer

The average value of the function on the given interval is $$\frac{1}{b-a}\int_a^b f(x) \ dx$$So, for the given function and interval, we have$$\frac{1}{2-0}\int_0^2 (3 + \sin \pi t) \ dt = \frac{1}{2} \left[ 3t - \frac{1}{\pi} \cos \pi t \right]_0^2 = \frac{1}{2} \left( 6 - \frac{1}{\pi} \right) = \frac{6 - \sqrt{2}}{2}$$Therefore, the average value of the function on the given interval is $\frac{6 - \sqrt{2}}{2}$, which is choice A.

Question 5

Use a &#64257;nite approximation to estimate the area under the graph of the given function on the stated interval as&#10;instructed.&#10; &#10;A) 8.4375&#10;B) 12.5&#10;C) 16.875&#10;D) 3.375

Accepted Answer

The interval from 0 to 3 is divided into 2 rectangles of equal width, so each rectangle has a width of 1.5 (3 - 0 = 3, divided by 2). For a lower sum, we use the left end of each subinterval to calculate the height of the rectangles. The first rectangle is from 0 to 1.5, with a height of $f(0) = 0^2 = 0$. The second rectangle is from 1.5 to 3, with a height of $f(1.5) = 1.5^2 = 2.25$. The area of the first rectangle is $0 \times 1.5 = 0$, and the area of the second rectangle is $2.25 \times 1.5 = 3.375$. Adding these areas together gives $0 + 3.375 = 3.375$.

Question 6

Use a &#64257;nite approximation to estimate the area under the graph of the given function on the stated interval as&#10;instructed.&#10; &#10;A) 69&#10;B) 86&#10;C) 62&#10;D) 54

Accepted Answer

Using an upper sum with four rectangles of equal width, we have:
$\Delta x=\frac{6-2}{4}=1$ 
$x_i=2+i\Delta x$ for $i=0,1,2,3,4$.
The upper sum $U$ is given by
\begin{align*}
U&=\sum_{i=1}^{4} f(x_i)\Delta x\
&=f(x_1)\Delta x+f(x_2)\Delta x+f(x_3)\Delta x+f(x_4)\Delta x\
&=f(3)\Delta x+f(4)\Delta x+f(5)\Delta x+f(6)\Delta x\
&=3^2+4^2+5^2+6^2\
&=86.
\end{align*}
Therefore, the best choice is (B), 86.

Question 7

Estimate the value of the quantity.&#10; &#10;A) 385 ft&#10;B) 3850 ft&#10;C) 5800 ft&#10;D) 3950 ft

Accepted Answer

The distance can be estimated using the right-end point Riemann sum: $10(12 + 30 + 56 + 53 + 58 + 55 + 58 + 48 + 15 + 0) = 3850$ ft.

Question 8

Estimate the value of the quantity.&#10; &#10;A) 164 in.&#10;B) 154 in.&#10;C) 170 in.&#10;D) 160 in.

Accepted Answer

The distance is estimated by summing the products of velocity and time interval for each subinterval using right-end points: (8*1) + (21*1) + (28*1) + (27*1) + (25*1) + (28*1) + (23*1) + (4*1) = 164 in.

Question 9

Use a &#64257;nite approximation to estimate the area under the graph of the given function on the stated interval as&#10;instructed.&#10; &#10;A) 174&#10;B) 126&#10;C) 149&#10;D) 165

Accepted Answer

The interval from 4 to 8 is divided into 4 equal parts, each with a width of 1 ((8-4)/4). The x-values at the left endpoints for the rectangles are 4, 5, 6, and 7. The area of each rectangle is the product of its width (1) and the height, which is the value of the function at the left endpoint. So, the total area is calculated as follows: $$f(4) \times 1 + f(5) \times 1 + f(6) \times 1 + f(7) \times 1 = 4^2 + 5^2 + 6^2 + 7^2 = 16 + 25 + 36 + 49 = 126.$$

Question 10

Use a &#64257;nite approximation to estimate the area under the graph of the given function on the stated interval as&#10;instructed.&#10;

Accepted Answer

The answer of Use a &#64257;nite approximation to estimate the...

Question 11

Use a &#64257;nite approximation to estimate the area under the graph of the given function on the stated interval as&#10;instructed.&#10; &#10;A) 4.5&#10;B) 2.5&#10;C) 1&#10;D) 5

Accepted Answer

The answer of Use a &#64257;nite approximation to estimate the...

Question 12

Use a &#64257;nite approximation to estimate the area under the graph of the given function on the stated interval as&#10;instructed.&#10;

Accepted Answer

The answer of Use a &#64257;nite approximation to estimate the...

Question 13

Estimate the value of the quantity.&#10;

Accepted Answer

The answer of Estimate the value of the quantity.&#10; ...

Question 14

Use a &#64257;nite approximation to estimate the area under the graph of the given function on the stated interval as&#10;instructed.&#10; &#10;A) )125&#10;B) )3145&#10;C) )625&#10;D) )75

Accepted Answer

The answer of Use a &#64257;nite approximation to estimate the...

Question 15

Estimate the value of the quantity.&#10; &#10;A) 164 in.&#10;B) 154 in.&#10;C) 328 in.&#10;D) 169 in.

Accepted Answer

The answer of Estimate the value of the quantity.&#10; &#10;A)...

Question 16

Use a &#64257;nite sum to estimate the average value of the function on the given interval by partitioning the interval and&#10;evaluating the function at the midpoints of the subintervals.&#10;

Accepted Answer

The answer of Use a &#64257;nite sum to estimate the...

Question 17

Use a &#64257;nite sum to estimate the average value of the function on the given interval by partitioning the interval and&#10;evaluating the function at the midpoints of the subintervals.&#10;

Accepted Answer

The answer of Use a &#64257;nite sum to estimate the...

Question 18

Use a &#64257;nite approximation to estimate the area under the graph of the given function on the stated interval as&#10;instructed.&#10;

Accepted Answer

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Question 19

Use a &#64257;nite sum to estimate the average value of the function on the given interval by partitioning the interval and&#10;evaluating the function at the midpoints of the subintervals.&#10;

Accepted Answer

The answer of Use a &#64257;nite sum to estimate the...

Question 20

Use a &#64257;nite sum to estimate the average value of the function on the given interval by partitioning the interval and&#10;evaluating the function at the midpoints of the subintervals.&#10; &#10;A) 173&#10;B) 293&#10;C) 1172&#10;D) 58.6

Accepted Answer

The answer of Use a &#64257;nite sum to estimate the...

Question 21

Write the sum without sigma notation and evaluate it.&#10;

Accepted Answer

The answer of Write the sum without sigma notation and...

Question 22

Write the sum without sigma notation and evaluate it.&#10;

Accepted Answer

The answer of Write the sum without sigma notation and...

Question 23

Write the sum without sigma notation and evaluate it.&#10;

Accepted Answer

The answer of Write the sum without sigma notation and...

Question 24

Write the sum without sigma notation and evaluate it.&#10;

Accepted Answer

The answer of Write the sum without sigma notation and...

Question 25

Provide an appropriate response.&#10;

Accepted Answer

The answer of Provide an appropriate response.&#10; ...

Question 26

Write the sum without sigma notation and evaluate it.&#10;

Accepted Answer

The answer of Write the sum without sigma notation and...

Question 27

Express the sum in sigma notation.&#10;

Accepted Answer

The answer of Express the sum in sigma notation.&#10; ...

Question 28

Provide an appropriate response.&#10;

Accepted Answer

The answer of Provide an appropriate response.&#10; ...

Question 29

Provide an appropriate response.&#10;

Accepted Answer

The answer of Provide an appropriate response.&#10; ...

Question 30

Estimate the value of the quantity.&#10; &#10;A) 193.8 gallons&#10;B) 32.3 gallons&#10;C) 145.2 gallons&#10;D) 253.8 gallons

Accepted Answer

The answer of Estimate the value of the quantity.&#10; &#10;A)...

Question 31

Estimate the value of the quantity.&#10; &#10;A) 463.4 m&#10;B) 926.8 m&#10;C) 887.6 m&#10;D) 443.8 m

Accepted Answer

The answer of Estimate the value of the quantity.&#10; &#10;A)...

Question 32

Write the sum without sigma notation and evaluate it.&#10;

Accepted Answer

The answer of Write the sum without sigma notation and...

Question 33

Write the sum without sigma notation and evaluate it.&#10;

Accepted Answer

The answer of Write the sum without sigma notation and...

Question 34

Write the sum without sigma notation and evaluate it.&#10;

Accepted Answer

The answer of Write the sum without sigma notation and...

Question 35

Estimate the value of the quantity.&#10; &#10;A) 34.15 hours&#10;B) 54.88 hours&#10;C) 28.15 hours&#10;D) -85.00 hours

Accepted Answer

The answer of Estimate the value of the quantity.&#10; &#10;A)...

Question 36

Estimate the value of the quantity.&#10;

Accepted Answer

The answer of Estimate the value of the quantity.&#10; ...

Question 37

Write the sum without sigma notation and evaluate it.&#10;

Accepted Answer

The answer of Write the sum without sigma notation and...

Question 38

Provide an appropriate response.&#10;

Accepted Answer

The answer of Provide an appropriate response.&#10; ...

Question 39

Estimate the value of the quantity.&#10; &#10;A) 244 ft&#10;B) 224 ft&#10;C) 122 ft&#10;D) 221 ft

Accepted Answer

The answer of Estimate the value of the quantity.&#10; &#10;A)...

Question 40

Write the sum without sigma notation and evaluate it.&#10;

Accepted Answer

The answer of Write the sum without sigma notation and...

Question 41

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Accepted Answer

The answer of 11eb49f8_e003_462b_b71a_13c204685c02_TB6591_0011eb49f8_e003_6d3e_b71a_1b64de533f44_TB6591_00 11eb49f8_e003_6d3f_b71a_4fa9720e3e5e_TB6591_00...

Question 42

Evaluate the sum.&#10; &#10;A) 164&#10;B) 59&#10;C) 204&#10;D) 199

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The answer of Evaluate the sum.&#10; &#10;A) 164&#10;B) 59&#10;C) 204&#10;D)...

Question 43

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The answer of 11eb49f8_e003_462b_b71a_13c204685c02_TB6591_0011eb49f8_e003_462d_b71a_33301e89e241_TB6591_00...

Question 44

Find the value of the speci&#64257;ed &#64257;nite sum.&#10; &#10;A) -9&#10;B) -7&#10;C) 7&#10;D) 10

Accepted Answer

The answer of Find the value of the speci&#64257;ed &#64257;nite...

Question 45

Express the sum in sigma notation.&#10;

Accepted Answer

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Question 46

Express the sum in sigma notation.&#10;

Accepted Answer

The answer of Express the sum in sigma notation.&#10; ...

Question 47

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Accepted Answer

The answer of 11eb49f8_e003_462b_b71a_13c204685c02_TB6591_0011eb49f8_e003_9456_b71a_3991c41fff6e_TB6591_00 11eb49f8_e003_9457_b71a_61714c3826b9_TB6591_00...

Question 48

Find the value of the speci&#64257;ed &#64257;nite sum.&#10; &#10;A) 9&#10;B) -19&#10;C) -9&#10;D) 19

Accepted Answer

The answer of Find the value of the speci&#64257;ed &#64257;nite...

Question 49

Evaluate the sum.&#10; &#10;A) 60&#10;B) 56&#10;C) 51&#10;D) 48

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Question 50

Evaluate the sum.&#10; &#10;A) 1584&#10;B) 506&#10;C) 1331&#10;D) 4356

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Question 51

Evaluate the sum.&#10;

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Question 52

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Accepted Answer

The answer of 11eb49f8_e003_462b_b71a_13c204685c02_TB6591_0011eb49f8_e003_9452_b71a_2795e78ea120_TB6591_00 11eb49f8_e003_9453_b71a_f3fd507ec2ae_TB6591_00...

Question 53

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Accepted Answer

The answer of 11eb49f8_e003_462b_b71a_13c204685c02_TB6591_0011eb49f8_e003_9454_b71a_3f1af954537c_TB6591_00 11eb49f8_e003_9455_b71a_ffbb3a083fe6_TB6591_00...

Question 54

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Accepted Answer

The answer of 11eb49f8_e003_462b_b71a_13c204685c02_TB6591_0011eb49f8_e003_6d40_b71a_671e750ed4d4_TB6591_00 11eb49f8_e003_6d41_b71a_ef00514e7115_TB6591_00...

Question 55

Find the value of the speci&#64257;ed &#64257;nite sum.&#10; &#10;A) -1&#10;B) 1&#10;C) 0&#10;D) 25

Accepted Answer

The answer of Find the value of the speci&#64257;ed &#64257;nite...

Question 56

Find the value of the speci&#64257;ed &#64257;nite sum.&#10; &#10;A) -21&#10;B) 21&#10;C) 1&#10;D) -1

Accepted Answer

The answer of Find the value of the speci&#64257;ed &#64257;nite...

Question 57

Find the value of the speci&#64257;ed &#64257;nite sum.&#10; &#10;A) 32&#10;B) 4&#10;C) -4&#10;D) -32

Accepted Answer

The answer of Find the value of the speci&#64257;ed &#64257;nite...

Question 58

Express the sum in sigma notation.&#10;

Accepted Answer

The answer of Express the sum in sigma notation.&#10; ...

Question 59

Express the sum in sigma notation.&#10;

Accepted Answer

The answer of Express the sum in sigma notation.&#10; ...

Question 60

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Accepted Answer

The answer of 11eb49f8_e003_462b_b71a_13c204685c02_TB6591_0011eb49f8_e003_462c_b71a_f9bea296c34d_TB6591_00...

Question 61

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Accepted Answer

The answer of 11eb49f8_e003_462b_b71a_13c204685c02_TB6591_0011eb49f8_e003_bb69_b71a_6df8370c700c_TB6591_00...

Question 62

Express the limit as a de&#64257;nite integral.&#10;

Accepted Answer

The answer of Express the limit as a de&#64257;nite integral.&#10;...

Question 63

Express the limit as a de&#64257;nite integral.&#10;

Accepted Answer

The answer of Express the limit as a de&#64257;nite integral.&#10;...

Question 64

Find the norm of the partition P.&#10;P = {1, 2.1, 2.5, 3.1, 4.7, 5}&#10;A) 0.4&#10;B) 1.1&#10;C) 0.6&#10;D) 1.6

Accepted Answer

The answer of Find the norm of the partition P.&#10;P...

Deck 6: Integrals