Question 1

Use a ﬁnite sum to estimate the average value of the function on the given interval by partitioning the interval and
evaluating the function at the midpoints of the subintervals.
-$f ( t ) = 1 e ^ { - t } \cos 2 \pi t$ on $[ - 2,3 ]$ divided into 5 subintervals (Round to the nearest hundredth.)

Accepted Answer

To estimate the average value of the function $f(t) = \frac{1}{e^{-t}} \cos 2\pi t$ on the interval $[-2,3]$ using 5 subintervals, we first calculate the width of each subinterval: $\Delta t = \frac{3 - (-2)}{5} = 1$. The midpoints of these subintervals are $-1.5, -0.5, 0.5, 1.5,$ and $2.5$. Evaluating the function at these midpoints and averaging the results gives an approximation of the average value of the function over the interval. The calculation involves substituting each midpoint into the function, summing these values, and then dividing by 5. The result, rounded to the nearest hundredth, is approximately -1.41.

Question 2

Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed. -f(x) = x² between x = 0 and x = 1 using the "midpoint rule" with two rectangles of equal width.

Accepted Answer

The midpoint rule with two rectangles involves evaluating the function at the midpoints of the intervals. The interval [0,1] is divided into two subintervals: [0,0.5] and [0.5,1]. The midpoints are 0.25 and 0.75. The function values at these points are f(0.25) = 0.25^2 = 0.0625 and f(0.75) = 0.75^2 = 0.5625. The width of each rectangle is 0.5. The estimated area is 0.5*(0.0625 + 0.5625) = 0.3125.

Question 3

Use a ﬁnite sum to estimate the average value of the function on the given interval by partitioning the interval and
evaluating the function at the midpoints of the subintervals.
-$\mathrm { f } ( \mathrm { x } ) = 6 + \sin \pi \mathrm { x }$ on $[ 0,2 ]$ divided into 4 subintervals

Accepted Answer

D

Question 4

Use a ﬁnite sum to estimate the average value of the function on the given interval by partitioning the interval and
evaluating the function at the midpoints of the subintervals.
-$f ( t ) = 8 - \left( \cos \frac { \pi t } { 2 } \right) ^ { 2 }$ on $[ 0,4 ]$ divided into 4 subintervals

Accepted Answer

To estimate the average value of $f(t) = 8 - \left( \cos \frac{\pi t}{2} \right)^2$ on $[0,4]$ with 4 subintervals, we divide the interval into 4 equal parts: $[0,1], [1,2], [2,3], [3,4]$. The midpoints are $0.5, 1.5, 2.5, 3.5$. Evaluating $f(t)$ at these midpoints and averaging gives us the estimate. Calculating $f(0.5), f(1.5), f(2.5), f(3.5)$ and averaging them gives $\frac{15}{2}$.

Question 5

Use a ﬁnite sum to estimate the average value of the function on the given interval by partitioning the interval and
evaluating the function at the midpoints of the subintervals.
-$\mathrm { f } ( \mathrm { x } ) = \frac { 5 } { \mathrm { x } }$ on $\left[ \frac { 1 } { 2 } , \frac { 11 } { 2 } \right]$ divided into 5 subintervals

Accepted Answer

The average value of a function $f(x)$ over an interval $[a, b]$ can be estimated by dividing the interval into $n$ subintervals, finding the midpoint of each subinterval, evaluating the function at these midpoints, and then averaging these values. Here, the interval $\left[\frac{1}{2}, \frac{11}{2}\right]$ is divided into 5 subintervals. The length of each subinterval is $\frac{\frac{11}{2} - \frac{1}{2}}{5} = 1$. The midpoints for these subintervals are $1, 2, 3, 4,$ and $5$. Evaluating $f(x) = \frac{5}{x}$ at these midpoints and averaging the results gives $\frac{1}{5} \left( \frac{5}{1} + \frac{5}{2} + \frac{5}{3} + \frac{5}{4} + \frac{5}{5} \right) = \frac{137}{60}$.

Question 6

Use a ﬁnite sum to estimate the average value of the function on the given interval by partitioning the interval and
evaluating the function at the midpoints of the subintervals.
-$f ( x ) = x ^ { 2 } - 2$ on $[ - 3,7 ]$ divided into 5 subintervals

Accepted Answer

We will use the midpoint rule to estimate the average value of the function. Dividing the interval into 5 subintervals gives us:

$$-3, \quad -3+2(2), \quad -3+2(4), \quad -3+2(6), \quad 7$$

or

$$-3, \quad -1, \quad 1, \quad 3, \quad 7$$

The width of each subinterval is 2. The midpoint of each subinterval is found by averaging the endpoints:

$$\frac{-3+(-1)}{2}=-2, \quad \frac{-1+1}{2}=0, \quad \frac{1+3}{2}=2, \quad \frac{3+7}{2}=5$$

Now we can evaluate the function at each midpoint:

$$f(-2)=(-2)^2-2=2$$
$$f(0)=0^2-2=-2$$
$$f(2)=2^2-2=2$$
$$f(5)=5^2-2=23$$

So the average value is approximately:

$$\frac{1}{5}\left(2-2+2+23 \right)=\frac{25}{5}=5$$

Therefore, the best choice is C) 10 since 5 is halfway between -3 and 7, the endpoints of the interval.

Question 7

Use a finite approximation to estimate the area under the graph of the given function on the stated interval as
instructed.
-$f ( x ) = \frac { 1 } { x }$ between $x = 1$ and $x = 8$ using the "midpoint rule" with two rectangles of equal width.

Accepted Answer

The answer of Use a finite approximation to estimate the...

Question 8

Use a finite approximation to estimate the area under the graph of the given function on the stated interval as
instructed.
-$$f ( x ) = x ^ { 2 } \text { between } x = 4 \text { and } x = 8 \text { using an upper sum with four rectangles of equal width. }$$

Accepted Answer

The interval from 4 to 8 is divided into 4 equal parts, each with a width of 1 (8 - 4 = 4, 4/4 = 1). For an upper sum, we use the right endpoint of each interval to calculate the area of each rectangle. The rectangles will be based at x = 5, 6, 7, and 8. The area of each rectangle is the function value at these points times the width (1). So, the total area is $(5^2 + 6^2 + 7^2 + 8^2) * 1 = (25 + 36 + 49 + 64) = 174$.

Question 9

Use a finite approximation to estimate the area under the graph of the given function on the stated interval as
instructed.
-$$f ( x ) = x ^ { 2 } \text { between } x = 4 \text { and } x = 8 \text { using a lower sum with four rectangles of equal width. }$$

Accepted Answer

The interval from $x = 4$ to $x = 8$ is divided into 4 equal parts, each with a width of $1$ unit ($\frac{8 - 4}{4} = 1$). For a lower sum, we use the left end of each interval to calculate the area of each rectangle. The x-values at these points are 4, 5, 6, and 7. The area of each rectangle is the function value at these points times the width (1 unit). So, the total area is $$f(4)\times1 + f(5)\times1 + f(6)\times1 + f(7)\times1 = 4^2 + 5^2 + 6^2 + 7^2 = 16 + 25 + 36 + 49 = 126.$$

Question 10

Use a finite approximation to estimate the area under the graph of the given function on the stated interval as
instructed.
-$f ( x ) = \frac { 1 } { x }$ between $x = 3$ and $x = 6$ using a upper sum with two rectangles of equal width.

Accepted Answer

The answer of Use a finite approximation to estimate the...

Question 11

Use a ﬁnite sum to estimate the average value of the function on the given interval by partitioning the interval and
evaluating the function at the midpoints of the subintervals.
-$f ( x ) = 3 x ^ { 5 }$ on $[ 1,3 ]$ divided into 4 subintervals

Accepted Answer

First, we need to find the length of each subinterval:

$$\Delta x = \frac{3-1}{4} = 0.5$$

Next, we need to find the midpoint of each subinterval:

$$x_1 = 1.25$$
$$x_2 = 1.75$$
$$x_3 = 2.25$$
$$x_4 = 2.75$$

Then, we can estimate the average value of the function using the formula:

$$\frac{1}{b-a}\sum_{i=1}^{n} f(x_i) \Delta x$$

Plugging in the values, we get:

$$\frac{1}{3-1}[(3(1.25)^5)+(3(1.75)^5)+(3(2.25)^5)+(3(2.75)^5)](0.5) \approx 175.8$$

Therefore, the best choice is B, 175.8.

Question 12

Use a ﬁnite sum to estimate the average value of the function on the given interval by partitioning the interval and
evaluating the function at the midpoints of the subintervals.
-$f ( x ) = \frac { \sqrt { x } } { 4 }$ on [3,7] divided into 4 subintervals (Round to the nearest hundredth.)

Accepted Answer

The interval [3,7] is divided into 4 subintervals, each of width 1. Evaluating the function at the midpoints (3.5, 4.5, 5.5, 6.5) and averaging gives approximately 0.56.

Question 13

Use a finite approximation to estimate the area under the graph of the given function on the stated interval as
instructed.
-$f ( x ) = \frac { 1 } { x }$ between $x = 4$ and $x = 5$ using a lower sum with two rectangles of equal width.

Accepted Answer

The answer of Use a finite approximation to estimate the...

Question 14

Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed. -f(x) = 1 - x² between x = -1 and x = 1 using the "midpoint rule" with two rectangles of equal width.

Accepted Answer

The answer of Use a finite approximation to estimate the...

Question 15

Use a ﬁnite sum to estimate the average value of the function on the given interval by partitioning the interval and
evaluating the function at the midpoints of the subintervals.
-$f ( x ) = 5 ( 2 ) ^ { x }$ on $[ 1,7 ]$ divided into 3 subintervals

Accepted Answer

To estimate the average value of $f(x) = 5(2)^x$ on the interval $[1,7]$ using 3 subintervals, we first find the width of each subinterval: $\Delta x = \frac{7-1}{3} = 2$. The midpoints of these subintervals are at $x = 2, 4, 6$. Evaluating $f(x)$ at these midpoints gives us $f(2) = 5(2)^2 = 20$, $f(4) = 5(2)^4 = 80$, and $f(6) = 5(2)^6 = 320$. The sum of these values is $20 + 80 + 320 = 420$. To find the average value, we divide this sum by the number of subintervals: $\frac{420}{3} = 140$.

Question 16

Use a finite approximation to estimate the area under the graph of the given function on the stated interval as
instructed.
-$$f ( x ) = x ^ { 2 } \text { between } x = 2 \text { and } x = 6 \text { using the "midpoint rule" with four rectangles of equal width. }$$

Accepted Answer

The answer of Use a finite approximation to estimate the...

Question 17

Use a finite approximation to estimate the area under the graph of the given function on the stated interval as
instructed.
-$$f ( x ) = x ^ { 2 } \text { between } x = 0 \text { and } x = 2 \text { using an upper sum with two rectangles of equal width. }$$

Accepted Answer

The interval from 0 to 2 is divided into two equal parts, each with a width of 1. For an upper sum, we use the right endpoint of each interval to calculate the area of the rectangles. The first rectangle uses the value of the function at x=1, which is $1^2 = 1$, and the second rectangle uses the value of the function at x=2, which is $2^2 = 4$. The total area is $1*1 + 4*1 = 5$.

Question 18

Use a ﬁnite sum to estimate the average value of the function on the given interval by partitioning the interval and
evaluating the function at the midpoints of the subintervals.
-$\mathrm { f } ( \mathrm { x } ) = 5 \mathrm { x } \sin \pi \mathrm { x }$ on $[ 0,4 ]$ divided into 4 subintervals

Accepted Answer

We have to use a finite sum to estimate the average value of the function on the given interval by partitioning the interval and evaluating the function at the midpoints of the subintervals.

The interval [0,4] is divided into 4 subintervals: [0,1], [1,2], [2,3], [3,4]

The length of each subinterval is:

$$\Delta x = \frac{4-0}{4} = 1$$

The midpoints of each subinterval are:

$$x_1 = \frac{1}{2}, x_2 = \frac{3}{2}, x_3 = \frac{5}{2}, x_4 = \frac{7}{2}$$

The sum we need to evaluate is:

$$\frac{1}{4} \sum_{i=1}^{4} f(x_i) \Delta x$$

where

$$f(x) = 5x\sin(\pi x)$$

So,

\begin{align*}
\frac{1}{4} \sum_{i=1}^{4} f(x_i) \Delta x &= \frac{1}{4}[(5\cdot \frac{1}{2}\sin(\pi \cdot \frac{1}{2})) + (5\cdot \frac{3}{2}\sin(\pi \cdot \frac{3}{2}))\
&\quad+ (5\cdot \frac{5}{2}\sin(\pi \cdot \frac{5}{2})) + (5\cdot \frac{7}{2}\sin(\pi \cdot \frac{7}{2}))] \cdot 1\
&= \frac{5}{4}[\sin(\frac{\pi}{2}) + 3\sin(\frac{3\pi}{2}) + 5\sin(\frac{5\pi}{2}) + 7\sin(\frac{7\pi}{2})]\
&= \frac{5}{4}[-1 + 3(-1) + 5(1) + 7(-1)]\
&= \frac{5}{4}(-1 - 3 + 5 - 7)\
&= -\frac{5}{2}
\end{align*}

Therefore, the best choice is (C), i.e., $-\frac{5}{2}$.

Question 19

Use a finite approximation to estimate the area under the graph of the given function on the stated interval as
instructed.
-$$f ( x ) = x ^ { 2 } \text { between } x = 0 \text { and } x = 1 \text { using a lower sum with two rectangles of equal width. }$$

Accepted Answer

To estimate the area using a lower sum with two rectangles, divide the interval [0,1] into two equal subintervals: [0,0.5] and [0.5,1]. The width of each rectangle is 0.5. The height of each rectangle is determined by the function value at the left endpoint of each subinterval. For [0,0.5], the height is f(0) = 0^2 = 0. For [0.5,1], the height is f(0.5) = (0.5)^2 = 0.25. The area is (0.5)(0) + (0.5)(0.25) = 0.125.

Question 20

Use a ﬁnite sum to estimate the average value of the function on the given interval by partitioning the interval and
evaluating the function at the midpoints of the subintervals.
-$f ( x ) = 8 | x |$ on $[ - 3,3 ]$ divided into 4 subintervals

Accepted Answer

The interval $[-3,3]$ is divided into 4 subintervals: $[-3,-1.5]$, $[-1.5,0]$, $[0,1.5]$, and $[1.5,3]$. The midpoints of these intervals are $-2.25$, $-0.75$, $0.75$, and $2.25$, respectively. Evaluating the function $f(x) = 8|x|$ at these midpoints gives $18$, $6$, $6$, and $18$, respectively. The average value is then $\frac{1}{4}(18 + 6 + 6 + 18) = 12$.

Quiz 5: Integration

Use a ﬁnite sum to estimate the average value of the function on the given interval by partitioning the interval and evaluating the function at the midpoints of the subintervals. - $f ( t ) = 1 e ^ { - t } \cos 2 \pi t$ on $[ - 2,3 ]$ divided into 5 subintervals (Round to the nearest hundredth.)

Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed. -f(x) = x² between x = 0 and x = 1 using the "midpoint rule" with two rectangles of equal width.

Use a ﬁnite sum to estimate the average value of the function on the given interval by partitioning the interval and evaluating the function at the midpoints of the subintervals. - $\mathrm { f } ( \mathrm { x } ) = 6 + \sin \pi \mathrm { x }$ on $[ 0,2 ]$ divided into 4 subintervals

Use a ﬁnite sum to estimate the average value of the function on the given interval by partitioning the interval and evaluating the function at the midpoints of the subintervals. - $f ( t ) = 8 - \left( \cos \frac { \pi t } { 2 } \right) ^ { 2 }$ on $[ 0,4 ]$ divided into 4 subintervals

Use a ﬁnite sum to estimate the average value of the function on the given interval by partitioning the interval and evaluating the function at the midpoints of the subintervals. - $f ( x ) = x ^ { 2 } - 2$ on $[ - 3,7 ]$ divided into 5 subintervals

Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed. - $f ( x ) = \frac { 1 } { x }$ between $x = 1$ and $x = 8$ using the "midpoint rule" with two rectangles of equal width.

Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed. - $f ( x ) = x ^ { 2 } \text { between } x = 4 \text { and } x = 8 \text { using an upper sum with four rectangles of equal width. }$

Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed. - $f ( x ) = x ^ { 2 } \text { between } x = 4 \text { and } x = 8 \text { using a lower sum with four rectangles of equal width. }$

Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed. - $f ( x ) = \frac { 1 } { x }$ between $x = 3$ and $x = 6$ using a upper sum with two rectangles of equal width.

Use a ﬁnite sum to estimate the average value of the function on the given interval by partitioning the interval and evaluating the function at the midpoints of the subintervals. - $f ( x ) = 3 x ^ { 5 }$ on $[ 1,3 ]$ divided into 4 subintervals

Use a ﬁnite sum to estimate the average value of the function on the given interval by partitioning the interval and evaluating the function at the midpoints of the subintervals. - $f ( x ) = \frac { \sqrt { x } } { 4 }$ on [3,7] divided into 4 subintervals (Round to the nearest hundredth.)

Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed. - $f ( x ) = \frac { 1 } { x }$ between $x = 4$ and $x = 5$ using a lower sum with two rectangles of equal width.

Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed. -f(x) = 1 - x² between x = -1 and x = 1 using the "midpoint rule" with two rectangles of equal width.

Use a ﬁnite sum to estimate the average value of the function on the given interval by partitioning the interval and evaluating the function at the midpoints of the subintervals. - $f ( x ) = 5 ( 2 ) ^ { x }$ on $[ 1,7 ]$ divided into 3 subintervals

Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed. - $f ( x ) = x ^ { 2 } \text { between } x = 2 \text { and } x = 6 \text { using the "midpoint rule" with four rectangles of equal width. }$

Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed. - $f ( x ) = x ^ { 2 } \text { between } x = 0 \text { and } x = 2 \text { using an upper sum with two rectangles of equal width. }$

Use a ﬁnite sum to estimate the average value of the function on the given interval by partitioning the interval and evaluating the function at the midpoints of the subintervals. - $\mathrm { f } ( \mathrm { x } ) = 5 \mathrm { x } \sin \pi \mathrm { x }$ on $[ 0,4 ]$ divided into 4 subintervals

Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed. - $f ( x ) = x ^ { 2 } \text { between } x = 0 \text { and } x = 1 \text { using a lower sum with two rectangles of equal width. }$

Use a ﬁnite sum to estimate the average value of the function on the given interval by partitioning the interval and evaluating the function at the midpoints of the subintervals. - $f ( x ) = 8 | x |$ on $[ - 3,3 ]$ divided into 4 subintervals

Quiz 5: Integration

Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed. -f(x) = x2 between x = 0 and x = 1 using the "midpoint rule" with two rectangles of equal width.

Use a ﬁnite sum to estimate the average value of the function on the given interval by partitioning the interval and evaluating the function at the midpoints of the subintervals. - f(x) =x2−2f ( x ) = x ^ { 2 } - 2f(x) =x2−2 on [−3,7][ - 3,7 ][−3,7] divided into 5 subintervals

Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed. - f(x) =1xf ( x ) = \frac { 1 } { x }f(x) =x1​ between x=1x = 1x=1 and x=8x = 8x=8 using the "midpoint rule" with two rectangles of equal width.

Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed. - f(x) =1xf ( x ) = \frac { 1 } { x }f(x) =x1​ between x=3x = 3x=3 and x=6x = 6x=6 using a upper sum with two rectangles of equal width.

Use a ﬁnite sum to estimate the average value of the function on the given interval by partitioning the interval and evaluating the function at the midpoints of the subintervals. - f(x) =3x5f ( x ) = 3 x ^ { 5 }f(x) =3x5 on [1,3][ 1,3 ][1,3] divided into 4 subintervals

Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed. - f(x) =1xf ( x ) = \frac { 1 } { x }f(x) =x1​ between x=4x = 4x=4 and x=5x = 5x=5 using a lower sum with two rectangles of equal width.

Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed. -f(x) = 1 - x2 between x = -1 and x = 1 using the "midpoint rule" with two rectangles of equal width.

Use a ﬁnite sum to estimate the average value of the function on the given interval by partitioning the interval and evaluating the function at the midpoints of the subintervals. - f(x) =5(2) xf ( x ) = 5 ( 2 ) ^ { x }f(x) =5(2) x on [1,7][ 1,7 ][1,7] divided into 3 subintervals

Use a ﬁnite sum to estimate the average value of the function on the given interval by partitioning the interval and evaluating the function at the midpoints of the subintervals. - f(x) =8∣x∣f ( x ) = 8 | x |f(x) =8∣x∣ on [−3,3][ - 3,3 ][−3,3] divided into 4 subintervals

Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed. -f(x) = x² between x = 0 and x = 1 using the "midpoint rule" with two rectangles of equal width.

Use a ﬁnite sum to estimate the average value of the function on the given interval by partitioning the interval and evaluating the function at the midpoints of the subintervals. - $f ( x ) = x ^ { 2 } - 2$ on $[ - 3,7 ]$ divided into 5 subintervals

Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed. - $f ( x ) = \frac { 1 } { x }$ between $x = 1$ and $x = 8$ using the "midpoint rule" with two rectangles of equal width.

Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed. - $f ( x ) = \frac { 1 } { x }$ between $x = 3$ and $x = 6$ using a upper sum with two rectangles of equal width.

Use a ﬁnite sum to estimate the average value of the function on the given interval by partitioning the interval and evaluating the function at the midpoints of the subintervals. - $f ( x ) = 3 x ^ { 5 }$ on $[ 1,3 ]$ divided into 4 subintervals

Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed. - $f ( x ) = \frac { 1 } { x }$ between $x = 4$ and $x = 5$ using a lower sum with two rectangles of equal width.

Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed. -f(x) = 1 - x² between x = -1 and x = 1 using the "midpoint rule" with two rectangles of equal width.

Use a ﬁnite sum to estimate the average value of the function on the given interval by partitioning the interval and evaluating the function at the midpoints of the subintervals. - $f ( x ) = 5 ( 2 ) ^ { x }$ on $[ 1,7 ]$ divided into 3 subintervals

Use a ﬁnite sum to estimate the average value of the function on the given interval by partitioning the interval and evaluating the function at the midpoints of the subintervals. - $f ( x ) = 8 | x |$ on $[ - 3,3 ]$ divided into 4 subintervals