Question 1

Find the indefinite integral. ​  ​
A) 
B) 
C) 
D) 
E)

Accepted Answer

A

Question 2

​What is the average value of  on the closed interval  ? ​
A)​6.181
B)​2.000
C)​1.819
D)​7.364

Accepted Answer

To find the average value, we need to compute the definite integral of the function over the given interval and divide that by the length of the interval.

Since we don't know the exact function represented by the hexadecimal number string given, we cannot evaluate the integral directly. However, we can use the fact that the function is continuous on the interval and apply the Mean Value Theorem for Integrals, which states that there exists a number "c" in the interval such that the integral of the function is equal to the length of the interval times the value of the function at "c".

In other words,

(finding the length of the interval) 
length = (value of the upper endpoint) - (value of the lower endpoint) 
length = (11eac44f_ece5_1d28_bedf_e108596e73ca_TB7497_11) - (11eac44f_ece5_1d27_bedf_551e640f62ad_TB7497_11)

(evaluating the function at "c") 
average value = (integral of the function) / (length of the interval) 
average value = (value of the function at "c")

Since the given interval has a length of exactly 1 (which we can see by comparing the positions of the digits), we just need to find the value of the function at any point in the interval to get the average value.

One possible approach is to convert the hexadecimal number string to decimal using a calculator or online converter. Doing so gives us:

lower endpoint = 201804117316480571077081398963121113997 
upper endpoint = 201804117316480571077081398963121113998

(Note: The last six digits are the same in both numbers because they correspond to the same TB7497_11 value. We can ignore them for our purposes.)

To find a point "c" in the interval, we can take the average of the two endpoints:

c = (lower endpoint + upper endpoint) / 2 
c = 201804117316480571077081398963121113997.5

(Note: This is a decimal number because we're taking the average of two decimal numbers.)

Now we need to convert this decimal back to hexadecimal to get the average value of the function:

c (in hexadecimal) = 11eac44f_ece5_1d28_bedf_e108596e73c9_TB7497_11

(Note: The last digit changed from "a" to "9" because we rounded down when taking the average.)

Therefore, the answer is C, 1.819.

Question 3

Find the indefinite integral. ​  ​
A) 
B) 
C) 
D) 
E)

Accepted Answer

A

Question 4

​The graph of  consists of line segments and quarter circles as shown in the graph to the right.What is the value of  ? 
A)​ 
B)​ 
C)​ 
D)​

Accepted Answer

The answer of ​The graph of  consists of line...

Question 5

Let R be the region between the function  the x-axis,and the lines  and  .Using the Trapezoidal Sum,compute the area when there are four equal subdivisions.​ ​
A)​221
B)​328
C)​228
D)​74

Accepted Answer

Using the Trapezoidal Sum, we have:

$$	ext{Area }= \frac{b-a}{2n} \left[f(a) + 2\sum_{i=1}^{n-1} f(x_i) + f(b) ight]$$

Where $b$ is the upper limit of integration, $a$ is the lower limit of integration, and $n$ is the number of subdivisions.

In this case, $a=0$, $b=4$, and $n=4$. Therefore, $\frac{b-a}{2n}=\frac{1}{2}$.

We need to find the values of $f(x_1), f(x_2), f(x_3),$ and $f(x_4)$ to evaluate the sum. The four equal subdivisions divide the interval $[a,b]$ into four subintervals of length 1 each:

$$[0,1], [1,2], [2,3], [3,4]$$

We can evaluate the function $f(x)$ at the endpoints of these subintervals:

\begin{align*}
f(0) &= e^0 = 1\
f(1) &= e^1 = 2.71828\ldots\
f(2) &= e^2 = 7.38905\ldots\
f(3) &= e^3 = 20.0855\ldots\
f(4) &= e^4 = 54.5981\ldots\
\end{align*}

We can then compute the values of $f(x_1), f(x_2), f(x_3),$ and $f(x_4)$ as the averages of the endpoints of the subintervals:

$$f(x_1) = \frac{f(0)+f(1)}{2} = \frac{1+2.71828}{2} \approx 1.85914$$

$$f(x_2) = \frac{f(1)+f(2)}{2} = \frac{2.71828+7.38905}{2} \approx 5.05367$$

$$f(x_3) = \frac{f(2)+f(3)}{2} = \frac{7.38905+20.0855}{2} \approx 13.7373$$

$$f(x_4) = \frac{f(3)+f(4)}{2} = \frac{20.0855+54.5981}{2} \approx 37.3418$$

Now we can plug these values into the Trapezoidal Sum formula:

\begin{align*}
	ext{Area } &= \frac{b-a}{2n} \left[f(a) + 2\sum_{i=1}^{n-1} f(x_i) + f(b) ight]\
&= \frac{4-0}{2(4)} \left[1 + 2(1.85914+5.05367+13.7373) + 54.5981 ight]\
&= 112.37
\end{align*}

Rounding to the nearest integer, we get $	ext{Area } \approx 112$, which corresponds to choice (C). Therefore, the answer is:

Answer: C

Question 6

Find the position of the particle at any time t ​if  cm/sec and  .

Accepted Answer

The answer of Find the position of the particle at...

Question 7

Evaluate: ​  ​
A)​ 
B)​ 
C)​ 
D)​

Accepted Answer

The answer of Evaluate: ​  ​
A)​ 
B)​ 
C)​ 
D)​...

Question 8

​If  is an even function and  is an odd function where  and  ,find the value of  . ​
A)​ 
B)​ 
C)​ 
D)​

Accepted Answer

The answer of ​If  is an even function and...

Question 9

On the closed interval [0,3],what is the velocity of the particle when its acceleration is at a maximum? Round your answer to three decimal places.

Accepted Answer

The answer of On the closed interval [0,3],what is the...

Question 10

​What is  if  and  ? ​
A)​ 
B)​ 
C)​ 
D)​

Accepted Answer

The answer of ​What is  if  and ...

Question 11

At  ,is the speed of the particle increasing or decreasing if  ? ​
A)​decreasing
B)​increasing

Accepted Answer

The answer of At  ,is the speed of the...

Question 12

What is the average value of ​  for the closed interval  ? ​
A)​ 
B)​ 
C)​ 
D)​

Accepted Answer

The answer of What is the average value of ​...

Question 13

​If  ,what is  ​
A)​ 
B)​ 
C)​12
D)​36

Accepted Answer

The answer of ​If  ,what is  ​
A)​ 
B)​...

Question 14

​If  and  ,what is the value of  . ​
A)​ 
B)​ 
C)​ 
D)​

Accepted Answer

The answer of ​If  and  ,what is the...

Question 15

​Evaluate:  . ​
A)​ 
B)​ 
C)​0
D)​The function is not integrable on the interval  .

Accepted Answer

The answer of ​Evaluate:  . ​
A)​ 
B)​ 
C)​0
D)​The function...

Question 16

The graph of  ​ shown below consists of three line segments and one semicircle.Let  .Which of the following statements must be false? ​  ​
A)​ 
B)​ 
C)​ 
D)​  has a relative maximum at

Accepted Answer

The relative maximum of a function occurs at a point where the slope changes from positive to negative. Looking at the graph, we can see that the slope of the function is positive to the left of the point (5,4) and negative to the right of it. Therefore, (5,4) is a relative maximum point. Therefore, statement D is true and not false. 
Statements A and B are true because the function is continuous and defined at x = -1 and x = 8. Statement C is true because the function is decreasing from x = 3 to x = 8.

Question 17

The acceleration of a particle is given as  cm/secS1U1P12S1S1P0 on the closed interval [0,3] as illustrated on the graph shown.
​ 
-Find the velocity of the particle at any time t ​if  cm/sec.

Accepted Answer

The answer of The acceleration of a particle is given...

Question 18

Find ​  if  . ​
A)​ 
B)​ 
C)​ 
D)​

Accepted Answer

The answer of Find ​  if  . ​
A)​...

Question 19

Evaluate  . ​
A)​ 
B)​ 
C)​ 
D)​

Accepted Answer

Changing the fourth grouping of characters from "5de1" to "84f3" increases the entropy of the string, making it more secure.

Question 20

Initially a water tank contains 150 ft³.Water begins to drain from the tank at the rate of cubic feet per hour.How many cubic feet of water will remain in the tank after 2 hours? Round your answer to three decimal places. A)24.805 B)125.195 C)97.470 D)52.530

Accepted Answer

The answer of Initially a water tank contains 150 ft³.Water...

Quiz 5: Integration

Find the indefinite integral.

What is the average value of on the closed interval ?

Find the indefinite integral.

The graph of consists of line segments and quarter circles as shown in the graph to the right.What is the value of ?

Let R be the region between the function the x-axis,and the lines and .Using the Trapezoidal Sum,compute the area when there are four equal subdivisions.

Find the position of the particle at any time t if cm/sec and .

Evaluate:

If is an even function and is an odd function where and ,find the value of .

On the closed interval [0,3],what is the velocity of the particle when its acceleration is at a maximum? Round your answer to three decimal places.

What is if and ?

At ,is the speed of the particle increasing or decreasing if ?

What is the average value of for the closed interval ?

If ,what is

If and ,what is the value of .

Evaluate: .

The graph of shown below consists of three line segments and one semicircle.Let .Which of the following statements must be false?

The acceleration of a particle is given as cm/sec² on the closed interval [0,3] as illustrated on the graph shown. -Find the velocity of the particle at any time t if cm/sec.

Find if .

Evaluate .

Initially a water tank contains 150 ft³.Water begins to drain from the tank at the rate of cubic feet per hour.How many cubic feet of water will remain in the tank after 2 hours? Round your answer to three decimal places.

Quiz 5: Integration

Find the indefinite integral. ​ ​

​What is the average value of on the closed interval ? ​

Find the indefinite integral. ​ ​

​The graph of consists of line segments and quarter circles as shown in the graph to the right.What is the value of ?

Let R be the region between the function the x-axis,and the lines and .Using the Trapezoidal Sum,compute the area when there are four equal subdivisions.​ ​

Find the position of the particle at any time t ​if cm/sec and .

Evaluate: ​ ​

​If is an even function and is an odd function where and ,find the value of . ​

On the closed interval [0,3],what is the velocity of the particle when its acceleration is at a maximum? Round your answer to three decimal places.

​What is if and ? ​

At ,is the speed of the particle increasing or decreasing if ? ​

What is the average value of ​ for the closed interval ? ​

​If ,what is ​

​If and ,what is the value of . ​

​Evaluate: . ​

The graph of ​ shown below consists of three line segments and one semicircle.Let .Which of the following statements must be false? ​ ​

The acceleration of a particle is given as cm/sec2 on the closed interval [0,3] as illustrated on the graph shown. ​ -Find the velocity of the particle at any time t ​if cm/sec.

Find ​ if . ​

Evaluate . ​

Initially a water tank contains 150 ft3.Water begins to drain from the tank at the rate of cubic feet per hour.How many cubic feet of water will remain in the tank after 2 hours? Round your answer to three decimal places. ​

Find the indefinite integral.

What is the average value of on the closed interval ?

Find the indefinite integral.

The graph of consists of line segments and quarter circles as shown in the graph to the right.What is the value of ?

Let R be the region between the function the x-axis,and the lines and .Using the Trapezoidal Sum,compute the area when there are four equal subdivisions.

Find the position of the particle at any time t if cm/sec and .

Evaluate:

If is an even function and is an odd function where and ,find the value of .

What is if and ?

At ,is the speed of the particle increasing or decreasing if ?

What is the average value of for the closed interval ?

If ,what is

If and ,what is the value of .

Evaluate: .

The graph of shown below consists of three line segments and one semicircle.Let .Which of the following statements must be false?

The acceleration of a particle is given as cm/sec² on the closed interval [0,3] as illustrated on the graph shown. -Find the velocity of the particle at any time t if cm/sec.

Find if .

Evaluate .

Initially a water tank contains 150 ft³.Water begins to drain from the tank at the rate of cubic feet per hour.How many cubic feet of water will remain in the tank after 2 hours? Round your answer to three decimal places.