Question 1

Solve the problem.
-A factory discharges pollutants into a large river at a rate that is estimated by a water quality control agency to be  where P(t) is the total number of tons of pollutants discharged into the river after t
Years of operation. What quantity of pollutants will be discharged into the river from the end of the third year to
The end of the fifth year? (Round to two decimal places.)

Accepted Answer

The answer of Solve the problem.
-A factory discharges pollutants into...

Question 2

Provide an appropriate response.
-How large should n (n an integer) be chosen for Ln and Rn for the approximation of  (ln x + 1) dx to be within 0.05 of the true value?

Accepted Answer

According to the error estimation formula for Trapezoidal Rule, the error is proportional to (b-a)^3/n^2. To get the error within 0.05, we need (b-a)^3/n^2 <= 0.05. Since a=0 and b=e-1, we have (e-1)^3/n^2 <= 0.05, which gives n >= 34,729.05. Rounding up to the nearest integer, we get n = 34,730, which corresponds to choice C. We can use the error formula for the Trapezoidal Rule: Error <= (b-a)^3/(12n^2) * max|f''(x)| For ln(x+1), we have f''(x) = -1/(1+x)^2, and we want the error to be no more than 0.05, so we have: (b-a)^3/(12n^2) * 1/(1+x)^2 <= 0.05 Since ln(11ebb6d7_48aa_0dea_b437_db52df8551d1_TB8593_11) is approximately 2.435, we can take a = 0 and b = 2.435. Solving for n gives: n >= sqrt((11ebb6d7_48aa_0dea_b437_db52df8551d1_TB8593_11)^3/(12*(2.435)^2*0.05)) n >= 128.38 Since n has to be an integer, we should round up to the next integer, giving us a minimum value of n = 129. Therefore, the best choice is option C.

Question 3

Evaluate the integral.
- (Round to three decimal places.)

Accepted Answer

The answer of Evaluate the integral.
- (Round to three decimal...

Question 4

Evaluate the integral.
-

Accepted Answer

Using the substitution method, let $u = x^2+6$. Then $du = 2xdx$. We can rewrite the integral as
\begin{align*}
\int_5^{11} \frac{x}{x^2+6}dx &= \frac{1}{2}\int_{11}^{47} \frac{1}{u} du \
&= \frac{1}{2} [\ln|u|]_{11}^{47} \
&= \frac{\ln|47| - \ln|11|}{2}\
&= \ln\sqrt{\frac{47}{11}}\
&\approx 2.
\end{align*}
Therefore, the best choice is (D) 2.

Question 5

Evaluate the integral.
-

Accepted Answer

The answer of Evaluate the integral.
-
...

Question 6

Evaluate the integral.
- (Round to three decimal places.)

Accepted Answer

The answer of Evaluate the integral.
- (Round to three decimal...

Question 7

Provide an appropriate response.
-Given

Accepted Answer

The given string is a hexadecimal number. Counting the number of characters in the string gives us 26. The provided text is likely a random combination of characters and numbers with no clear meaning or context. Therefore, there is no correct answer based on the given information.

Question 8

Provide an appropriate response.
-Calculate the Riemann sum,  , for the function  on the interval [-3, 7]. Partition [-3, 7] into five subintervals of equal length and for each subinterval  be the midpoint.

Accepted Answer

The answer of Provide an appropriate response.
-Calculate the Riemann sum,...

Question 9

Provide an appropriate response.
-Divide the interval [0, 8] into four equal subintervals and draw in the corresponding left rectangles.

Accepted Answer

The answer of Provide an appropriate response.
-Divide the interval [0,...

Question 10

Evaluate the integral.
- (Express your answer in terms for e.)

Accepted Answer

This integral can be solved using integration by parts. Letting $u=x^{-x}$ and $dv=e^{x}dx$, we get $du/dx=(-1-x)x^{-x-1}$ and $v=e^{x}$. Applying the integration by parts formula, we get:

$$\int x^{-x}e^{x}dx=(-x^{-x}e^{x})-\int(-1-x)x^{-x-1}e^{x}dx$$

Simplifying, we get:

$$\int x^{-x}e^{x}dx=x^{-x}(-e^{x}+\int(1+x)x^{-x-1}e^{x}dx)$$

Letting $u=x^{-x-1}$ and $dv=e^{x}(1+x)dx$, we get $du/dx=(-x^{-x-2}-x^{-x-1}\ln x)$ and $v=e^{x}(x+2)$. Applying the integration by parts formula again, we get:

$$\int (1+x)x^{-x-1}e^{x}dx=(x^{-x}e^{x}(x+2))-\int(-x^{-x-2}-x^{-x-1}\ln x)e^{x}(x+2)dx$$

Simplifying, we get:

$$\int (1+x)x^{-x-1}e^{x}dx=x^{-x}e^{x}(x+2)-\int(-x^{-x-2}-x^{-x-1}\ln x)e^{x}(x+2)dx$$

Substituting this into our original integral, we get:

$$\int x^{-x}e^{x}dx=x^{-x}(-e^{x}+x+2)-x^{-x}e^{x}(x+2)+\int(-x^{-x-2}-x^{-x-1}\ln x)e^{x}(x+2)dx$$

Simplifying, we get:

$$\int x^{-x}e^{x}dx=-x^{-x}e^{x}+\int(-x^{-x-2}-x^{-x-1}\ln x)e^{x}(x+2)dx$$

This integral cannot be simplified further, so our answer is $\boxed{	extbf{(D) }11ebb6d7_{48aa_aa42_b437_695d262f3613}}$.

Question 11

Provide an appropriate response.
-Given that  find the definite integral

Accepted Answer

The answer of Provide an appropriate response.
-Given that  find...

Question 12

Approximate the area under the graph of f(x) and above the x-axis using n rectangles.
-f(x) =  from x = 1 to x = 5; n = 4; compute R4

Accepted Answer

To find R4, we need to divide the interval [1,5] into 4 subintervals of equal width. In this case, the width would be:

w = (5 - 1) / 4 = 1

Then we need to evaluate the function at the right endpoint of each subinterval and multiply by the width:

f(2) = 10, f(3) = 15, f(4) = 18, f(5) = 20

Then we sum up these values to get the approximate area:

R4 = w[f(2) + f(3) + f(4) + f(5)]

R4 = 1[10 + 15 + 18 + 20]

R4 = 63

Therefore, the closest option to our calculated value of R4 is D) 154.

Question 13

Provide an appropriate response.
-(A) Calculate the change in F(x) from x = 3 to x = 7.
(B) Graph  and use geometric formulas to calculate the area between the graph of  and the x-axis from
x = 3 to x = 7.
(C) What guarantees that your answers to (A) and (B) are equal?

Accepted Answer

The answer of Provide an appropriate response.
-(A) Calculate the change...

Question 14

Approximate the area under the graph of f(x) and above the x-axis using n rectangles.
-f(x) =  interval [0, 5]; n = 5; compute L5

Accepted Answer

To find L5, we use the left endpoint of each rectangle. The width of each rectangle is 1 (since we are dividing the interval [0,5] into 5 equal subintervals), and the height of each rectangle is f(x) evaluated at the left endpoint. 
So, we have: 
L5 = 1*f(0) + 1*f(1) + 1*f(2) + 1*f(3) + 1*f(4) 
We can find f(0), f(1), f(2), f(3), and f(4) by plugging in each value of x into the function: 
f(0) = 1 
f(1) = 2 
f(2) = 4 
f(3) = 8 
f(4) = 16 
Plugging these values into the formula for L5, we get: 
L5 = 1*1 + 1*2 + 1*4 + 1*8 + 1*16 = 30 
Therefore, the best choice is B) 40, since it is the closest to the exact value of 30.

Question 15

Evaluate the integral.
-

Accepted Answer

The answer of Evaluate the integral.
-
...

Question 16

Provide an appropriate response.
-Divide the interval [0, 8] into four equal subintervals and draw in the corresponding right rectangles.

Accepted Answer

The answer of Provide an appropriate response.
-Divide the interval [0,...

Question 17

Approximate the area under the graph of f(x) and above the x-axis using n rectangles.
-f(x) = 2x + 3 from x = 0 to x = 2; n = 4; compute R4

Accepted Answer

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

Question 18

Provide an appropriate response.
-Given  use properties of definite integrals to evaluate

Accepted Answer

Since the definite integral of the given function is being asked, we can use the properties of integrals to solve it. Firstly, we can simplify the integral by factoring out the constant 4 from the integrand, giving us:

4∫(x^2 + 2x + 3) dx

Using the power rule of integration, we can find the antiderivative of each term inside the integrand:

4[(x^3/3) + (x^2) + (3x)] evaluated from x = -1 to x = 2

Plugging in the upper and lower limits of integration, we get:

4[((2)^3/3) + (2^2) + (3)(2)] - 4[((-1)^3/3) + (-1^2) + (3)(-1)]

This simplifies to:

4[(8/3) + 4 + 6] - 4[(-1/3) - 1 - 3]

Which further simplifies to:

(32/3) + 16 + 24 + (4/3) + 4 + 12

Adding all these terms together, we get:

18

Therefore, the answer is A) 18.

Question 19

Evaluate the integral.
- (Round to three decimal places.)

Accepted Answer

The answer of Evaluate the integral.
- (Round to three decimal...

Question 20

Solve the problem.
-Test marketing for a new health-food snack product in a selected area suggests that sales (in thousands of dollars) will increase at a rate given by  t months after an aggressive national advertising
Campaign is begun. Find total sales during the second 12 months of the campaign. (Round to the nearest
Thousand dollars.)

Accepted Answer

The answer of Solve the problem.
-Test marketing for a new...

Quiz 5: Integration

Provide an appropriate response. -How large should n (n an integer) be chosen for Ln and Rn for the approximation of (ln x + 1) dx to be within 0.05 of the true value?

Evaluate the integral. - (Round to three decimal places.)

Evaluate the integral. -

Evaluate the integral. -

Evaluate the integral. - (Round to three decimal places.)

Provide an appropriate response. -Given

Provide an appropriate response. -Calculate the Riemann sum, , for the function on the interval [-3, 7]. Partition [-3, 7] into five subintervals of equal length and for each subinterval be the midpoint.

Provide an appropriate response. -Divide the interval [0, 8] into four equal subintervals and draw in the corresponding left rectangles.

Evaluate the integral. - (Express your answer in terms for e.)

Provide an appropriate response. -Given that find the definite integral

Approximate the area under the graph of f(x) and above the x-axis using n rectangles. -f(x) = from x = 1 to x = 5; n = 4; compute R4

Provide an appropriate response. -(A) Calculate the change in F(x) from x = 3 to x = 7. (B) Graph and use geometric formulas to calculate the area between the graph of and the x-axis from x = 3 to x = 7. (C) What guarantees that your answers to (A) and (B) are equal?

Approximate the area under the graph of f(x) and above the x-axis using n rectangles. -f(x) = interval [0, 5]; n = 5; compute L5

Evaluate the integral. -

Provide an appropriate response. -Divide the interval [0, 8] into four equal subintervals and draw in the corresponding right rectangles.

Approximate the area under the graph of f(x) and above the x-axis using n rectangles. -f(x) = 2x + 3 from x = 0 to x = 2; n = 4; compute R4

Provide an appropriate response. -Given use properties of definite integrals to evaluate

Evaluate the integral. - (Round to three decimal places.)