Question 1

Decide whether the integral. If the integral converges, compute its value. 
A)converges to 
B)converges to converges to 
C)converges to 
D)converges to 
E)diverges

Accepted Answer

The answer of Decide whether the integral. If the integral...

Question 2

The Laplace transform  of a function  is given by the formula.  Find  if  ( m constant).
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of The Laplace transform  of a function...

Question 3

Decide whether the integral converges. If the integral converges, compute its value. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Decide whether the integral converges. If the...

Question 4

Decide whether the integral converges. If the integral converges, compute its value. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

We can use the limit comparison test with $f(x) = \frac{1}{x^2}$ since it also satisfies the conditions of the test: $f(x)$ is positive, continuous, and decreasing for $x \geq 1$.

$$\lim_{x 	o \infty} \frac{\frac{\cos x}{x^2}}{\frac{1}{x^2}} = \lim_{x 	o \infty} \cos x = 	ext{DNE}$$

Since the limit is greater than zero and finite, the integral $\int_1^\infty \frac{\cos x}{x^2} \, dx$ converges if and only if $\int_1^\infty \frac{1}{x^2} \, dx$ converges. But we know that $\int_1^\infty \frac{1}{x^2} \, dx$ does converge, so the given integral also converges. To compute its value, we can use integration by parts with $u = \cos x$ and $dv = \frac{1}{x^2} \, dx$:

$$\int_1^{\infty} \frac{\cos x}{x^2} \, dx = \left[ -\frac{\cos x}{x} ight]_1^\infty + \int_1^{\infty} \frac{\sin x}{x} \, dx$$

The limit as $x 	o \infty$ of $-\frac{\cos x}{x}$ is $0$ since $-1 \leq \cos x \leq 1$ and $\int_1^{\infty} \frac{\sin x}{x} \, dx$ is the well-known Dirichlet integral, which converges to $\frac{\pi}{2}$. Therefore:

$$\int_1^{\infty} \frac{\cos x}{x^2} \, dx = 0 + \frac{\pi}{2} = \frac{\pi}{2}$$

Thus, our answer is D.

Question 5

Decide whether the integral converges. If the integral converges, compute its value. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Decide whether the integral converges. If the...

Question 6

Decide whether the integral converges. If the integral converges, compute its value. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Decide whether the integral converges. If the...

Question 7

Find the total value of the given income stream and also find its present value (at the beginning of the given interval)using the given interest rate.  Please enter your answer in the form TV = , PV = . Give the answer to the nearest cent if necessary. TV = $__________ PV = $__________

Accepted Answer

The answer of Find the total value of the given...

Question 8

Decide whether the integral converges. If the integral converges, compute its value. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Decide whether the integral converges. If the...

Question 9

The Laplace transform  of a function  is given by the formula.  Find  if 
A) 
B) 
C) 
D) 
E)

Accepted Answer

We can rearrange the given formula to get:

$f(t) = \frac{1}{2\pi i} \int_{\sigma-i\infty}^{\sigma+i\infty} F(s) e^{st} ds$

where $F(s)$ is the Laplace transform.

Substituting $s=i\omega$ and $\omega=\frac{\pi}{2t}$, we get:

$f(t) = \frac{1}{\pi} \int_{-\infty}^{\infty} F(i\omega) e^{i\pi \frac{\omega}{2t}} d\omega$

Comparing with the given formula, we see that $F(i\omega) = \frac{1}{1+\omega^2}$.

Substituting $\omega=1$, we get:

$F(i) = \frac{1}{1+1^2} = \frac{1}{2}$

Therefore, using the formula for $f(t)$ and $F(s)$, we get:

$f(1) = \frac{1}{2\pi i} \int_{\sigma-i\infty}^{\sigma+i\infty} \frac{e^s}{1+s^2} ds = \frac{1}{2\pi i} \oint_C \frac{e^z}{1+z^2} dz$

where $C$ is a contour in the complex plane including the imaginary axis and enclosing the pole at $z=i$.

By the residue theorem, this integral is equal to $2\pi i$ times the residue of the integrand at $z=i$, which is:

$\lim_{z	o i} (z-i) \frac{e^z}{1+z^2} = \frac{e^i}{2i}$

Therefore:

$f(1) = \frac{1}{2} e^i$

Taking the real part, we get:

$f(1) = \frac{1}{2} \cos 1 + \frac{i}{2} \sin 1$

The best choice is A since it is the closest to the value of $\frac{1}{2} \cos 1$.

Question 10

The weekly demand for your company s Lo-Cal Mousse is modeled by the equation  where t is time from now in weeks and  is the number of gallons sold per week. Evaluate the integral  .
A)3.427 gallons
B)1.312 gallons
C)2.624 gallons
D)0.097 gallons
E)6.854 gallons

Accepted Answer

The answer of The weekly demand for your company s...

Question 11

Decide whether the integral converges. If the integral converges, compute its value. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Decide whether the integral converges. If the...

Question 12

The normal distribution curve, which models the distributions of data in a wide range of applications, is given by the function   and  and  are constants called the standard deviation and the mean, respectively. With  and  , approximate  .
A)1
B)4
C)-1
D)5
E)-5

Accepted Answer

The answer of The normal distribution curve, which models the...

Question 13

Decide whether the integral converges. If the integral converges, compute its value. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Decide whether the integral converges. If the...

Question 14

If the integral converges, compute its value. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of If the integral converges, compute its value....

Question 15

Decide whether the integral converges. If the integral converges, compute its value. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Decide whether the integral converges. If the...

Question 16

Sales of the text "Calculus and You" have been declining continuously at an annual rate of 6%. Assuming that "Calculus and You" currently sell 6,000 copies per year and that sales will continue this pattern of decline, calculate total future sales of the text. Use the formula for continuously compounded interest with a negative rate.
A)2,500,000 copies
B)600,000 copies
C)100,000 copies
D)1,000,000 copies
E)5,000,000 copies

Accepted Answer

The formula for continuously compounded interest is A = P * e^(rt). Here, P = 6000, r = -0.06, and t approaches infinity. The integral of sales over time gives a total of 100,000 copies.

Question 17

Decide whether the integral. If the integral converges, compute its value. If the integral diverges, enter your answer as diverges .

Accepted Answer

The answer of Decide whether the integral. If the integral...

Question 18

Decide whether the integral converges. If the integral converges, compute its value. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

This integral can be evaluated using integration by parts or the substitution $u=x^2+1$, with limits from $0$ to $1$. Both methods give the result $\frac{\pi}{4e}+\frac{1}{2}\int_0^1 \frac{x^2}{x^2+1}\,dx=\frac{\pi}{4e}+\frac{1}{2}\int_0^1 \left(1-\frac{1}{x^2+1}\right)\,dx=\frac{\pi}{4e}+\frac{1}{2}\left[x-\arctan(x)\right]_0^1=\frac{\pi}{4e}+\frac{1}{2}-\frac{\pi}{4}=\boxed{\frac{1}{2}+\frac{\pi}{4(e-1)}}$.

Question 19

Find the general solution of the differential equation.  Solve for  as a function of  . NOTE: Use the symbol C to enter the constant.
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find the general solution of the differential...

Question 20

Decide whether the integral converges. If the integral converges, compute its value. 
A)converges to 4
B)converges to 0
C)converges to 5
D)converges to 1
E)diverges

Accepted Answer

The answer of Decide whether the integral converges. If the...

Quiz 7: Further Integration Techniques and Applications of the Integral

Decide whether the integral. If the integral converges, compute its value.

The Laplace transform of a function is given by the formula. Find if ( m constant) .

Decide whether the integral converges. If the integral converges, compute its value.

Decide whether the integral converges. If the integral converges, compute its value.

Decide whether the integral converges. If the integral converges, compute its value.

Decide whether the integral converges. If the integral converges, compute its value.

Find the total value of the given income stream and also find its present value (at the beginning of the given interval)using the given interest rate. Please enter your answer in the form TV = , PV = . Give the answer to the nearest cent if necessary. TV = $ PV = $

Decide whether the integral converges. If the integral converges, compute its value.

The Laplace transform of a function is given by the formula. Find if

The weekly demand for your company s Lo-Cal Mousse is modeled by the equation where t is time from now in weeks and is the number of gallons sold per week. Evaluate the integral .

Decide whether the integral converges. If the integral converges, compute its value.

The normal distribution curve, which models the distributions of data in a wide range of applications, is given by the function and and are constants called the standard deviation and the mean, respectively. With and , approximate .

Decide whether the integral converges. If the integral converges, compute its value.

If the integral converges, compute its value.

Decide whether the integral converges. If the integral converges, compute its value.

Decide whether the integral. If the integral converges, compute its value. If the integral diverges, enter your answer as diverges .

Decide whether the integral converges. If the integral converges, compute its value.

Find the general solution of the differential equation. Solve for as a function of . NOTE: Use the symbol C to enter the constant.

Decide whether the integral converges. If the integral converges, compute its value.