$$\int_{0}^{\frac{\pi}{2}} \frac{1}{2} [3 \sin(2\theta)]^2 d\theta$$

Question 1

Evaluate the definite integral $$\int_{0}^{1} x e$$^{x^2}$$ dx.$$

Accepted Answer

$$\frac{1}{2}(e^1$ - 1)$$

Question 2

Which of the following is the correct indefinite integral of $$\int \frac{1}{x^2$ + 4} dx$$?

Accepted Answer

$$\frac{1}{2} \arctan\left(\frac{x}{2}\right) + C$$

Question 3

Evaluate the definite integral $$\int_{0}^{\pi/2} \sin^2$(x) dx.$$

Accepted Answer

$$\frac{\pi}{4}$$

Question 4

Determine whether the series $$\sum_{n=1}^{\infty} \frac{(-1)^n}{n} is $$absolutely convergent, conditionally convergent, or divergent.

Accepted Answer

Conditionally convergent

Question 5

Find the interval of convergence for the power series $$\sum_{n=1}^{\infty} \frac{x^n}{n^2$}.$$

Accepted Answer

$$|x| \leq 1$$

Question 6

Find the area inside one leaf of the polar rose $$r = 3 \sin(2	heta).$$

Accepted Answer

$$\frac{9\pi}{4}$$

Question 7

Solve the initial value problem $$\frac{dy}{dx} + 2y = 10$$ with $$y(0) = 1.$$

Accepted Answer

$$y(x) = 5 - 4e$$^{-2x}$$

Question 8

Find the Maclaurin series for $$e^{x^2} up to $$the $$x^4$$ term.

Accepted Answer

$$1 + x^2$ + \frac{x^4$}{2}$$

Question 9

Find the fourth degree Taylor polynomial for $$\sin(x)$$ about $$x=0.$$

Accepted Answer

$$x - \frac{x^3$}{6}$$

Question 10

Evaluate the integral $$\int_{0}^{1} \frac{1}{\sqrt{1 - x^2$}} dx.$$

Accepted Answer

$$\frac{\pi}{2}$$

Question 11

Which test is most appropriate to determine the convergence of the series $$\sum_{n=1}^{\infty} \frac{1}{n^2$}$$?

Accepted Answer

p-test

Question 12

Find the length of the portion of the helix $$\vec{r}(t) = (\cos t, \sin t, t)$$ over the interval $$[0,4\pi].$$

Accepted Answer

$$4\pi \sqrt{2}$$

Question 13

Which of the following series is absolutely convergent?

Accepted Answer

$$\sum_{n=1}^{\infty} \frac{1}{n^2$}$$

Question 14

Given $$x = 2 \sin(3t)$$, find $$\frac{dx}{dt}.$$

Accepted Answer

$$6 \cos(3t)$$

Question 15

Which of the following is the correct integral for the area inside one leaf of $$r = 3 \sin(2	heta)$$?

Accepted Answer

$$\int_{0}^{\frac{\pi}{2}} \frac{1}{2} [3 \sin(2	heta)]^2 d	heta$$

Question 16

If $$y = x^2$ e$$^{x}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$e^{x}(x^2$ + 2x)$

Question 17

Which of the following is the correct Maclaurin series for $$\sin(x) up to $$the $$x^3$$ term?

Accepted Answer

$$x - \frac{x^3$}{6}$$

Question 18

Evaluate $$\int_{0}^{1} \frac{1}{\sqrt{4 - x^2$}} dx.$$

Accepted Answer

$$\frac{\pi}{6}$$

Question 19

Which of the following is the solution to the differential equation $$\frac{dy}{dx} = 2y$$ with $$y(0) = 3$$?

Accepted Answer

$$y(x) = 3e$$^{2x}$$

Calculus Practice Final Exam Study Notes

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