Q: The area of an ellipse defined by $$\frac{x^2$}{a^2$} + \frac{y^2$}{b^2$} = 1$$ can be found using which substitution?

Let $$x = a \cos \theta$$, $$y = b \sin \theta.$$

Question 1

Which of the following statements best describes the Fundamental Theorem of Calculus?

Accepted Answer

It connects differentiation and integration, stating that if $$F(x) is an $$antiderivative of $$f(x)$$, then $$\int_a^b f(x) \, dx = F(b) - F(a).$$

Question 2

A region bounded by $$y = x^2$ and y = 2$ is revolved around the y$-axis. Which integral represents the volume using the shell method?

Accepted Answer

$$\$$int_0$$^{\sqrt{2}} 2\pi x (2 - x^2$) \, dx$$

Question 3

Evaluate $$\int \frac{x^3$ + 2x^2$ + 1}{x^2$ + 1} \, dx$$ using long division and partial fractions. What is the correct first step?

Accepted Answer

Divide $$x^3$$ + $$2x^2 + 1$ by x^2$ + 1 to $$get $$x + 2$$ with remainder $$-x + 1.$$

Question 4

The area of an ellipse defined by $$\frac{x^2$}{a^2$} + \frac{y^2$}{b^2$} = 1$$ can be found using which substitution?

Accepted Answer

Let $$x = a \cos 	heta$$, $$y = b \sin 	heta.$$

Question 5

Find the third-order Taylor polynomial for $$f(x) = \sin x$$ centered at $$x = 0.$$

Accepted Answer

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

Question 6

Zeno's Paradox involves moving half the remaining distance to a target at each step. What is the sum of the infinite series representing the total distance traveled if the initial distance is 1?

Accepted Answer

$$1$$

Question 7

Which of the following series converges by the comparison test?

Accepted Answer

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

Question 8

Use the integral test to determine the convergence of $$\sum_{n=2}^{\infty} \frac{1}{n \ln n}. $$What is the result?

Accepted Answer

The series diverges because $$\$$int_2$$^{\infty} \frac{1}{x \ln x} \, dx$$ diverges.

Question 9

Which of the following is a necessary condition for the Alternating Series Test to guarantee convergence?

Accepted Answer

The terms must decrease in absolute value and approach zero.

Question 10

Does the series $$\sum_{n=1}^{\infty} \frac{(-1$$)^{n+1}$$}{n}$$ converge absolutely, conditionally, or diverge?

Accepted Answer

It converges conditionally.

Question 11

For the power series $$\sum_{n=1}^{\infty} \frac{x^n}{n^2$}$$, what is the radius of convergence?

Accepted Answer

$$1$$

Question 12

Which test is most appropriate to determine the convergence of $$\sum_{n=1}^{\infty} \frac{1}{n^p}$$ for $$p \u003e 0$$?

Accepted Answer

The $$p-$$test.

Question 13

Given $$f(x) = \sin x$$, which of the following is the correct Maclaurin series expansion up to the $$x^5$$ term?

Accepted Answer

$$x - \frac{x^3$}{6} + \frac{x^5$}{120}$$

Question 14

Which of the following series diverges by the limit comparison test?

Accepted Answer

$$\sum_{n=1}^{\infty} \frac{3n-2}{n}$$

Question 15

For the series $$\sum_{n=1}^{\infty} \frac{(-1$$)^{n+1}$$}{n^2$}$$, does it converge absolutely or conditionally?

Accepted Answer

Absolutely.

Question 16

Which of the following is the correct interval of convergence for the power series $$\sum_{n=1}^{\infty} \frac{x^n}{n^2$}$$?

Accepted Answer

$$[-1, 1]$$

Question 17

Which of the following is true about the convergence of $$\sum_{n=1}^{\infty} \frac{1}{n}$$?

Accepted Answer

It diverges because the terms do not approach zero fast enough.

Question 18

Which of the following is the correct Taylor polynomial of order 2 for $$f(x) = e^x$$ centered at $$x = 0$$?

Accepted Answer

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

Question 19

Which of the following integrals can be used to test the convergence of $$\sum_{n=2}^{\infty} \frac{1}{n (\ln n)^2$}$$?

Accepted Answer

$$\$$int_2$$^{\infty} \frac{1}{x (\ln x)^2$} \, dx$$

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