Q: Which of the following is the correct formula for the arc length of a curve given by $$ r = f(\theta) $$ from $$ \theta = a to$$ $$ \theta = b in $$polar coordinates?

$$ \int_a^b \sqrt{[f(\theta)]^2 + [f'(\theta)]^2} \, d\theta $$

Question 1

Which of the following improper integrals converges?

Accepted Answer

$$ \$$int_0$$^1 \frac{1}{\sqrt{x}} \, dx $$

Question 2

Suppose $$ \$$int_1$$^\infty f(x) \, dx $$ diverges and $$ 0 \leq g(x) \leq f(x) $$ for all $$ x \geq 1 . $$What can you conclude about $$ \$$int_1$$^\infty g(x) \, dx $$?

Accepted Answer

It diverges by the Comparison Test.

Question 3

Solve the separable differential equation $$ \frac{dy}{dx} = 3y $$ with initial condition $$ y(0) = 2 . $$What is $$ y(x) $$?

Accepted Answer

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

Question 4

Which of the following sequences converges?

Accepted Answer

$$ a_n = \frac{1}{n} $$

Question 5

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

Accepted Answer

Integral Test

Question 6

Consider the series $$ \sum_{n=1}^\infty \frac{(-1$$)^{n+1}$$}{n} . $$Which of the following is true?

Accepted Answer

It converges conditionally.

Question 7

Which of the following series can be tested for convergence using the Ratio Test?

Accepted Answer

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

Question 8

Find the radius of convergence for the power series $$ \sum_{n=0}^\infty x^n .$$

Accepted Answer

$$ 1 $$

Question 9

What is the Maclaurin series for $$ \sin(x) $$?

Accepted Answer

$$ \sum_{n=0}^\infty \frac{(-1)^n x$$^{2n+1}$$}{(2n+1)!} $$

Question 10

Which of the following is the Taylor series for $$ f(x) = e^x $$ centered at $$ x = 0 $$?

Accepted Answer

$$ \sum_{n=0}^\infty \frac{x^n}{n!} $$

Question 11

A particle moves along a path defined by the parametric equations $$ x = t^2$ $$, $$ y = t^3$ . $$What is $$ \frac{dy}{dx}  in $$terms of $$ t $$?

Accepted Answer

$$ \frac{3t^2$}{2t} = \frac{3t}{2} $$

Question 12

Which of the following is the polar equation of a circle centered at the origin with radius $$ a $$?

Accepted Answer

$$ r = a $$

Question 13

Find the area enclosed by one loop of the polar curve $$ r = \sin(2	heta) .$$

Accepted Answer

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

Question 14

Which conic section is represented by the polar equation $$ r = \frac{e}{1 - e \cos 	heta} $$ for $$ 0 \u003c e \u003c 1 $$?

Accepted Answer

Ellipse

Question 15

Suppose you want to approximate $$ \ln(1.1) $$ using the first three nonzero terms of its Maclaurin series. Which of the following is the correct approximation?

Accepted Answer

$$ 0.1 - \frac{0.1^2$}{2} + \frac{0.1^3$}{3} $$

Question 16

Evaluate $$ \$$int_0$$^\infty e$$^{-x}$$ \, dx .$$

Accepted Answer

$$ 1 $$

Question 17

Which of the following statements about the sequence $$ a_n = \frac{(-1)^n}{n}  is $$true?

Accepted Answer

It converges to $$ 0 .$$

Question 18

Which of the following is the correct formula for the arc length of a curve given by $$ r = f(	heta) $$ from $$ 	heta = a  to$$ $$ 	heta = b  in $$polar coordinates?

Accepted Answer

$$ \int_a^b \sqrt{[f(	heta)]^2 + [f'(	heta)]^2} \, d	heta $$

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