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## Calculus 2 - Exam B2

You CAN Ace Calculus

17calculus > exam list > calc 2 exam B2

 l'hôpital's rule improper integrals infinite series parametrics (including calculus) polar coordinates (including calculus)

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exam list

This is the second exam for second semester single variable calculus.

 Downloads Exam Details Tools Time 2 hours Calculators no Questions 15 Formula Sheet(s) none Total Points 100 Other Tools none download one page list of these questions

Instructions:
- This exam is in four main parts, labeled sections 1-4, with different instructions for each section.
- For each problem, correct answers are worth 1 point. The remaining points are earned by showing calculations and giving reasoning that justify your conclusions.
- Correct notation counts (i.e. points will be taken off for incorrect notation).

Section 1

Evaluate each of the limits or explain why it doesn't exist. Each question in this section is worth 5 points.

Question 1

$$\displaystyle{ \lim_{x \to \infty}{\frac{\ln(x)}{x}} }$$

solution

Question 2

$$\displaystyle{ \lim_{x \to 0}{\frac{\sec(x)-1}{x^2}} }$$

solution

Question 3

$$\displaystyle{ \lim_{x \to 0^+}{\left[ \frac{1}{x} - \frac{1}{\sin(x)} \right]} }$$

solution

Section 2

Evaluate each improper integral or show that it diverges. Each question in this section is worth 5 points.

Question 4

$$\displaystyle{ \int_{0}^{\infty}{\frac{3x}{(x^2+9)^2}dx} }$$

solution

Question 5

$$\displaystyle{ \int_0^2{\frac{x}{\sqrt{4-x^2}}dx} }$$

solution

Question 6

$$\displaystyle{ \int_0^{\infty}{xe^{-5x}dx} }$$

solution

Section 3

Determine whether these series converge or diverge. At the end of each question, state your result and which test you used to determine your answer. Each question in this section is worth 5 points.

Question 7

$$\displaystyle{ \sum_{n=1}^{\infty}{\frac{n}{n+100}} }$$

solution

Question 8

$$\displaystyle{ \sum_{n=1}^{\infty}{\frac{1}{n\ln(n)}} }$$

solution

Question 9

$$\displaystyle{ \sum_{n=1}^{\infty}{\frac{5n^2}{11n^3+4n^2+6n+3}} }$$

solution

Question 10

$$\displaystyle{ \sum_{n=1}^{\infty}{\frac{(-1)^{n+1}}{n+1}} }$$

solution

Section 4

Solve the following problems. Each problem in this section is worth 10 points.

Question 11

Determine the convergence set of the power series $$\displaystyle{ \sum_{n=1}^{\infty}{\frac{x^n}{n}} }$$

solution

Question 12

Find the power series for the function $$\displaystyle{ f(x) = \frac{1}{(1-3x)^2} }$$ Hint: $$\displaystyle{ \sum_{n=0}^{\infty}{3^n x^n} }$$ is the geometric series with ratio $$r=3x$$.

solution

Question 13

Taylors Formula with remainder applied to $$\displaystyle{ f(x) = \ln(2+3x) }$$ centered about $$a=0$$ shows that $$\displaystyle{ \ln(2+3x) = P_2(x) + R_2(x) }$$ where $$P_2(x)$$ is the Maclaurin polynomial of degree 2 and $$R_2(x)$$ is the remainder term. Determine $$P_2(x)$$ and $$R_2(x)$$.

solution

Question 14

Find the tangent line to the parametric curve $$x = t^3 + t, ~~~ y = t^2 -1$$; $$0 < t < 2$$ at the point $$(2,0)$$.

Calculate the area of one leaf of the 3-leaved rose $$r=5\sin(3\theta)$$.