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Calculus 2 - Exam B2

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17calculus > exam list > calc 2 exam B2

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

Exam Details

Tools

Time

2 hours

Calculators

no

Questions

15

Formula Sheet(s)

none

Total Points

100

Other Tools

none

 

Downloads

download one page list of these questions

download multiple page exam with space to work out the solutions

Instructions:
- This exam is in four main parts, labeled sections 1-4, with different instructions for each section.
- Show all your work.
- 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).
- Give exact answers.

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

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

answer

solution

Question 2

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

answer

solution

Question 3

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

answer

solution

Section 2

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

Question 4

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

answer

solution

Question 5

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

answer

solution

Question 6

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

answer

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

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

answer

solution

Question 8

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

answer

solution

Question 9

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

answer

solution

Question 10

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

answer

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

answer

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 \( r=3x \).

answer

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 \( R_2(x)\).

answer

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 \( (2,0) \).

answer

solution

Question 15

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

answer

solution

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