Preparation Problems#

Complete all preparation problems before class time on the scheduled date.

Week 15#

Day 15A: Monday, April 29th

No prep problem assigned for today.

Day 15B: Wednesday, May 1st

No prep problem assigned for today.

Day 15C: Friday, May 3rd

No prep problem assigned for today.

Week 14#

Day 14A: Monday, April 22nd

No prep problem for today.

Day 14B: Wednesday, April 24th

Find a power series representation for the following function.

\[ \dfrac{1}{1+x^2} \]
Day 14C: Friday, April 26th

Find a power series representation for the following function.

\[ \dfrac{1}{2+x} \]

Week 13#

Day 13A: Monday, April 15th

Evaluate the following power series at \(x=\dfrac{1}{4}\) and \(x=\dfrac{1}{3}\).

\[ f(x)=\sum_{n=0}^{\infty} 5 x^n \]
Day 13B: Wednesday, April 17th

Complete the following table for the various tests on series \(\sum a_n\).

Test

Necessary Conditions

What we check

Conclusions

Test for Divergence

none

\(\displaystyle \lim_{n\to \infty} a_n\)

  • Series is divergent if limit \(\neq0\) or DNE
  • Inconclusive if limit \(=0\)

Integral Test

  • \(a_n=f(n)\)
  • \(f(x)\) is positive
  • \(f(x)\) is continuous
  • \(f(x)\) is decreasing

\(\displaystyle \int_{1}^{\infty} f(x) \; dx\)

  • Series is convergent if improper integral is convergent
  • Series is divergent if improper integral is divergent

Direct Comparison Test

Limit Comparison Test

Alternating Series Test

Ratio Test

Root Test

Absolute Convergence Test

Day 13C: Friday, April 19th

  1. Use the Integral Test to determine if the following series is convergent or divergent. \(\displaystyle \sum_{n=1}^{\infty} \dfrac{1}{(5n+7)^{3/2}}\)

  2. Use the Liimit Comparison Test to determine if the follwoing series is convergent or divergent. \(\displaystyle \sum_{n=1}^{\infty} \dfrac{6n+7}{2n^5+4n}\)

Week 12#

Day 12A: Monday, April 8th

No preparation problem for today.

Day 12B: Wednesday, April 10th

ST5 Use the alternating series test to show the following series is convergent.

\[ \sum_{n=1}^{\infty} \dfrac{(-1)^n 3n}{n^2+1} \]
Day 11C: Friday, April 5th

ST8 Use the absolute convergence test to determine if the following series is convergent or divergent.

\[ \sum_{n=1}^{\infty} \dfrac{(-1)^{n-1}}{5n^3} \]

Week 11#

Day 11A: Monday, April 1st

No prep problem for today.

Day 11B: Wednesday, April 3rd

ST1 What conclusion can you make about each of the following series using only the Test for Divergence?

  1. \(\displaystyle \sum_{n=1}^{\infty} \dfrac{5n^3}{2n^3-7}\)

  2. \(\displaystyle \sum_{n=1}^{\infty} \dfrac{5n^2}{2n^3-7}\)

Day 11C: Friday, April 5th

ST4 Use the limit comparions test to determine if the following series is convergent or divergent.

\[ \sum_{n=1}^{\infty} \dfrac{5n^2}{2n^3+7} \]

Week 10#

Day 10A: Monday, March 25th

S1 Write out the first four terms for each of the following sequences:

  1. \(\left\{ \dfrac{4n}{n^2+1}\right\}_{n=1}^{\infty}\)

  2. \(\left\{ \dfrac{(-1)^{n+1}4n}{n^2+1}\right\}_{n=0}^{\infty}\)

Day 10B: Wednesday, March 27th

S3 Write out the first four terms of the sequence of partial sums for:

\[ \sum_{n=1}^{\infty} \dfrac{5}{n^2} \]
Day 10C: Friday, March 29th

S4 Determine if the following geometric series are convergent or divergent. If convergent, find the sum.

  1. \(\sum_{n=0} 6 \left( \dfrac{1}{3} \right)^n\)

  2. \(\sum_{n=0} 6 \left( \dfrac{1}{3} \right)^{n+1}\)

Week 9#

Day 9A: Monday, March 18th

Spring Break - No prep problem for today.

Day 9B: Wednesday, March 20th

Spring Break - No prep problem for today.

Day 9C: Friday, March 22nd

Spring Break - No prep problem for today.

Week 8#

Day 8A: Monday, March 11th

No preparation problem assigned for today.

Day 8B: Wednesday, March 13th

Sketch each polar region that satisfies both inequalities:

  1. \(0\leq r \leq 2 \quad \text{and} \quad 0 \leq \theta \leq \pi/2\)

  2. \(0\leq r \leq 3 \quad \text{and} \quad \pi/4 \leq \theta \leq 3\pi/4\)

Day 8C: Friday, March 15th

Graph each equation on the \(r\theta\) rectangular plane. Do not graph the polar curve.

  1. \(r=1+\cos \thetat\)

  2. \(r=\sin 2\theta\)

Week 7#

Day 7A: Monday, March 4th

No preparation problem assigned for today.

Day 7B: Wednesday, March 6th

No preparation problem assigned for today.

Day 7C: Friday, March 8th

No preparation problem assigned for today.

Week 6#

Day 6A: Monday, February 26th

Evaluate the following improper integral:

\[ \int_1^{\infty} \dfrac{1}{e^{3x}} \; dx \]
Day 6B: Wednesday, February 28th

Determine if the following improper integral is convergent or divergent:

\[ \int_1^{\infty} \dfrac{1}{x} \; dx \]
Day 6C: Friday, March 1st

No preparation problem for today.

Week 5#

Day 5A: Monday, February 19th

No preparation problem for today.

Day 5B: Wednesday, February 21st

AI6: Find the partial fraction decomposition of the function: \(f(x)=\dfrac{1}{(2x-4)(x-3)}\)

Day 5C: Friday, February 23rd

AI7: Calculate the following integrals:

  1. \( \displaystyle \int \dfrac{6}{7x+1} \; dx\)

  2. \( \displaystyle \int \dfrac{5}{(7x+1)^3} \; dx\)

  3. \( \displaystyle \int \dfrac{3x+4}{x^2+6} \; dx\)

Week 4#

Day 4A: Monday, February 12th

AI1: Calculate the following trigonometric integral:

\[ \displaystyle \int \sin^3 x\; dx \]
Day 4B: Wednesday, February 14th

AI2: Calculate the following trigonometric integral:

\[ \displaystyle \int \big( 1- sin^2 4x \big) \; dx \]
Day 4C: Friday, February 16th

AI4: Use a trigonometric substitution to convert the following integral into a trigonometric integral. Simplify the trig functions as much as possible. You do not need to evaluate the integral.

\[ \displaystyle \int \dfrac{\sqrt{x^2+4}}{x} \; dx \]

Week 3#

Day 3A: Monday, February 5th

AD1: Find the area bounded between the following two curves.

\[ y=x^2+2x+8 \qquad \qquad y=-x^2+10x+2 \]
Day 3B: Wednesday, February 7th

AD2: Find the volume of the solid obtained by revolving the given region about the \(x\)-axis.

\[ \text{Region below} \quad y=\sin(\pi x) \qquad \text{from } x=0 \text{ to } x=1 \]

Hint: See the homework page for a hint about the integration (this hint is also helpful for the I6 homework).

Day 3C: Friday, February 9th

AD3: Set up the integral that would give the length of the following curve:

\[ \text{Curve} \quad y=\sin(\pi x) \qquad \text{from } x=0 \text{ to } x=1 \]

Week 2#

Day 2A: Monday, January 29th

Use the substitution rule to calculate the following integral:

\[ \int_0^1 \left( 2x^4+3 \right)^6 \cdot 4x^3 \; dx \]
Day 2B: Wednesday, January 31st

Use the integration by parts to calculate the following integral:

\[ \int 2x\cdot e^{3x} \; dx \]
Day 2C: Friday, February 2nd

Use the integration by parts to calculate the following integral:

\[ \int_0^{\pi/2} 2x\cdot \sin(4x) \; dx \]

Week 1#

Day 1A: Monday, January 22nd

No preparation problems assigned for today.

Day 1B: Wednesday, January 24th

No preparation problems assigned for today.

Day 1C: Friday, January 26th

Use the substitution rule to calculate the following integral:

\[ \int \left( 4x^2+3 \right)^9 \cdot 8x \; dx \]