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Pre-Class Assignments, Math 104 Calculus II, Spring 2024

This page uses MathJax to display mathematical notation, so please let me know if any part isn't clear.

Be sure to check back, because this page will be updated often during the semester.

Week 1: January 24 - 26

The Fundamental Theorem of Calculus
u-substitution

We'll spend the first week reviewing this material from Calculus I. No Pre-Class Questions to submit this week.
To Skim for Review
Optional Videos to Watch

Week 2: Due Sunday January 28 @ 11:59 pm

Inverse Trig Functions and Integration by Parts

To Read
Optional Videos to Watch

Pre-Class Questions

  1. Why do you think we are studying the inverse trig functions now?
  2. Find an antiderivative of \( f(x) = \displaystyle \frac{3x^2}{ 1 + x^6}\)
  3. Integration by parts attempts to undo one of the techniques of differentiation. Which one is it?
  4. Use integration by parts to find an antiderivative of \(f(x) = 2x e^{x}\)
  5. Is there anything else you'd like me to know?
Submit answers through Canvas

Week 3: Due Sunday February 4 @ 11:59 pm

Numeric Integration and Volume by Revolution

To Read
Optional Videos to Watch

Pre-Class Questions

  1. Why would you ever want to numerically approximate an integral?
  2. Let \( \mathcal{I} = \displaystyle\int_0^{\pi} \sin(x^2) dx\).
    1. Which would you expect to be MOST accurate in approximating \( \mathcal{I} \) : a Right Hand approximation \( R_n\), a Trapezoidal approximation \(T_n\), or a Simpson's approximation \(S_n\)? Why?
    2. Which would you expect to be LEAST accurate in approximating \( \mathcal{I} \) : a Right Hand approximation \( R_n\), a Trapezoidal approximation \(T_n\), or a Simpson's approximation \(S_n\)? Why?
  3. Let R be the rectangle formed by the x-axis, the y-axis, and the lines y=1 and x=4. Sketch R and describe the shape of the solid formed when R is rotated about the x-axis.
  4. Is there anything else you'd like me to know?
Submit answers through Canvas

Week 4: Due Sunday February 11 @ 11:59 pm

Arc Length and Improper Integrals

To Read
Optional Videos to Watch

Pre-Class Questions

  1. Set up the integral that gives the length of the curve \( y=\sin(2x)\) from \(x=0\) to \( x=2\pi\).
  2. Explain why \( \displaystyle\int_1^{\infty} \frac{1}{x^2} dx \) is improper.
  3. Explain why \( \displaystyle\int_0^1 \frac{1}{x^2} dx \) is improper.
  4. Is there anything else you'd like me to know?
Submit answers through Canvas

Week 5: Due Sunday February 18 @ 11:59 pm

Sequences and Series

To Read
Optional Videos to Watch

Pre-Class Questions

  1. Does the following sequence converge or diverge? Explain.
    \[ 1, 3, 5, 7, 9, 11, 13, \ldots \]
  2. There are two sequences associated with every series. What are they?
  3. Does the geometric series \( \displaystyle \sum_{n=0}^{\infty} \left( \frac{1}{4}\right)^n\) converge or diverge? Why?
  4. Is there anything else you'd like me to know?
Submit answers through Canvas

Week 6: February 26 - March 1

Integral and Comparison Tests for Infinite Series

To Read
Optional Videos to Watch

Pre-Class Questions

Think about these, but no need to submit with the Takehome part of Exam 1 due Monday.
  1. What does the nth-Term Theorem tell you about the series \( \displaystyle \sum 2^n \)?
  2. What does the nth-Term Theorem tell you about the series \( \displaystyle \sum \frac{1}{n} \)?
  3. What does the Integral Test tell you about the series \( \displaystyle \sum \frac{1}{n^3} \)?

  4. What does the Integral Test tell you about the series \( \displaystyle \sum \frac{1}{\sqrt{n}} \)?
  5. Is there anything else you'd like me to know?

Week 7: Due Sunday March 3 @ 11:59 pm

Alternating and Power Series

To Read
Optional Videos to Watch

Pre-Class Questions

  1. Explain why the alternating series \( \displaystyle\sum_{n=1}^\infty (-1)^{n+1} \frac{1}{n}\) converges.
  2. How closely does \(\displaystyle S_{50}\), the 50th partial sum, approximate the value of the series \(\displaystyle \sum_{n=1}^\infty (-1)^{n+1} \frac{1}{n}\)? Why?
  3. How do power series differ from other series we have looked at up to this point?
  4. Is there anything else you'd like me to know?
Submit answers through Canvas

Week 8: Due Sunday March 17 @ 11:59 pm

Taylor Series

To Read
Optional Videos to Watch

Pre-Class Questions

  1. What is the difference between a Taylor polynomial and a Taylor series?
  2. What is the difference between a Taylor series and a Maclaurin series?
  3. Why would you ever want to compute a Taylor series for a function like f(x)=cos(x)?
  4. Is there anything else you'd like me to know?
Submit answers through Canvas

Week 9: Due Sunday March 24 @ 11:59 pm

Multivariable Functions

To Read
Optional Videos to Watch
I know this looks like alot, but these are fairly short, so it's under 35 minutes total

Pre-Class Questions

  1. Describe the level curves of the function \(f(x,y)= x^2 + y^2\) for c= 4, 0, and -1.
  2. If \( g(x,y)= x^2-y^2\), what is \( g_x(x,y) \), the partial derivative of \( g \) with respect to \( x \)?
  3. If \( g(x,y)= x^2-y^2\), what is \( g_x(2,1)? \)? What geometric information does this give you?
  4. Is there anything else you'd like me to know?
Submit answers through Canvas

Week 10: April 1 - 5

The Dot Product

To Read
Optional Videos to Watch

Pre-Class Questions

Think about these, but no need to submit with the Takehome part of Exam 2 due Monday.

Let \( \vec{\,v_1}=\langle 2,3 \rangle\) and \( \vec{\,v_2}=\langle -6,4 \rangle\)

  1. Give the unit vector in the same direction as \( \vec{\,v_1} \)
  2. What is \( \vec{\,v_1} \cdot \vec{\,v_2}\ \)? What geometric information does this give you about the vectors?

Week 11: Due Sunday April 7 @ 11:59 pm

Directional Derivatives
Multivariable Optimization

To Read
Optional Videos to Watch

Pre-Class Questions

  1. What does the directional derivative \( D_{\vec{\,u}} f(a,b)\) measure?
  2. If \(f(x,y) = 3xy^2 + 2x-4y^2\), what is \(\nabla f(x,y)\) ?
  3. Where can the local extrema of a function f(x,y) occur?
  4. Is there anything else you'd like me to know?
Submit answers through Canvas

Week 12: Due Sunday April 14 @ 11:59 pm

Multivariable Optimization

Double Integrals

To Read
Optional Videos to Watch

Pre-Class Questions

  1. Why would you want to switch the order of integration in an iterated integral?
  2. If \(f(x,y)\) is a function of two variables and \(R\) is a rectangle in the xy-plane, what does \( \iint_R f(x,y)\, dA\) measure?
  3. Explain the idea of Fubini's Theorem in a couple of sentences in your own words.
  4. Is there anything else you'd like me to know?
Submit answers through Canvas

Week 13: April 22 - 26

Focus on reviewing and prepping for Exam 3 this week.

Week 14: April 29 - May 3

Polar Coordinates and Double Integrals

To Read

Pre-Class Questions

Think about these, but no need to submit with the Takehome part of Exam 3 due Monday.
  1. Is the graph of the polar function \( r = \cos(2\theta) \) the graph of a function y=f(x)? Explain.
  2. Describe the shape of a polar "rectangle."
  3. Why would you ever want to use polar coordinates to evaluate a double integral?

Maintained by: ratliff_thomas@wheatoncollege.edu