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Pre-Class Assignments, Math 104 Calculus II, Spring 2021

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

All section numbers refer to APEX Calculus, Version 4.0.
Be sure to check the Errata for corrections to the text.

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

Week 1: February 3 - 5

The Fundamental Theorem of Calculus
u-substitution

We'll spend the first week reviewing this material from Calculus I
For Wednesday February 3
For Thursday February 4
For Friday February 5

Week 2: Due Sunday February 7 @ midnight

Inverse Trig Functions and Integration by Parts

To Read
To Watch

Pre-Class Questions

  1. Find an antiderivative of \( f(x) = \displaystyle \frac{3x^2}{ 1 + x^6}\)
  2. Integration by parts attempts to undo one of the techniques of differentiation. Which one is it?
  3. Use integration by parts to find an antiderivative of \(f(x) = 2x e^{x}\)
Submit answers through onCourse

Week 3: Due Sunday February 14 @ midnight

Numeric Integration and Volume by Revolution

To Read
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. Describe the shape of the solid formed when R is rotated about the x-axis.
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Week 4: Due Sunday February 21 @ midnight

Arc Length and Improper Integrals

To Read
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.
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Week 5: Due Sunday February 28 @ midnight

Sequences and Series

To Read
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. What assignments are due this week? When are they due?
Submit answers through onCourse

Week 6: Due Sunday March 7 @ midnight

Integral and Comparison Tests for Infinite Series

To Read
To Watch

Pre-Class Questions

  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 \( \displaystyle \sum \frac{1}{\sqrt{n}} \)?
Submit answers through onCourse

Week 7: Due Sunday March 14 @ midnight

Alternating and Power Series

To Read
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?
Submit answers through onCourse

Week 8: Due Sunday March 21 @ midnight

Taylor Series

To Read
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)=sin(x)?
Submit answers through onCourse

Week 9: Due Sunday March 28 @ midnight

Multivariable Functions

To Read
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?
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Week 10: Due Sunday April 4 @ midnight

The Dot Product

To Read
To Watch

Pre-Class Questions

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?
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Week 11: Due Sunday April 11 @ midnight

Directional Derivatives

To Read
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 \(\nabla f(x,y)\) ?
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Week 12: Due Sunday April 18 @ midnight

Multivariable Optimization

To Read
To Watch

Pre-Class Questions

  1. Where can the local extrema of a function f(x,y) occur?
  2. In Example 12.8.3, why does it make sense that the critical point (1,2) is called a "saddle point"?
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Week 13: Due Sunday April 25 @ midnight

Double Integrals

To Read
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.
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