# 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 Calculusu-substitution

We'll spend the first week reviewing this material from Calculus I
##### For Wednesday February 3
• Watch:
• Welcome to Calc II! (Echo360)
• Overview of Calc II Content (Echo360)
• Submit:
##### For Thursday February 4
• Skim: Section 5.4 The Fundamental Theorem of Calculus, for review
• Watch: The Fundamental Theorem of Calculus (Echo360), for review

### Week 2: Due Sunday February 7 @ midnight

#### Inverse Trig Functions and Integration by Parts

• Section 2.7 Derivatives of Inverse Functions
• Section 6.2 Integration by Parts

#### 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 $$f(x) = 2x e^{x}$$

### Week 3: Due Sunday February 14 @ midnight

#### Numeric Integration and Volume by Revolution

• Section 5.5 Numerical Integration
Focus on the intuitive ideas behind Ln, Rn, Tn, and Sn and the statement of Theorem 5.5.1 that gives error bounds for Tn, and Sn. We'll use technology to calculate the approximations.
• Section 7.2 Volume by Cross-Sectional Area; Disk and Washer

#### 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.

### Week 4: Due Sunday February 21 @ midnight

#### Arc Length and Improper Integrals

• Section 7.4 Arc Length and Surface Area
Focus on pp 378-381
• Section 6.8 Improper Integration

#### 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.

### Week 5: Due Sunday February 28 @ midnight

#### Sequences and Series

• Section 8.1 Sequences
• Section 8.2 Infinite Series

#### 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?

### Week 6: Due Sunday March 7 @ midnight

#### Integral and Comparison Tests for Infinite Series

• Section 8.3 Integral and Comparison Tests
##### To Watch
• The Integral Test for Series (Echo360)
• The Direct Comparison Test for Series (Echo360)

#### 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 $$\displaystyle \sum \frac{1}{\sqrt{n}}$$?

### Week 7: Due Sunday March 14 @ midnight

#### Alternating and Power Series

• Section 8.5 Alternating Series
• Section 8.6 Power Series

#### 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?

### Week 8: Due Sunday March 21 @ midnight

#### Taylor Series

• Section 8.7 Taylor Polynomials
• Section 8.8 Taylor Series
##### To Watch
• Taylor Polynomials (Echo360)
• A catalog of Taylor Series (Echo360)

#### 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)?

### Week 9: Due Sunday March 28 @ midnight

#### Multivariable Functions

• Section 12.1 Introduction to Multivariable Functions
• Section 12.3 Partial Derivatives
##### 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?

### Week 10: Due Sunday April 4 @ midnight

#### The Dot Product

• Section 10.2 An Introdution to Vectors
• Sectxion 10.3 The Dot Product

#### 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?

### Week 11: Due Sunday April 11 @ midnight

#### Directional Derivatives

• Section 12.6 Directional Derivatives
##### To Watch
• Directional Derivatives (Echo360)

#### 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 $$\nabla f(x,y)$$ ?

### Week 12: Due Sunday April 18 @ midnight

#### Multivariable Optimization

• Section 12.8 Extreme Values
##### To Watch
• Multivariable Optimization Overview (Echo360)
• Multivariable Optimization Example (Echo360)

#### 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"?

### Week 13: Due Sunday April 25 @ midnight

#### Double Integrals

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?