$$\def\R{\mathbb{R}} \def\N{\mathbb{N}} \def\e{\epsilon} \def\dst{\displaystyle}$$

### Reading Assignments - Math 104 Calculus II - Fall 2017

Be sure to check back, because this may change during the semester.

All numbers indicate sections from APEX Calculus, Version 3.0, and check the Errata for corrections to the text.

#### For Friday September 1 (Due 8/31 @ 8:00 pm)

Section 6.1 Substitution

1. Substitution attempts to undo one of the techniques of differentiation. Which one is it?
2. Use u-substitution to find an antiderivative of $$f(x) = 3x^2 \cos(x^3)$$
3. Explain why $$\dst \int \cos(x) \sin(x)^2 dx$$ and $$\dst\int \frac{\ln(x)^2}{x} dx$$ are essentially the same integral after performing a substitution.

#### For Monday September 4

Labor Day. No class meeting or reading assignment due.

#### For Wednesday September 6 (Due 9/5 @ 8:00 pm)

Section 2.7 Derivatives of Inverse Functions

1. Why do you think we are studying the inverse trig functions now?
2. Find an antiderivative of $$f(x) = \dst \frac{x^2}{ 1 + x^6}$$

#### For Friday September 8 (Due 9/7 @ 8:00 pm)

Section 6.2 Integration by Parts

1. Integration by parts attempts to undo one of the techniques of differentiation. Which one is it?
2. Use integration by parts to find an antiderivative of $$f(x) = 2x e^{4x}$$

#### For Monday September 11 (Due 9/10 @ 8:00 pm)

Section 6.2 Integration by Parts

Reading Questions Would you use u-substitution or integration by parts to find each anti-derivative? Find the antiderivative and explain why you chose the method you did.

1. $$\int \cos(x) \sin(x) dx$$
2. $$\int e^x x^2 dx$$

#### For Wednesday September 13 (Due 9/12 @ 8:00 pm)

Section 5.5 Numerical Integration

1. Why would you ever want to numerically approximate an integral?
2. Which would you expect to be MOST accurate: a Right Hand approximation, a Trapezoidal approximation, or a Simpson's approximation? Why?
3. Which would you expect to be LEAST accurate: a Right Hand approximation, a Trapezoidal approximation, or a Simpson's approximation? Why?

#### For Friday September 15 (Due 9/14 @ 8:00 pm)

Section 7.2 Volume by Cross-Sectional Area; Disk and Washer

1. 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.
2. Let T be the triangle formed by the lines y=2x, x=2 and the x-axis. Describe the shape of the solid formed when T is rotated about the line y = -1.

#### For Monday September 18

Section 7.2 Volume by Cross-Sectional Area; Disk and Washer

#### For Wednesday September 20 (Due 9/19 @ 8:00 pm)

Section 7.4 Arc Length and Surface Area

1. Set up the integral that gives the length of the curve $$y=\sin(2x)$$ from $$x=0$$ to $$x=2\pi$$.
2. Set up the integral that gives the surface area of the surface formed when the curve $$y=x^2 + 2$$ from $$x=0$$ to $$x=3$$ is rotated about the x-axis.

#### For Friday September 22 (Due 9/21 @ 8:00 pm)

Section 6.7 L'Hopital's Rule

1. Does l'Hopital's Rule apply to $$\dst \lim_{x \to \infty} \frac{x^2}{e^x}$$ ? Why or why not?
2. Does l'Hopital's Rule apply to $$\dst \lim_{x\to\infty} \frac{x^2}{\sin(x)}$$? Why or why not?
3. For each limit in #1 and #2 where l'Hopital's applies, use it to find the limit.

#### For Monday September 25 (Due 9/24 @ 8:00 pm)

Section 6.8 Improper Integration

1. Explain why $$\dst\int_1^{\infty} \frac{1}{x^2} dx$$ is improper.
2. Explain why $$\dst\int_0^1 \frac{1}{x^2} dx$$ is improper.
3. Explain why $$\dst\int_{-1}^1 \frac{1}{x^2} dx$$ is improper.

#### For Wednesday September 27 (Due 9/26 @ 8:00 pm)

Section 6.8 Improper Integration

Suppose f and g are continuous and $$0 < f(x) < g(x)$$ for $$x > 0$$.

1. If the improper integral $$\int_1^{\infty} g(x) dx$$ converges, what can you conclude about the improper integral $$\int_1^{\infty} f(x) dx$$?
2. If the improper integral $$\int_1^{\infty} f(x) dx$$ diverges, what can you conclude about the improper integral $$\int_1^{\infty} g(x) dx$$ ?
3. If the improper integral $$\int_1^{\infty} f(x) dx$$ converges, what can you conclude about the improper integral $$\int_1^{\infty} g(x) dx$$ ?

#### For Friday September 29 (Due 9/28 @ 8:00 pm)

Section 8.1 Sequences

1. Does the following sequence converge or diverge? Be sure to explain your answer.
1, 3, 5, 7, 9, 11, 13, . . .
2. Find a symbolic expression for the general term ak of the sequence
1, 2, 4, 8, 16, 32, . . .
3. Is the following sequence bounded? Is it monotone? Explain. $1, -\frac{1}{2}, \frac{1}{4}, -\frac{1}{8}, \frac{1}{16}, -\frac{1}{32}, \ldots$

#### For Monday October 2 (Due 10/1 @ 8:00 pm)

Section 8.2 Infinite Series

1. There are two sequences associated with every series. What are they?
2. Does the geometric series $$\dst \sum \left( \frac{1}{4}\right)^k$$ converge or diverge? Why?
3. Does the geometric series $$\dst \sum \left( \frac{\pi}{e}\right)^k$$ converge or diverge? Why?

#### For Wednesday October 4 (Due 10/3 @ 8:00 pm)

Q & A for Exam 1. No Reading Assignment for today.

#### For Friday October 6 (Due 10/5 @ 8:00 pm)

Section 8.2 Infinite Series

1. What does the nth-Term Theorem tell you about the series $$\dst \sum 2^k$$?
2. What does the nth-Term Theorem tell you about the series $$\dst \sum \frac{1}{k}$$?

#### For Monday October 9

Fall Break. No class meeting or reading assignment due.

#### For Wednesday October 11 (Due 10/10 @ 8:00 pm)

Section 8.3 Integral and Comparison Tests

1. What does the Integral Test tell you about the series $$\dst \sum \frac{1}{k^3}$$?

2. What does the Integral Test tell you about the series $$\dst \sum \frac{1}{\sqrt{k}}$$?
3. What does the Direct Comparison Test tell you about the series $$\dst \sum \frac{1}{k^3 + \sqrt{k}}$$?

#### For Friday October 13 (Due 10/12 @ 8:00 pm)

Section 8.3 Integral and Comparison Tests

1. Use the Limit Comparison Test to show that $$\dst\sum \frac{1}{n^2 - n}$$ converges.
2. Explain why it would have been more difficult to apply the Direct Comparison Test to this series.

#### For Monday October 16 (Due 10/15 @ 8:00 pm)

Section 8.5 Alternating Series and Absolute Convergence

Consider the series $$\sum_{k=1}^\infty (-1)^k \frac{1}{k^2}$$

1. Why does this series converge?
2. How closely does $$S_{50}$$, the 50th partial sum, approximate the value of the series? Why?

#### For Wednesday October 18

Section 8.5 Alternating Series and Absolute Convergence

#### For Friday October 20 (Due 10/19 @ 8:00 pm)

Section 8.6 Power Series

1. How do power series differ from the series we have looked at up to this point?
2. What is the interval of convergence of a power series? Explain in your own words.

#### For Monday October 23 (Due 10/22 @ 8:00 pm)

Section 8.7 Taylor Polynomials

Explain the basic idea of constructing the n-th degree Taylor polynomial for a function f(x). Do not give the formula, but describe the idea in your own words in a few sentences.

#### For Wednesday October 25 (Due 10/24 @ 8:00 pm)

Section 8.8 Taylor Series

1. What is the difference between a Taylor series and a Maclaurin series?
2. Why would you ever want to compute a Taylor series for a function like sin(x)?

#### For Friday October 27 (Due 10/26 @ 8:00 pm)

Section 12.1 Introduction to Multivariable Functions

1. Describe the level curves of the function $$f(x,y)= x^2 - y$$
2. Describe the level surfaces of the function $$g(x,y,z)=x^2+y^2+z^2$$

#### For Monday October 30 (Due 10/29 @ 8:00 pm)

Section 12.3 Partial Derivatives

1. For a function $$f(x,y)$$, what information does $$f_x(2,3)$$ give?
2. How many second-order partial derivatives does a function $$g(x,y)$$ have? Why?

#### For Wednesday November 1 (Due 10/31 @ 8:00 pm)

Q & A for Exam 2. No Reading Assignment for today.

#### For Friday November 3

Moving Monday's class (10/30) here due to power outage on campus.

#### For Monday November 6 (Due 11/5 @ 8:00 pm)

Section 10.2 An Introduction to Vectors
Section 10.3 The Dot Product

1. Give the unit vector in the same direction as $$\vec{\,v}=\langle 2,3 \rangle$$
2. If the dot product $$\vec{\,u}\cdot \vec{\,v}=0$$, what does this tell you about the vectors?

#### For Wednesday November 8 (Due 11/7 @ 8:00 pm)

Section 12.6 Directional Derivatives

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

#### For Thursday November 9

Section 12.8 Extreme Values

1. Where can the local extrema of a function f(x,y) occur?
2. Why does the term "saddle point" make sense?

#### For Friday November 10

Section 12.8 Extreme Values

#### For Monday November 13 (Due 11/12 @ 8:00 pm)

Section 13.1 Iterated Integrals and Area

1. Why would you want to calculate an iterated integral?
2. Why would you want to switch the order of integration in an iterated integral?

#### For Wednesday November 15 (Due 11/14 @ 8:00 pm)

Section 13.2 Double Integration and Volume

1. If $$f(x,y)$$ is a function of two variables and $$R$$ is a rectangle in the xy-plane, what does $$\int\int_R f(x,y)\, dA$$ measure?
2. Explain the idea of Fubini's Theorem in a couple of sentences in your own words.

#### For Friday November 17

No class. I will be at a conference.

#### For Monday November 20

Section 13.2 Double Integration and Volume

#### For Wednesday November 22

Thanksgiving Break. No class meetings or reading assignments due.

#### For Friday November 24

Thanksgiving Break. No class meetings or reading assignments due.

#### For Monday November 27 (Due 11/26 @ 8:00 pm)

Section 9.4 Introduction to Polar Coordinates

1. What do the coordinates $$(r,\theta)$$ in polar coordinates measure?
2. Why do you think we are studying polar coordinates now?
3. Is the graph of the polar function $$r = 4\cos(\theta)$$ is the graph of a function y=f(x)? Explain.

#### For Wednesday November 29 (Due 11/28 @ 8:00 pm)

Q & A for Exam 3. No Reading Assignment for today.

#### For Friday December 1 (Due 11/30 @ 8:00 pm)

Section 13.3 Double Integration with Polar Coordinates

1. Describe the shape of a polar "rectangle."
2. Why would you ever want to use polar coordinates to evaluate a double integral?

#### For Monday December 4 (Due 12/3 @ 8:00 pm)

Section 13.3 Double Integration with Polar Coordinates

#### For Wednesday December 6 (Due 12/5 @ 8:00 pm)

Section 13.4 Center of Mass

2. When using a double integral to calculate $$M_x$$, the moment about the x-axis, what is the reason for multiplying the density function $$\delta(x,y)$$ by $$y$$ rather than by $$x$$?