# Reading AssignmentsMath 104 Calculus II, Spring 2019

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 there will certainly be some changes during the semester.

All numbers indicate sections from APEX Calculus, Version 4.0, and check the Errata for corrections to the text.

### For Friday January 25 (Due 1/24 @ 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 $$f(x) = 5x^4 \sin(x^5)$$
3. Explain why $$\dst \int 2x\cos(x^2) \sin(x^2)^2 dx$$ and $$\dst\int \frac{(\ln(x)+1)^2}{x} dx$$ are essentially the same integral after performing a substitution.

### For Monday January 28 (Due 1/27 @ 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 Wednesday January 30 (Due 1/29 @ 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 Friday February 1 (Due 1/31 @ 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 Monday February 4 (Due 2/3 @ 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 Wednesday February 6 (Due 2/5 @ 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 Friday February 8

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

### For Monday February 11 (Due 2/10 @ 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 Wednesday February 13 (Due 2/12 @ 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 Friday Feburary 15 (Due 2/14 @ 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 Monday February 18

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

### For Wednesday February 20 (Due 2/19 @ 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 Friday February 22 (Due 2/21 @ 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 Monday February 25 (Due 2/24 @ 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 Wednesday February 27 (Due 2/26 @ 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 March 1 (Due 2/28 @ 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 March 4 (Due 3/3 @ 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 March 6

Section 8.5 Alternating Series and Absolute Convergence

Re-read the section, but no Reading Questions for today. Monday's snow day messed up the schedule a bit.

### For Friday March 8

Section 8.5 Alternating Series and Absolute Convergence

### March 11 - 15

Spring Break. Surprisingly, no Reading Assignments.

### For Monday March 18 (Due 3/17 @ 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 Wednesday March 20 (Due 3/19 @ 8:00 pm)

Section 8.7 Taylor Polynomials
Section 8.8 Taylor Series

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 sin(x)?

### For Friday March 22

Section 8.8 Taylor Series

### For Monday March 25

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

### For Wednesday March 27 (Due 3/26 @ 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 Friday March 29 (Due 3/28 @ 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 Monday April 1

Section 12.3 Partial Derivatives

### For Wednesday April 3 (Due 4/2 @ 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 Friday April 5 (Due 4/4 @ 8:00 pm)

Section 11.2 Calculus and Vector-Valued Functions
Section 11.3 The Calculus of Motion

Let $$\vec{\;r}(t) = \langle \sin(2t),t^2 \rangle$$

1. Find $$\vec{\;r}\ ' (1)$$
2. What is the velocity of $$\vec{\;r}(t)$$ at time $$t=1$$?
3. What is the speed of $$\vec{\;r}(t)$$ at time $$t=1$$?

### For Monday April 8 (Due 4/7 @ 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 Wednesday April 10

Flex Day. No Reading Assignment due.

### For Friday April 12

No Class today, so no Reading Assignment.

### For Monday April 15 (Due 4/14 @ 8:00 pm)

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 Wednesday April 17

Section 12.8 Extreme Values

### For Friday April 19 (Due 4/18 @ 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 Monday April 22 (Due 4/21 @ 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 Wednesday April 24

Section 13.2 Double Integration and Volume

### For Friday April 26 (Due 4/25 @ 8:00 pm)

Section 13.2 Double Integration and Volume

### For Monday April 29

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

### For Wednesday May 1

Section 13.4 Center of Mass