Math 104 - Calculus II - Fall 2006

Reading Assignments

For Friday September 1

Section 5.1 Areas and Integrals
Section 5.2 The Area Function
Section 5.3 The Fundamental Theorem of Calculus
Section 3.4 Inverse Functions and Their Derivatives (at the back of the book)

To read : Sections 5.1, 5.2, and 5.3 should be review, so you can skim these to remind yourself of the Fundamental Theorem of Calculus. You can skim the beginning of Section 3.4, but read the section Working with inverse trigonometric functions beginning on page S-8 carefully.

Be sure to understand : The statements of both forms of the Fundamental Theorem of Calculus. The derivatives of the inverse trig functions.

Reading Questions :

1. What is the domain of the function arccos(x)? Why?
2. Why do you think we studying the inverse trig functions now?
3. Find one antiderivative of 1 / (1+x²).

For Wednesday September 6

Section 5.4 Finding Antiderivatives; The Method of Substitution

To read : All

Be sure to understand : Examples 6, 7, 10, and 13

Reading Questions :

1. Explain the difference between a definite integral and an indefinite integral.
2. What are the three steps in the process of substitution?
3. Substitution attempts to undo one of the techniques of differentiation. Which one is it?

For Friday September 8

Section 5.6 Approximating Sums; The Integral as a Limit

To read : All

Be sure to understand : Examples 2 and 3

Reading Questions :

1. When approximating an integral, which would you expect to be more accurate, L₁₀ or L₁₀₀? Why?
2. Give an example of a partition of the interval [0,3].
3. Explain the idea of a Riemann sum in your own words.

For Monday September 11

Section 6.1 Approximating Integrals Numerically

To read : All

Be sure to understand : The statements of Theorem 1 and Theorem 2

Reading Questions :

1. Why would we want to approximate an integral?
2. Let f(x)=x² and I=int( f(x), x= -1. . 2). Does Theorem 1 apply to I? Explain.
3. Let f(x)=x² and I=int( f(x), x= -1. . 2). Does Theorem 2 apply to I? Explain.

For Wednesday September 13

Section 6.2 Error Bounds for Approximating Sums

To read : All

Be sure to understand : The statement of Theorem 3 and Example 6.

Reading Questions :

1. Explain in words what K₁ is in Theorem 3.
2. Explain in words what K₂ is in Theorem 3.
3. Let I=int( x³, x= -1. . 3). Is 2 a valid value for K₁ in Theorem 3? Explain.

For Friday September 15

Section 6.2 Error Bounds for Approximating Sums

To read : Reread the section, but no Reading Questions for today.

For Monday September 18

Section 7.1 Measurement and the Definite Integral; Arc Length

To read : All

Be sure to understand : The Fact on page 416, Example 5, the Fact on page 419, and Example 8.

Reading Questions :
Let f(x)=sin(x)+10 and g(x)=2x-5.

1. Set up the integral that determines the area of the region bounded by y=f(x) and y=g(x) between x=-1 and x=3.
2. Set up the integral that gives the length of the curve y=g(x) from x=-1 to x=3.

For Wednesday September 20

For Friday September 22

Section 7.2 Finding Volumes by Integration

To read : All

Be sure to understand : The section Solids of revolution

Reading Questions :

1. Let R be the rectangle formed by the x-axis, the y-axis, and the lines y=1 and x=3. What is 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=x, x=1 and the x-axis. What is the shape of the solid formed when T is rotated about the x-axis?

For Monday September 25

Section 7.2 Finding Volumes by Integration

To read : Reread the section, but no Reading Questions for today.

For Wednesday September 27

For Friday September 29

Section 7.3 Work

To read : All

Be sure to understand : The section Work as an integral and Examples 2 and 5.

Reading Question :

Why do we need to use calculus to calculate work when the force varies?

For Monday October 2

Section 8.1 Integration by Parts

To read : All

Be sure to understand : Theorem 1. Be warned that Examples 8 and 9 can be a little slippery.

Reading Questions :

1. Integration by parts attempts to undo one of the techniques of differentiation. Which one is it?
2. Pick values for u and dv in the integral int( x * sin(x), x). Use parts to find an antiderivative for x * sin(x).

For Wednesday October 4

Section 8.1 Integration by Parts

To read : Reread the section for today

Reading Questions : Each integral can be evaluated using u-substitution or integration by parts. Which technique would you use in each case? Do not evaluate the integral, but explain your choice.

1. int( x*cos(x), x)
2. int(x*cos(x²),x)

For Friday October 6

Section 9.1 Taylor Polynomials

To read : All, but you can skip the section Trigonometric polynomials: Another nice family.

Be sure to understand : The statement of Theorem 1, Example 7, and the definition of the Taylor polynomial.

Reading Question :

Explain the basic idea of the Taylor polynomial for a function f(x) at x=x₀ in your own words.

For Monday October 9

Fall Break. No Reading Assignment.

For Wednesday October 11

Section 9.2 Taylor's Theorem: Accuracy Guarantees for Taylor Polynomials

To read : All, but you can skip the section Proving Taylor's theorem.

Be sure to understand : The statement of Theorem 2 and Examples 2 and 3.

Reading Questions :

What is the point of Theorem 2? Explain in your own words.

For Friday October 13

For Monday October 16

Section 10.1 Improper Integrals: Ideas and Definitions

To read : All

Be sure to understand : The section Convergence and divergence: Formal definitions and Examples 1 - 5.

Reading Questions :

1. What are the two ways in which an integral may be improper?
2. Explain why int( 1/x², x=1..infty) is improper.
3. Explain why int( 1/x², x=0..1) is improper.

For Wednesday October 18

Section 10.2 Detecting Convergence, Estimating Limits

To read : All

Be sure to understand : The statements of Theorems 1 and 2 and Example 4.

Reading Questions : Suppose that 0 < f(x) < g(x).

1. If int(f(x), x=1. .infty) diverges, what can you conclude about int( g(x), x=1. . infty)?
2. If int(g(x), x=1. .infty) diverges, what can you conclude about int( f(x), x=1. . infty)?
3. If int(f(x), x=1. .infty) converges, what can you conclude about int( g(x), x=1. . infty)?

For Friday October 20

No class today.

For Monday October 23

Section 10.2 Detecting Convergence, Estimating Limits

To read : Reread the section for today.

Be sure to understand : Example 5.

Reading Questions :

1. If 0 < f(x) < g(x) and int( g(x), x=1. . infty) converges, will int(f(x), x=1. .infty) converge or diverge? Why?
2. There are two types of errors that arise in Example 4 for approximating int( 1/(x⁵ +1), x=1..infty). What are the sources of these errors?

For Wednesday October 25

Section 4.2 More on Limits: Limits Involving Infinity and l'Hopital's Rule
Section 11.1 Sequences and Their Limits

To read : The section l'Hopital's rule: finding limits by differentiation that begins on page S-19 and all of Section 11.1.

Be sure to understand : The statement of l'Hopital's rule and the section Terminology and basic examples in Section 11.1.

Reading Questions :

1. Does l'Hopital's Rule apply to lim_{(x -> infty)} x² / e^x ? Why or why not?
2. Does l'Hopital's Rule apply to lim_{(x -> infty)} x² / sin(x) ? Why or why not?
3. Does the following sequence converge or diverge? Be sure to explain your answer.
   1, 3, 5, 7, 9, 11, 13, . . .
4. Find a symbolic expression for the general term a_k of the sequence 1, 2, 4, 8, 16, 32, . . .

For Friday October 27

Section 11.2 Infinite Series, Convergence, and Divergence

To read : Through Example 4. This can be tough going.

Be sure to understand : The sections "Why series matter: A look ahead" and "Definitions and terminology".

Reading Questions :

1. There are two sequences associated with every series. What are they?
2. Does the geometric series (1/4)^k converge or diverge? Why?

For Monday October 30

Section 11.2 Infinite Series, Convergence, and Divergence

To read : Finish the section and reread through Example 4.

Be sure to understand : The nth term test.

Reading Questions : What does the nth Term Test tell you about each series? Explain.

1. 2^k
2. 1/k

For Wednesday November 1

The Big Picture before Exam 2. No Reading Assignment for today.

For Friday November 3

Section 11.3 Testing for Convergence; Estimating Limits

To read : Through Example 5.

Be sure to understand : The statements of the Comparison test and Integral test for non-negative series.

Reading Question :

Explain in a couple of sentences why you think the Comparison Test should hold.

For Monday November 6

Section 11.3 Testing for Convergence; Estimating Limits

To read : Finish the section for today.

Be sure to understand : The statement of the Ratio test.

Reading Question:

What does the Ratio test remind you of?

For Wednesday November 8

For Friday November 10

No class today.

For Monday November 13

Section 11.4 Absolute Convergence; Alternating Series

To read : All

Be sure to understand : The statement of the Alternating series test.

Reading Questions :

1. Give an example of a series that is conditionally convergent. Explain.
2. Give an example of a series that is absolutely convergent. Explain.

For Wednesday November 15

Section 11.4 Absolute Convergence; Alternating Series

Reread the section for today, but no Reading Questions.

For Friday November 17

Section 11.5 Power Series

To read : All

Be sure to understand : Examples 4 and 5.

Reading Questions :

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 November 20

Section 11.5 Power Series
Reread the section, but no Reading Questions for today.

For Monday November 27

Section 11.5 Power Series
Reread the section, but no Reading Questions for today.

For Wednesday November 29

The Big Picture before Exam 3. No Reading Assignment for today.

For Friday December 1

Section 11.6 Power Series as Functions

To read : All

Be sure to understand : The section "Writing known functions as power series"

Reading Question :

Give two good reasons for writing a known function ( such as cos(x) ) as a power series.

For Monday December 4

Section 11.6 Power Series as Functions
Reread the section, but no Reading Questions for today.

For Wednesday December 6

We'll talk about why 1/k² = Pi²/6 today. No Reading Assignment.

For Friday December 8

The REALLY BIG PICTURE for the course. No Reading Assignment.