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

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

All numbers indicate sections from Calculus by Taalman and Kohn.

#### For Friday January 27 (Due 1/26 @ 8:00 pm)

Section 5.1 Integration by Substitution

• Topics:
• The Art of Integration
• Undoing the Chain Rule
• Choosing a Useful Substitution
• Finding Definite Integrals by Using Substitution
• Examples: 1 - 6

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) = 3x2 cos(x3)
3. Explain why ∫ cos(x) sin(x)2 dx and ∫ ln(x)2 / x dx are essentially the same integral after performing a substitution.

#### For Monday January 30 (Due 1/29 @ 8:00 pm)

Section 2.6 Inverse Trigonometric Functions

• Topics:
• Derivatives of Inverse Trigonometric Functions
• Example: 3

1. Why do you think we are studying the inverse trig functions now?
2. Find an antiderivative of f(x) = x2 / ( 1 + x6 )

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

Section 5.2 Integration by Parts

• Topics:
• Undoing the Product Rule
• Strategies for Applying Integration by Parts
• Finding Definite Integrals by Using Integration by Parts
• Examples: 1 - 4

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 e4x

#### For Friday February 3 (Due 2/2 @ 8:00 pm)

Section 5.2 Integration by Parts

Would you use u-substitution or integration by parts to find each anti-derivative? Find the antiderivative and explain why the other method would not work.

1. ∫ cos(x) sin(x) dx
2. ∫ ex cos(x) dx

#### For Monday February 6 (Due 2/5 @ 8:00 pm)

Section 5.6 Improper Integrals

• Topics:
• Integrating over an Unbounded Interval
• Integrating Unbounded Functions
• Improper Integrals of Power Functions
• Examples: 1, 2, 3

1. Explain why ∫1 1/x2 dx is improper.
2. Explain why ∫01 1/x2 dx is improper.
3. Explain why ∫-11 1/x2 dx is improper.

#### For Wednesday February 8 (Due 2/7 @ 8:00 pm)

Section 5.6 Improper Integrals

• Topics:
• Determining Convergence or Divergence with Comparisons
• Examples: 4

Suppose f and g are continuous and 0 ≤ f(x) ≤ g(x) for x>0. 1. If the improper integral ∫1 g(x) dx converges, what can you conclude about the improper integral ∫1 f(x) dx ?
2. If the improper integral ∫1 f(x) dx diverges, what can you conclude about the improper integral ∫1 g(x) dx ?
3. If the improper integral ∫1 f(x) dx converges, what can you conclude about the improper integral ∫1 g(x) dx ?

#### For Friday February 10 (Due 2/9 @ 8:00 pm)

Section 5.7 Numeric Integration

• Topics:
• Approximations and Error
• Error in Left and Right Sums
• Example: 1

1. Why would we want to approximate a definite integral?
2. When approximating an integral, which would you expect to be more accurate, LEFT(5) or LEFT(20)? Why?
3. If a function f(x) is decreasing on an interval, will LEFT(n) underestimate or overestimate the integral? Why?

#### For Monday February 13 (Due 2/12 @ 8:00 pm)

Section 5.7 Numeric Integration

• Topics:
• Error in Trapezoid and Midpoint Sums
• Example: 2

1. If a function f(x) is concave down on an interval, will TRAP(n) overestimate or underestimate the integral?
2. Consider the integral ∫-21 x3 dx. Is 4 a valid value for M in Theorem 5.27? Why or why not?

#### For Wednesday February 15 (Due 2/14 @ 8:00 pm)

Section 6.1 Volumes by Slicing

• Topics:
• Approximating Volume by Slicing
• Volume as a Definite Integral of Cross-Sectional Area
• Volumes by Disks and Washers
• Examples: 1, 2, 3
• Another optional resource are the videos at Kahn Academy on Solids of Revolution. The first two, "Disk method around x-axis" and "Generalizing disc method around x-axis", are relevant for today.

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

#### For Friday February 17 (Due 2/16 @ 8:00 pm)

Section 6.1 Volumes by Slicing

• Re-read the topics from Wednesday
• Example: 5

1. Let R be the rectangle formed by the x-axis, the y-axis, and the lines y=1 and x=3. Describe the shape of the solid formed when R is rotated about the line y=-5.
2. Let T be the triangle formed by the lines y=2x, y=6 and the y-axis. Describe the shape of the solid formed when T is rotated about the x-axis.

#### For Monday February 20 (Due 2/19 @ 8:00 pm)

Section 3.6 l'Hopital's Rule

• Topics:
• Geometrical Motivation for l'Hopital's Rule
• l'Hopital's Rule for the Indeterminate Forms 0/0 and ∞/∞
• Using Logarithms for the Indeterminate Forms 00, 1, and ∞0
• Examples: 1, 2, 3

1. Does l'Hopital's Rule apply to lim(x -> ∞) x2 / ex ? Why or why not?
2. Does l'Hopital's Rule apply to lim(x -> ∞) x2 / 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 Wednesday February 22 (Due 2/21 @ 8:00 pm)

Section 7.1 Sequences
Section 7.2 Limits of Sequences

• Topics in 7.1:
• Sequences of Numbers
• Recursively Defined Sequences
• Geometric and Arithmetic Sequences
• Monotonic Sequences
• Bounded Sequences
• Examples in 7.1: 1, 2, 3, 5

• Topics in 7.2:
• Convergence or Divergence of a Sequence
• Convergence and Divergence of Basic Sequences
• Bounded Monotonic Sequences
• Examples in 7.2: 1, 2, 3

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, -1/2, 1/4, -1/8, 1/16, -1/32, . . .

#### For Friday February 24 (Due 2/23 @ 8:00 pm)

Section 7.3 Series

• Topics:
• Adding Up Sequences to Get Series
• Convergence and Divergence of Series
• The Algebra of Series
• Geometric Series
• Examples: 1, 3, 4

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

#### For Monday February 27

Q&A for Exam 1. No Reading Assignment.

#### For Wednesday March 1 (Due 2/28 @ 8:00 pm)

Section 7.4 Introduction to Convergence Tests

• Topics:
• An Overview of Convergence Tests for Series
• The Divergence Test
• The Integral Test
• Convergence and Divergence of p-Series and the Harmonic Series
• Approximating a Convergent Series
• Examples: 1, 2, 3

1. What does the Divergence Theorem tell you about the series Σ 2k ?
2. What does the Divergence Theorem tell you about the series Σ 1/k ?
3. What does the Integral Test tell you about the series Σ 1/k2 ?
4. What does the Integral Test tell you about the series Σ 1/k ?

#### For Friday March 3 (Due 3/2 @ 8:00 pm)

Section 7.7 Alternating Series

• Topics:
• Alternating Series
• Absolute and Conditional Convergence
• The Curious Behavior of a Conditionally Convergent Series
• Examples: 1, 2

Consider the series Σ (-1)(k+1) / k2

1. Why does this series converge?
2. How closely does S50 approximate the value of the series? Why?

#### For Wednesday March 8 (Due 3/7 @ 8:00 pm)

Section 8.1 Power Series

• Topics:
• Power Series
• The Interval of Convergence of a Power Series
• Power Series in x - x0
• Examples: We'll go over a few different ones in class

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 Friday March 10 (Due 3/9 @ 8:00 pm)

Section 8.2 Maclaurin Series and Taylor Series

• Topics:
• Maclaurin Polynomials and Taylor Polynomials
• Taylor Series and Maclaurin Series
• Examples: 1, 2, 3

1. What is the basic idea of constructing the n-th degree Taylor polynomial for a function f(x)? Do not give the formula, but explain in your own words in a few sentences.
2. What is the difference between a Taylor series and a Maclaurin series?

#### March 13 - 17

Spring Break. Surprisingly, no Reading Assignments.

#### For Monday March 20

Re-read Section 8.2 Maclaurin Series and Taylor Series

#### For Friday March 24 (Due 3/23 @ 8:00 pm)

Section 12.1 Functions of Two and Three Variables

• Topics:
• Functions of Two Variables
• Graphing a Function of Two Variables
• Functions of Three Variables
• Level Curves and Level Surfaces
• Examples: 1, 4

1. Is the graph of the hyperboloid of one sheet x2 + y2 - z2 = 1 the graph of a function of two variables? (See page 785 for a sketch of this surface.) Explain.
2. Is the graph of the elliptic paraboloid z = x2 + y2 the graph of a function of two variables? (See page 785 for a sketch of this surface.) Explain.

#### For Monday March 27

Q&A for Exam 2. No Reading Assignment.

#### For Wednesday March 29

Re-read Section 12.1 Functions of Two and Three Variables

#### For Friday March 31 (Due 3/30 @ 8:00 pm)

Section 12.3 Partial Derivatives

• Topics:
• Partial Derivatives of Functions of Two and Three Variables
• Higher Order Partial Derivatives
• Finding a Function When the Partial Derivatives Are Given
• Examples: 1, 2, 4, 5

1. For a function f(x,y), what information does fx(1,0) give?
2. How many second-order partial derivatives does g(x,y,z) have? Why?

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

Section 12.6 Extreme Values

• Topics:
• You'll need to understand Definition 12.35 The Gradient on pg 957
• The Gradient at a Local Extremum
• Examples: 1 and 3. We'll talk about classifying extrema by methods other than the Second-Derivative Test.

1. If f(x,y)= x2y + 3 x y, give an expression for ∇ f(x,y).
2. If the partials fx and fy exist everywhere, at what points (x0, y0) can f(x,y) have a local max or a local min?

#### For Monday April 10 (Due 4/9 @ 8:00 pm)

Section 13.1 Double Integrals over Rectangular

• Topics:
• Volumes
• Double and Triple Summations
• Iterated Integrals and Fubini's Theorem (pay special attention here)
• Examples: 1 and 2

1. If f(x,y) is a function of two variables and R is a rectangle in the xy-plane, what does ∫ ∫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 12 (Due 4/11 @ 8:00 pm)

Section 13.2 Double Integrals over General Regions

• Topics:
• General Regions in the Plane
• Double Integrals over General Regions
• Algebraic Properties of Double Integrals
• Examples: 1, 2, 3

Let Ω be the region in the xy-plane bounded by the parabola y=x2 and the horizontal line y=9.

1. Give values for a, b, g1(x), and g2(x) so that ∫ ∫Ω f(x,y) dA = ab g1(x)g2(x) f(x,y) dy dx.

2. Give values for c, d, h1(y), and h2(y) so that ∫ ∫Ω f(x,y) dA = cd h1(y)h2(y) f(x,y) dx dy.

#### For Friday April 14 (Due 4/13 @ 8:00 pm)

Section 13.2 Double Integrals over General Regions

• Examples: Make sure to understand Example 4
Consider the integral 01 y1 ex2 dx dy.

1. Explain why you would want to reverse the order of integration for this integral.
2. Evaluate the integral by reversing the order of integration.

#### For Wednesday April 19 (Due 4/18 @ 8:00 pm)

Section 9.2 Polar Coordinates

• Topics:
• Plotting Points in Polar Coordinates
• Converting Between Polar and Rectangular Coordinates
• The Graphs of Some Simple Polar Coordinate Equations
• Examples: 1, 2, 3, 4

1. What do the coordinates (r, θ) 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(θ) is the graph of a function y=f(x)? Explain.

#### For Friday April 21 (Due 4/20 @ 8:00 pm)

Section 9.3 Graphing Polar Equations

• Topics:
• Using the θr - plane to Get Information About a Polar Graph
• Symmetry in Polar Graphs
• Examples: 1 - 6

Describe the shape of the graph of the following polar equations.

1. r = 2θ, θ≥ 0
2. r = 2 sin(θ)
3. r = sin(2θ)
4. r = sin(3θ)

#### For Monday April 24

Q&A for Exam 3. No Reading Assignment.

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

Section 13.3 Double Integrals Using Polar Coordinates

• Topics:
• Polar Coordinates and Double Integrals
• Double Integrals in Polar Coordinates over General Regions
• Examples: 1, 4, 5

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

#### For Friday April 28

Re-read Section13.3 Double Integrals Using Polar Coordinates

#### For Wednesday May 3

The BIG Picture. No Reading Assignment.

#### For Friday May 5

The BIG Picture. No Reading Assignment. Maintained by: ratliff_thomas@wheatoncollege.edu