Reading AssignmentsMath 104 Calculus II, Spring 2020

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 24 (Due 1/23 @ midnight)

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 27 (Due 1/26 @ midnight)

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 $$f(x) = \dst \frac{3x^2}{ 1 + x^6}$$

For Wednesday January 29 (Due 1/28 @ midnight)

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

For Monday February 3 (Due 2/2 @ midnight)

Section 5.5 Numerical Integration

1. Why would you ever want to numerically approximate an integral?
2. Let $$\mathcal{I} = \dst\int_0^{\pi} \sin(x^2) dx$$.
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?

For Wednesday February 5 (Due 2/4 @ midnight)

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 February 10 (Due 2/9 @ midnight)

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 12 (Due 2/11 @ midnight)

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 14 (Due 2/13 @ midnight)

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 17

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

For Wednesday February 19 (Due 2/18 @ midnight)

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 $$a_n$$ of the sequence $\{ a_n\} = \{1, 2, 4, 8, 16, 32, \ldots \}$

For Friday February 21 (Due 2/20 @ midnight)

Section 8.2 Infinite Series

1. There are two sequences associated with every series. What are they?
2. Does the geometric series $$\dst \sum_{n=0}^{\infty} \left( \frac{1}{4}\right)^n$$ converge or diverge? Why?
3. Does the geometric series $$\dst \sum_{n=0}^{\infty} \left( \frac{\pi}{e}\right)^n$$ converge or diverge? Why?

For Monday February 24 (Due 2/23 @ midnight)

Section 8.2 Infinite Series

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

For Wednesday February 26 (Due 2/25 @ midnight)

Section 8.3 Integral and Comparison Tests

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

2. What does the Integral Test tell you about the series $$\dst \sum \frac{1}{\sqrt{n}}$$?

For Monday March 2 (Due 3/1 @ midnight)

Section 8.5 Alternating Series and Absolute Convergence

Consider the series $$\sum_{n=1}^\infty (-1)^n \frac{1}{n^2}$$
1. Why does this series converge?
2. How closely does $$S_{50}$$, the 50th partial sum, approximate the value of the series? Why?

For Friday March 6 (Due 3/5 @ midnight)

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.

March 9 - 13

Spring Break. Surprisingly, no Reading Assignments.

For Monday March 16 (Due 3/15 @ midnight)

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 20 (Due 3/19 @ midnight)

Section 12.1 Introduction to Multivariable Functions

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

For Monday March 23

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

For Wednesday March 25 (Due 3/24 @ midnight)

Section 12.3 Partial Derivatives

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

For Monday March 30 (Due 3/29 @ midnight)

Section 10.2 An Introduction to Vectors Section 10.3 The Dot Product

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 does this tell you about the vectors?

For Wednesday April 1 (Due 3/30 @ midnight)

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 Friday April 3 (Due 4/2 @ midnight)

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 8 (Due 4/7 @ midnight)

Section 12.8 Extreme Values

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"?

For Monday April 13 (Due 4/12 @ midnight)

Section 13.1 Iterated Integrals and Area

1. What geometric value does the iterated intergral $$\dst\int_0^1 \int_{-x^2}^{x^2} 1 \ dy\ dx$$ measure?
2. Why would you want to switch the order of integration in an iterated integral?

For Wednesday April 15 (Due 4/14 @ midnight)

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 22 (Due 4/21 @ midnight)

Section 9.4 Introduction to Polar Coordinates

1. What do the coordinates $$(r,\theta)$$ in polar coordinates measure?
2. Is the graph of the polar function $$r = \cos(2\theta)$$ the graph of a function y=f(x)? Explain.

For Friday April 24 (Due 4/23 @ midnight)

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 April 29

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