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

### Reading Assignments - Math 101 Calculus I - Fall 2017

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 3.0, and check the Errata for corrections to the text.

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

Review of Exponential and Logarithmic Functions

To watch

The text doesn't include background material on exponential or logarithmic functions, so these are some videos that should serve as a good review. They were done by Ed Berger, formerly of Williams Collage and currently president of Southwestern University in Georgetown, Texas.

Depending on how comfortable you are with this material, you may be able to skip through parts of these. Also feel free to look for your own resources!

• Exponential Functions
• Logarithmic Functions

1. How are the graphs of the functions $$f(x)=2^x$$ and $$g(x)=\log_2(x)$$ related?
2. Solve for x in the equation $$\log_2(x) + \log_2(x^3)=4$$.

#### For Monday September 4

Labor Day. No class meeting or reading assignment due.

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

Review of Trigonometric Functions

To watch

As above, depending on how comfortable you are with this material, you may be able to skip through parts of these.

1. What is 120 degrees equal to in radians?
2. What is the period of the sine function? How can you tell from the graph?

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

Section 1.1 An Introduction to Limits

1. If $$f(x)=x^2$$, explain why $$\dst\lim_{x\to 2} f(x) = 4$$.
2. If $$f(x)=\dst\frac{1}{x}$$, explain why $$\dst\lim_{x\to 0} f(x)$$ does not exist.
3. How is the difference quotient related to calculus?

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

Section 1.3 Finding Limits Analytically

You can de-emphasize the section on the Squeeze Theorem.

1. Explain why $$\dst\lim_{x\to -3} \frac{x^2-9}{x+3} = -6$$
2. If $$f(x)=x^2$$, explain why $$\dst\lim_{h\to 0} \frac{f(5+h) -f(5)}{h} = 10$$
3. How is the last question related velocity?

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

Section 1.4 One Sided Limits
Section 1.5 Continuity

In Section 1.4, do not worry about the references to the $$\epsilon - \delta$$ definition of the limit, but try to think about the intuition behind one sided limits.

1. In Figure 1.21, explain why $$\dst\lim_{x\to 1^+}f(x) \ne f(1)$$
2. How can you tell from the graph of y=f(x) if the function f(x) is continuous?
3. Why is the Intermediate Value Theorem called the Intermediate Value Theorem?

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

Section 1.6 Limits Involving Infinity

Do not worry about the references to the $$\epsilon - \delta$$ definition of the limit.

1. Give an example of a function that has a vertical asymptote at x = 3. Explain.
2. Give an example of a function that has a horizontal asymptote at y = 3. Explain.

#### For Monday September 18 (Due 9/17 @ 8:00 pm)

Section 2.1 Instantaneous Rates of Change: The Derivative

1. Let $$f(x)=3x^2$$. Find $$f'(2)$$.
2. Use the graph of $$f(x)=|x|$$ to explain why $$f'(0)$$ does not exist.

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

Section 2.2 Interpretations of the Derivative

Let $$T(h)$$ give the temperature in degrees Farenheit in Norton $$h$$ hours after midnight on May 20, 2017.

1. What are the units for $$T'(h)$$?
2. Do you think $$T'(10)$$ is positive or negative? Explain.
3. Do you think $$T(10)$$ is positive or negative? Explain.

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

Section 2.3 Basic Differentiation Rules

1. If $$f(x)=x^8+2x^2$$, what is $$f'(x)$$?
2. If $$g(x)=e^x$$, what is the 42nd derivative of $$g(x)$$?

#### For Monday September 25

Section 2.3 Basic Differentiation Rules

#### For Wednesday September 27

Q & A for Exam 1. No reading assignment.

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

Section 2.4 The Product and Quotient Rules

Explain what is wrong with the following calculations and fix them.

1. If $$f(x)=(x^2+7x)(x^4 + 5 x^2 + 9)$$, then $$f'(x)=(2x+7)(4x^3+10x)$$.
2. If $$f(x)=\dst\frac{x^2+7x}{x^4 + 5 x^2 + 9}$$, then $$f'(x)=\dst\frac{2x+7}{4x^3+10x}$$.

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

Section 2.5 The Chain Rule

Explain what is wrong with the following calculations and fix them.

1. If $$f(x)=(x^2+2x)^{130}$$, then $$f'(x)=130(x^2+2x)^{129}$$.
2. If $$f(x)=\sin(x^2)$$, then $$f'(x)=\cos(2x)$$.

#### For Wednesday October 4

Section 2.5 The Chain Rule

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

Section 3.1 Extreme Values

Reading Questions Let $$f(x)=2 x^3-3 x^2-12 x+1$$.

1. Find the critical numbers of $$f(x)$$.
2. Find the extrema of $$f(x)$$ on the interval $$[-2,1]$$.

#### For Monday October 9

Fall Break. No class meeting or reading assignment due.

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

Section 3.2 The Mean Value Theorem

You can skip the proofs of the Mean Value Theorem and Rolle's Theorem.

1. Explain the Mean Value Theorem using "car talk" (that is, in terms of velocity).
2. Consider $$f(x)=5 x^3 - 2 x$$. Find $$c$$ in $$[1,4]$$ that satisfies the Mean Value Theorem.

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

Section 3.3 Increasing and Decreasing Functions

Let $$f(x)=\dst\frac{x^5}{5}-\frac{x^4}{4}-\frac{2 x^3}{3}+3$$

1. Verify that $$x=-1$$, $$x=0$$, and $$x=2$$ are the critical numbers of $$f(x)$$.
2. Find the intervals where $$f(x)$$ is increasing and decreasing.
3. Use the First Derivative Test to identify each critical number as a relative maximum, minimum, or neither.

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

Section 3.4 Concavity and the Second Derivative

Let $$f(x)=\dst\frac{x^5}{5}-\frac{x^4}{4}-\frac{2 x^3}{3}+3$$. Notice this is the same function from Friday.

1. Verify that $$x=0$$, $$x=\dst\frac{1}{8} \left(3-\sqrt{73}\right)$$, and $$x=\dst\frac{1}{8} \left(3+\sqrt{73}\right)$$ are the inflection points of $$f(x)$$.
2. Find the intervals where $$f(x)$$ is concave up and concave down.
3. Use the Second Derivative Test to identify each critical number of $$f(x)$$ as a relative maximum or minimum, if possible.

#### For Wednesday October 18

Section 3.5 Curve Sketching

Read all of this section and pay special attention to Examples 93, 94, and 95, but no Reading Questions for today.

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

Section 4.3 Optimization

Consider the following problem: Find the minimum sum of two non-negative numbers, $$a$$ and $$b$$, whose product is 100.

1. Write the quantity to be optimized in terms of $$a$$ and $$b$$. The text calls this the "fundamental equation."
2. Write the quantity from 1 in terms of just $$a$$.
3. How do you use the expression in 2 to find the minimum sum?

#### For Monday October 23

Section 4.3 Optimization

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

Section 8.7 Taylor Polynomials

You can de-emphasize the parts related to bounding the error $$R_n(x)$$ described in Theorem 76.

1. What is the purpose of finding the Taylor polynomial for a known function like $$f(x)=\sin(x)$$?
2. In your own words, explain the basic concept underlying the construction of a Maclaurin polynomial in a few sentences.

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

Section 5.1 Antiderivatives and Indefinite Integration

1. Evaluate $$\int 2x + \cos(x) dx$$
2. Verify that $$\int \ln(x) dx = x\ln(x)-x+C$$
3. Find $$f(x)$$ given that $$f'(x)=x^2$$ and $$f(2)=3$$

#### For Wednesday November 1

Q & A for Exam 2. No reading assignment.

#### For Friday November 3

Section 5.1 Antiderivatives and Indefinite Integration

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

Section 5.2 The Definite Integral

1. What is the difference between a definite integral and an indefinite integral?
2. Look at graph in Figure 5.9 on pg 205. Will $$\int_0^a f(t) dt$$ be positive or negative? How about $$\int_0^b f(t) dt$$? Explain.

#### For Wednesday November 8

Section 5.2 The Definite Integral

#### For Friday November 10 (Due 11/9 @ 8:00 pm)

Section 5.3 Riemann Sums

1. What is the purpose of a Riemann sum?
2. Will a Right Hand Rule sum overestimate or underestimate $$\int_0^2 x^2 dx$$? Explain.

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

Section 5.4 The Fundamental Theorem of Calculus

1. Does every continuous function have an antiderivative? Why or why not?
2. Find the area of the region above the $$x$$-axis and below the graph of $$f(x)= \dst\frac{4}{x} + \cos(x) + 1$$ between $$x=1$$ and $$x=10$$.

#### For Wednesday November 15

Section 5.4 The Fundamental Theorem of Calculus

#### For Friday November 17 (Due 11/16 @ 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? Explain.
3. Which would you expect to be LEAST accurate: a Right Hand approximation, a Trapezoidal approximation, or a Simpson's approximation? Explain.

#### For Monday November 11/20

Section 5.5 Numerical Integration

#### 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 6.1 Substitution

You can skip the parts related to the inverse trig functions.

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 $$\int\cos(x) \sin(x)^2\ dx$$ and $$\dst\int \frac{\ln(x)^2}{x}\ dx$$ are essentially the same integral after performing a substitution.

#### For Wednesday November 29

Q & A for Exam 3. No reading assignment.

#### For Friday December 1

Section 6.1 Substitution

#### For Monday December 4

Section 6.1 Substitution

#### For Wednesday December 6

The Big Picture. No reading assignment for today.

#### For Friday December 8

The Big Picture. No reading assignment for today.

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