# Reading AssignmentsMath 101 Calculus I, Fall 2018

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 August 31 (Due 8/30 @ 8:00 pm)

#### Review of Exponential and Logarithmic Functions

The text doesn't include background material on exponential or logarithmic functions, but I have posted two references, APEX Pre-Calculus and Essential Precalculus, to onCourse. For today's class, read

• Section 1.5 Logarithms and Exponential Functions from APEX Pre-Calculus
• Section 3.4 Exponential and Logarithmic Functions from Essential Precalculus
Depending on how comfortable you are with this material, you may be able to skip through parts of these sections. Also feel free to look for your own resources!

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 $$\log_2(x^3-11)=4$$.

### For Monday September 3

Labor Day. No class meeting or reading assignment due.

### For Wednesday September 5 (Due 9/4 @ 8:00 pm)

#### Review of Trigonometric Functions

• Section 3.4 Introduction to Trigonometric Functions from APEX Pre-Calculus
• Section 3.5 Trigonometric Functions and Triangles APEX Pre-Calculus
Chapter 4 Foundations of Trigonometry from Essential Precalculus is a more complete treatment that is another good reference.

1. What is 120 degrees equal to in radians?
2. What is the period of the sine function? How can you tell by using the unit circle to evaluate sine?

### For Friday September 7 (Due 9/6 @ 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 10 (Due 9/9 @ 8:00 pm)

#### Section 1.3 Finding Limits Analytically

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 12 (Due 9/11 @ 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 1.4.1, 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 14 (Due 9/13 @ 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 = 2. Explain.
2. Give an example of a function that has a horizontal asymptote at y = 3. Explain.

### For Monday September 17 (Due 9/16 @ 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 19 (Due 9/18 @ 8:00 pm)

#### Section 2.2 Interpretations of the Derivative

Let $$T(h)$$ give the temperature in degrees Celsius in Norton $$h$$ hours after midnight on August 18, 2018.
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 21 (Due 9/20 @ 8:00 pm)

#### Section 2.3 Basic Differentiation Rules

1. If $$f(x)=x^8+\frac{1}{x}-\sin(x)+3\cos(x)$$, what is $$f'(x)$$?
2. If $$g(x)=e^x$$, what is the 42nd derivative of $$g(x)$$?

### For Wednesday September 26

Q & A for Exam 1. No reading assignment.

### For Friday September 28 (Due 9/27 @ 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 1 (Due 9/30 @ 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 Friday October 5 (Due 10/4 @ 8:00 pm)

#### Section 3.1 Extreme Values

Let $$f(x)=2 x^3-3 x^2-12 x+1$$.
1. Find the critical numbers of $$f(x)$$.
2. Find the extreme values of $$f(x)$$ on the interval $$[-2,1]$$.

### For Monday October 8

Fall Break. No class meeting or reading assignment due.

### For Wednesday October 10 (Due 10/9 @ 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 12 (Due 10/11 @ 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 15 (Due 10/14 @ 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 $$x=\dst\frac{1}{8} \left(3+\sqrt{73}\right)$$ could be 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 17

#### Section 3.5 Curve Sketching

Read all of this section and pay special attention to Examples 3.5.1, 3.5.2, and 3.5.3, but no Reading Questions for today.

### For Friday October 19 (Due 10/18 @ 8:00 pm)

#### Section 6.7 L'Hôpital's Rule

1. Does L'Hôpital's Rule apply to $$\dst \lim_{x \to \infty} \frac{x^2}{e^x}$$ ? Why or why not?
2. Does L'Hôpital's Rule apply to $$\dst \lim_{x\to\infty} \frac{x^2}{\sin(x)}$$? Why or why not?
3. For each limit in #1 and #2 where L'Hôpital's applies, use it to find the limit.

### For Monday October 22 (Due 10/21 @ 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 29 (Due 10/28 @ 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 8.7.1.

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 Wednesday October 31

Q & A for Exam 2. No reading assignment.

### For Monday November 5 (Due 11/4 @ 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 7 (Due 11/6 @ 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 5.2.8 on pg 213. Will $$\int_0^a f(t) dt$$ be positive or negative? How about $$\int_0^b f(t) dt$$? Explain.

### For Monday November 12 (Due 11/11 @ 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 $$\dst\int_0^2 x^2 dx$$? Explain.

### For Wednesday November 14

#### 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 Monday November 11/19 (Due 11/18 @ 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 Wednesday November 21

Thanksgiving Break. No class meetings or reading assignments due.

### For Friday November 23

Thanksgiving Break. No class meetings or reading assignments due.

### For Wednesday November 28

Q & A for Exam 3. No reading assignment.

### For Friday November 30

#### 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 Friday December 8

The Big Picture. No reading assignment for today.