# Reading AssignmentsMath 101 Calculus I, Fall 2019

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.

• Since the main text doesn't include background material on exponentials, logarithms, or trig functions, I posted references for the first two weeks of class to onCourse.
• Beginning on Monday September 9, all numbers indicate sections from APEX Calculus, Version 4.0. Be sure to check the Errata for corrections to the text.

### For Friday August 30 (Due 8/29 @ midnight)

#### An intuitive introduction to derivatives

Read Section 1.4 Amount Functions and Rate Functions: The Idea of the Derivative from Ostebee/Zorn, pp. 35-44, posted to onCourse.

Look at the graphs of P(t) and V(t) in Figure 1 on page 37.
1. Is the derivative of P positive or negative at t=5 ? Explain.
2. Is the second derivative of P positive or negative at t=5 ? Explain.
3. Give a value of t where the derivative of P is zero.

### For Tuesday September 3 (Due 9/2 @ midnight)

#### Review of Exponential Functions

Read Section 3.4 Exponential and Logarithmic Functions, pp. 85-89, from Essential Precalculus, posted to onCourse.

1. All exponential functions $$f(x)=b^x$$ share a common point on their graphs. What is it?
2. Why are exponential functions useful?

### For Wednesday September 4 (Due 9/3 @ midnight)

#### Review of Logarithmic Functions

Read Section 3.4 Exponential and Logarithmic Functions, pp. 90-96, from Essential Precalculus, posted to onCourse.

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 Friday September 6 (Due 9/5 @ midnight)

#### Review of Trigonometric Functions

Read Section 4.1 The Unit Circle: Sine and Cosine from Essential Precalculus, posted to onCourse.

Use the unit circle definitions of sine and cosine to answer the following.
1. Is $$\sin(6\pi / 7)$$ positive or negative? Why?
2. Is $$\cos(6\pi / 7)$$ positive or negative? Why?
3. What is the period of the sine function? Why?

### For Monday September 9 (Due 9/8 @ midnight)

#### 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 Wednesday September 11 (Due 9/10 @ midnight)

#### 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 Friday September 13 (Due 9/12 @ midnight)

#### 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 Monday September 16 (Due 9/15 @ midnight)

#### 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 Wednesday September 18 (Due 9/17 @ midnight)

#### 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 Friday September 20 (Due 9/19 @ midnight)

#### 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 25

Q & A for Exam 1. No reading assignment.

### For Friday September 27 (Due 9/27 @ midnight)

#### 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 September 30 (Due 9/29 @ midnight)

#### 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 4 (Due 10/3 @ midnight)

#### Section 3.1 Extreme Values

Let $$f(x)=2x^3+3x^2-12x+5$$.
1. Find the critical numbers of $$f(x)$$.
2. Find the extreme values of $$f(x)$$ on the interval $$[-1,2]$$.

### For Monday October 7 (Due 10/6 @ midnight)

#### 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 $$[0,2]$$ that satisfies the Mean Value Theorem.

### For Wednesday October 9 (Due 10/8 @ midnight)

#### Section 3.3 Increasing and Decreasing Functions

Let $$f(x)=x^3 - 3 x^2 - 9 x + 7$$
1. Find 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 Friday October 11 (Due 10/10 @ midnight)

#### Section 3.4 Concavity and the Second Derivative

Let $$f(x)=x^3 - 3 x^2 - 9 x + 7$$. Notice this is the same function from Wednesday.
1. Find 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 Monday October 14

Fall Break. No class meeting or reading assignment due.

### 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 18 (Due 10/17 @ midnight)

#### 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 21 (Due 10/20 @ midnight)

#### 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 Friday October 25 (Due 10/24 @ midnight)

#### 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 30

Q & A for Exam 2. No reading assignment.

### For Friday November 1 (Due 10/31 @ midnight)

#### Section 5.1 Antiderivatives and Indefinite Integration

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

### For Monday November 4 (Due 11/3 @ midnight)

#### 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 Wednesday November 6 (Due 11/5 @ midnight)

#### 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 Friday November 8 (Due 11/7 @ midnight)

#### 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 $$f(x)=\cos(x) + 2$$ between $$x=1$$ and $$x=7$$.

### For Wednesday November 13 (Due 11/12 @ midnight)

#### 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 in approximating $$\int_1^3 \cos(x^4) dx$$: a Right Hand approximation, a Trapezoidal approximation, or a Simpson's approximation? Explain.
3. Which would you expect to be LEAST accurate in approximating $$\int_1^3 \cos(x^4) dx$$: a Right Hand approximation, a Trapezoidal approximation, or a Simpson's approximation? Explain.

### For Monday November 18 (Due 11/17 @ midnight)

#### 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 $$f(x) = 3x^2\cos(x^3)$$

### For Wednesday November 20

Q & A for Exam 3. No reading assignment.

### For Wednesday November 27

Thanksgiving Break. No class meetings or reading assignments due.

### For Friday November 29

Thanksgiving Break. No class meetings or reading assignments due.

### For December 2 - 6

For the last week of the semetser we'll be reviewing and looking at the Big Picture of Calculus I. No Reading Assignments for this week.