# Pre-Class Assignments, Math 101 Calculus I, Fall 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 this page will be updated often 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 7, all numbers indicate sections from APEX Calculus, Version 4.0.
Be sure to check the Errata for corrections to the text.

### Due Tuesday August 25 @ midnight

#### An intuitive introduction to derivatives

• Section 1.4 Amount Functions and Rate Functions: The Idea of the Derivative, from Ostebee/Zorn, pp. 35-44, posted to onCourse
##### To Watch
Links in "Echo360 and other course videos" topic in onCourse. ~10 minutes total

#### Pre-Class Questions

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.
4. Which day does your tutorial group meet this week?

### Due Sunday August 30 @ midnight

#### Review of exponentials and logarithms

• Section 3.4 Exponential and Logarithmic Functions, pp. 85-96, from Essential Precalculus, posted to onCourse
##### To Watch
• The Khan Academy Unit on Logarithms has lots of relevant videos, depending on how much you need to review. Specifically, these might be worth reviewing if you're feeling a little rusty.

#### Pre-Class Questions

1. All exponential functions $$f(x)=b^x$$ share a common point on their graphs. What is it?
2. How are the graphs of the functions $$f(x)=2^x$$ and $$g(x)=\log_2(x)$$ related?
3. Solve for x in the equation $$\log_2(x^3-11)=4$$.

### Due Tuesday September 1 @ midnight

#### Review of trigonometric functions

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

#### Pre-Class Questions

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?
4. Which day does your tutorial group meet this week?

### Due Sunday September 6 @ midnight

#### Finding limits graphically and analytically

• Section 1.1 An Introduction to Limits
• Section 1.3 Finding Limits Analytically
• Section 1.4 One Sided Limits

#### Pre-Class Questions

1. If $$f(x)=x^2$$, use the graph of $$y=f(x)$$ to explain why $$\displaystyle\lim_{x\to 2} f(x) = 4$$.
2. If $$f(x)=\displaystyle\frac{1}{x}$$, use the graph of $$y=f(x)$$ to explain why $$\displaystyle\lim_{x\to 0} f(x)$$ does not exist.
3. Explain why $$\displaystyle\lim_{x\to -3} \frac{x^2-9}{x+3} = -6$$
4. If $$f(x)=x^2$$, explain why $$\displaystyle\lim_{h\to 0} \frac{f(5+h) -f(5)}{h} = 10$$
5. How is the last question related velocity?

### Due Tuesday September 8 @ midnight

#### Continuity and limits at infinity

• Section 1.5 Continuity
• Section 1.6 Limits Involving Infinity

Don't worry about the references to the $$\epsilon - \delta$$ definition of the limit in these sections, but try to use your graphical intuition about limits.

#### Pre-Class Questions

1. In Figure 1.4.1, explain why $$\displaystyle\lim_{x\to 1^+}f(x) \ne f(1)$$
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?
4. Give an example of a function that has a vertical asymptote at x = 2. Explain.
5. Give an example of a function that has a horizontal asymptote at y = 3. Explain.
6. Which day does your tutorial group meet this week?

### Due Sunday September 13 @ midnight

#### The derivative

• Section 2.1 Instantaneous Rates of Change: The Derivative
##### To Watch
• Week 4: Definition of the derivative and the derivative of xn (Echo360)

#### Pre-Class Questions

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.

### Due Tuesday September 15 @ midnight

#### Finding formulas for derivatives

• Section 2.3 Basic Differentiation Rules
##### To Watch
• Week 4: Basic differentiation rules (Echo360)

#### Pre-Class Questions

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 $$g(x)$$?
3. Which day does your tutorial group meet this week?

### Due Sunday September 20 @ midnight

#### The product and quotient rules

• Section 2.4 The Product and Quotient Rules
##### To Watch
• Week 5: The product and quotient rules (Echo360)

#### Pre-Class Questions

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

### For Tuesday September 22 @ midnight

#### The chain rule

• Section 2.5 The Chain Rule
##### To Watch
• How to Use the Chain Rule by Calcvids
Don't get hung up on the derivative of r(x) in the video - we'll talk about derivatives of logarithms other than ln(x) later.
No questions to submit today because of Exam 1

### For Sunday September 27

#### Putting it all together

Review the techniques of differentiation, but no questions to submit for today.

### Due Tuesday September 29 @ midnight

#### Extreme values

• Section 3.1 Extreme Values
##### To Watch
• Week 6: Finding extreme values (Echo360)

#### Pre-Class Questions

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 $$[-1,2]$$.

### Due Sunday October 4 @ midnight

#### The Mean Value Theorem and First Derivative Test

• 3.2 The Mean Value Theorem
You can skip the proofs of the Mean Value Theorem and Rolle's Theorem.
• 3.3 Increasing and Decreasing Functions
##### To Watch
• Why we care about the Mean Value Theorem (Echo360)
• First Derivative Test by the Organic Chemistry Tutor

#### Pre-Class Questions

1. What is an important consequence of the Mean Value Theorem related to finding antiderivatives of a function f(x)?
2. Let f(x)=x3-3x2-9x+7.
1. Find the critical numbers of f(x)
2. Find the intervals where f(x) is increasing and the intervals where f(x) is decreasing
3. Use the First Derivative Test to identify each critical number as a relative maximum, minimum, or neither

### Due Tuesday October 6 @ midnight

#### Concavity and the Second Derivative

• 3.4 Concavity and the Second Derivative
• 3.5 Curve Sketching

#### Pre-Class Questions

Let f(x)=x3-3x2-9x+7. Notice this is the same question from Sunday.
1. Find the inflection points of f(x)
2. Find the intervals where f(x) is concave up and the intervals where f(x) is concave down
3. Use the Second Derivative Test to identify each critical number of f(x) as a relative maximum or minimum, if possible.
4. Which day does your tutorial group meet this week?

### Due Sunday October 11 @ midnight

#### L'Hôpital's Rule and Optimization

• 6.7 L'Hôpital's Rule
• 4.3 Optimization

#### Pre-Class Questions

1. Is the limit $$\displaystyle \lim_{x \to \infty} \frac{e^x}{x}$$ in indeterminant form? Explain.
2. Is the limit $$\displaystyle \lim_{x \to 0} \frac{\cos(3x)}{x}$$ in indeterminant form? Explain.
3. Look back at Problem 3 from the Week 6 Tutorial. What is the fundamental equation for this problem?

### Due Tuesday October 13 @ midnight

#### Optimization

Re-read Section 4.3. No questions to submit today, but be sure to check which day your tutorial group meets!

### Due Sunday October 18 @ midnight

#### Taylor and Maclaurin Polynomials

• 8.7 Taylor Polynomials
Focus on pages 473-476. We won't get into the details of error bounds this semester.
##### To Watch
• Why Taylor polynomials (Echo360)

#### Pre-Class Questions

1. What is the purpose of finding the Taylor polynomial for a known function like $$f(x)=\sin(x)$$?
2. What is the difference between a Taylor polynomial and a Maclaurin polynomial?
3. Consider forming the Maclaurin polynomial of degree 3 for $$f(x)=\sin(x)$$. Call this polynomial $$P_3(x)$$. Which derivatives of $$f(x)$$ do you need to form $$P_3(x)$$?

### Due Tuesday October 20 @ midnight

#### Pre-Class Questions

1. Why is a logistic model more accurate than an exponential model when modeling an epidemic?
2. Around the 5:00 mark of the "Exponential growth and epidemics" video, they mark a point as an inflection point. Explain why this point matches our calculus definition of an inflection point being a place where the second derivative is 0.
3. Which day does your tutorial group meet this week?

### Due Sunday October 25 @ midnight

#### Definite and Indefinite Integrals

• 5.1 Antiderivatives and Indefinite Integration
• 5.2 The Definite Integral

#### Pre-Class Questions

1. Evaluate $$\displaystyle\int 2x + \cos(x) dx$$
2. What is the difference between a definite integral and an indefinite integral?
3. Look at graph in Figure 5.2.8 on pg 213. Will $$\displaystyle \int_0^a f(t) dt$$ be positive or negative? How about $$\displaystyle\int_0^b f(t) dt$$? Explain.

### Due Tuesday October 27 @ midnight

#### Riemann Sums

• Section 5.3 Riemann Sums
##### To Watch
No questions to submit today because of Exam 2

### Due Sunday November 1 @ midnight

#### The Fundamental Theorem of Calculus

• 5.4 The Fundamental Theorem of Calculus
Focus on the concepts in the first four pages (pp 236 - 239)
##### To Watch
• The Fundamental Theorem of Calculus (Echo360)

#### Pre-Class Questions

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.

### Due Tuesday November 3 @ midnight

#### The Fundamental Theorem of Calculus

Re-read Section 5.4 and re-watch the video, but no questions to submit today.

### Due Sunday November 8 @ midnight

#### Antidifferentiation by Substitution

• 6.1 Substitution
You can skip the parts related to the inverse trig functions.

#### Pre-Class Questions

1. Substitution attempts to undo one of the techniques of differentiation. Which one is it?
2. Use $$u$$-substitution to find an antiderivative of $$f(x) = 3x^2\cos(x^3)$$