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
To Read
- 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- Evaluating compositions (Khan Academy), for review, if needed
- Week 1: Intuitive idea of derivatives (Echo360)
Pre-Class Questions
Look at the graphs of P(t) and V(t) in Figure 1 on page 37.- Is the derivative of P positive or negative at t=5 ? Explain.
- Is the second derivative of P positive or negative at t=5 ? Explain.
- Give a value of t where the derivative of P is zero.
- Which day does your tutorial group meet this week?
Due Sunday August 30 @ midnight
Review of exponentials and logarithms
To Read
- 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
- All exponential functions \( f(x)=b^x\) share a common point on their graphs. What is it?
- How are the graphs of the functions \( f(x)=2^x\) and \(g(x)=\log_2(x)\) related?
- Solve for x in the equation \(\log_2(x^3-11)=4\).
Due Tuesday September 1 @ midnight
Review of trigonometric functions
To Read
- Section 4.1 The Unit Circle: Sine and Cosine, from Essential Precalculus, posted to onCourse
To Watch
- The Khan Academy Unit on Trigonometry
has a lot that you can review. A few that you may want to especially pay attention to are:
- The unit circle definitions of trig functions
- The graph of sin(x) from the unit circle
Pre-Class Questions
Use the unit circle definitions of sine and cosine to answer the following.- Is \( \sin(6\pi / 7) \) positive or negative? Why?
- Is \( \cos(6\pi / 7) \) positive or negative? Why?
- What is the period of the sine function? Why?
- Which day does your tutorial group meet this week?
Due Sunday September 6 @ midnight
Finding limits graphically and analytically
To Read
- Section 1.1 An Introduction to Limits
- Section 1.3 Finding Limits Analytically
- Section 1.4 One Sided Limits
To Watch
- Limit at a point by Calcvids
Pre-Class Questions
- If \( f(x)=x^2\), use the graph of \(y=f(x)\) to explain why \(\displaystyle\lim_{x\to 2} f(x) = 4 \).
- 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.
- Explain why \( \displaystyle\lim_{x\to -3} \frac{x^2-9}{x+3} = -6 \)
- If \( f(x)=x^2\), explain why \( \displaystyle\lim_{h\to 0} \frac{f(5+h) -f(5)}{h} = 10\)
- How is the last question related velocity?
Due Tuesday September 8 @ midnight
Continuity and limits at infinity
To Read
- 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.
To Watch
- Infinite limits and asymptotes (Khan Academy)
Pre-Class Questions
- In Figure 1.4.1, explain why \( \displaystyle\lim_{x\to 1^+}f(x) \ne f(1)\)
- How can you tell from the graph of y=f(x) if the function f(x) is continuous?
- Why is the Intermediate Value Theorem called the Intermediate Value Theorem?
- Give an example of a function that has a vertical asymptote at x = 2. Explain.
- Give an example of a function that has a horizontal asymptote at y = 3. Explain.
- Which day does your tutorial group meet this week?
Due Sunday September 13 @ midnight
The derivative
To Read
- 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
- Let \( f(x)=3x^2\). Find \( f'(2)\).
- Use the graph of \(f(x)=|x|\) to explain why \( f'(0)\) does not exist.
Due Tuesday September 15 @ midnight
Finding formulas for derivatives
To Read
- Section 2.3 Basic Differentiation Rules
To Watch
- Week 4: Basic differentiation rules (Echo360)
Pre-Class Questions
- If \(f(x)=x^8+\frac{1}{x}-\sin(x)+3\cos(x)\), what is \(f'(x)\)?
- If \(g(x)=e^x\), what is the 42nd derivative of \(g(x)\)?
- Which day does your tutorial group meet this week?
Due Sunday September 20 @ midnight
The product and quotient rules
To Read
- 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.- If \( f(x)=(x^2+7x)(x^4 + 5 x^2 + 9)\), then \( f'(x)=(2x+7)(4x^3+10x)\).
- 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
To Read
- 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.
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
To Read
- Section 3.1 Extreme Values
To Watch
- Week 6: Finding extreme values (Echo360)
Pre-Class Questions
Let \(f(x)=2x^3+3x^2-12x+5\).- Find the critical numbers of \(f(x)\).
- 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
To Read
- 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
- What is an important consequence of the Mean Value Theorem related to finding antiderivatives of a function f(x)?
-
Let f(x)=x3-3x2-9x+7.
- Find the critical numbers of f(x)
- Find the intervals where f(x) is increasing and the intervals where f(x) is decreasing
- 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
To Read
- 3.4 Concavity and the Second Derivative
- 3.5 Curve Sketching
To Watch
- Concavity, Inflection Points, and Second Derivative by The Organic Chemistry Tutor
- Second Derivative Test by The Organic Chemistry Tutor
Pre-Class Questions
Let f(x)=x3-3x2-9x+7. Notice this is the same question from Sunday.- Find the inflection points of f(x)
- Find the intervals where f(x) is concave up and the intervals where f(x) is concave down
- Use the Second Derivative Test to identify each critical number of f(x) as a relative maximum or minimum, if possible.
- Which day does your tutorial group meet this week?
Due Sunday October 11 @ midnight
L'Hôpital's Rule and Optimization
To Read
- 6.7 L'Hôpital's Rule
- 4.3 Optimization
To Watch
- L'Hôpital's Rule by the Organic Chemistry Tutor (focus on first 6 minutes or so)
Pre-Class Questions
- Is the limit \( \displaystyle \lim_{x \to \infty} \frac{e^x}{x} \) in indeterminant form? Explain.
- Is the limit \( \displaystyle \lim_{x \to 0} \frac{\cos(3x)}{x} \) in indeterminant form? Explain.
- 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
To Read
- 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
- What is the purpose of finding the Taylor polynomial for a known function like \(f(x)=\sin(x)\)?
- What is the difference between a Taylor polynomial and a Maclaurin polynomial?
- 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
Logistic Growth
To Watch
- Exponential growth and epidemics by 3Blue1Brown
- More on logistic models (Echo360)
- This next video isn't required, but it's one of the best explanations I saw this spring. It's well worth your 23 minutes.
Simulating an epidemic by 3Blue1Brown
Pre-Class Questions
- Why is a logistic model more accurate than an exponential model when modeling an epidemic?
- 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.
- Which day does your tutorial group meet this week?
Due Sunday October 25 @ midnight
Definite and Indefinite Integrals
To Read
- 5.1 Antiderivatives and Indefinite Integration
- 5.2 The Definite Integral
To Watch
- Definite integrals intro (Khan Academy)
Pre-Class Questions
- Evaluate \( \displaystyle\int 2x + \cos(x) dx\)
- What is the difference between a definite integral and an indefinite integral?
- 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
To Read
- Section 5.3 Riemann Sums
To Watch
- Riemann approximation introduction (Khan Academy)
Due Sunday November 1 @ midnight
The Fundamental Theorem of Calculus
To Read
- 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
- Does every continuous function have an antiderivative? Why or why not?
- 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
To Read
- 6.1 Substitution
You can skip the parts related to the inverse trig functions.
To Watch
- Introduction to u-substition (Khan Academy)
- How to Integrate Using U-Substitution (NancyPi), for some more examples
Pre-Class Questions
- Substitution attempts to undo one of the techniques of differentiation. Which one is it?
- Use \(u\)-substitution to find an antiderivative of \(f(x) = 3x^2\cos(x^3)\)