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Pre-Class Assignments, Math 101 Calculus I, Spring 2022

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.

Week 1: Due Thursday January 27 @ 11:59 pm

An intuitive introduction to derivatives

To Read
To Watch
Optional Resources

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. Have you filled out the Background Questionairre yet? (link in onCourse)
Submit answers through onCourse

Week 2: Due Sunday January 30 @ 11:59 pm

Review of exponentials, logarithms, and trigonometric functions

To Read
Optional Resources

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. 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?
Submit answers through onCourse

Week 3: Due Sunday February 6 @ 11:59 pm

Limits, continuity, and asymptotes

To Read

Note that these sections are from APEX Calculus.

This is the longest assignment of the semester by far. Try to read/skim these sections for the main ideas, and don't get too hung up in the details in your first reading. We'll discuss all the most important points thoroughly during class. You can also ignore any references to the ε - δ definition of the limit in these sections.

Optional Resources

Pre-Class Questions

  1. Explain why \( \displaystyle\lim_{x\to -3} \frac{x^2-9}{x+3} = -6 \)
  2. If \( f(x)=x^2\), explain why \( \displaystyle\lim_{h\to 0} \frac{f(5+h) -f(5)}{h} = 10\)
  3. In Figure 1.4.1, explain why \( \displaystyle\lim_{x\to 1^+}f(x) \ne f(1)\)
  4. How can you tell from the graph of y=f(x) if the function f(x) is continuous?
  5. Give an example of a function that has a vertical asymptote at x = 2. Explain.
Submit answers through onCourse

Week 4: Due Sunday February 13 @ 11:59 pm

The derivative and basic differentiation rules

To Read
To Watch

Pre-Class Questions

  1. Let \( f(x)=3x^2\). Use the limit definition of the derivative (Definition 2.1.1 on page 62) to find \( f'(2)\).
  2. Use the graph of \(f(x)=|x|\) to explain why \( f'(0)\) does not exist.
  3. If \(f(x)=x^8+\frac{1}{x}-\sin(x)+3\cos(x)\), what is \(f'(x)\)?
  4. If \(g(x)=e^x\), what is the 42nd derivative of \(g(x)\)?
Submit answers through onCourse

Week 5: Due Sunday February 20 @ 11:59 pm

The product, quotient, and chain rules

To Read
To Watch
Optional Resources

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 \( f'(x)=\displaystyle\frac{2x+7}{4x^3+10x}\).

  3. If \(f(x)=(x^2+2x)^{130}\), then \(f'(x)=130(x^2+2x)^{129}\).
Submit answers through onCourse

Week 6: Due Sunday February 27 @ 11:59 pm

Extreme values and the Mean Value Theorem

To Read
To Watch

Pre-Class Questions

  1. 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 \([-1,2]\).
  2. What is an important consequence of the Mean Value Theorem related to finding antiderivatives of a function f(x)?
Submit answers through onCourse

Week 7: Due Sunday March 6 @ 11:59 pm

Derivatives, concavity, and curve sketching

To Read
Optional Resources

Pre-Class Questions

  1. 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
  2. Using the same function f(x),
    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.
Submit answers through onCourse

Week 8: Due Sunday March 20 @ 11:59 pm

L'Hôpital's Rule and Optimization

To Read
Optional Resources

Pre-Class Questions

  1. Is the limit \( \displaystyle \lim_{x \to \infty} \frac{e^x}{x} \) in indeterminate form? Explain.
  2. Is the limit \( \displaystyle \lim_{x \to 0} \frac{\cos(3x)}{x} \) in indeterminate form? Explain.
  3. No specific questions on optimization, but pay attention to what the text calls the fundamental equation in the examples in Section 4.3.
Submit answers through onCourse

Week 9: Due Sunday March 27 @ 11:59 pm

Taylor and Maclaurin Polynomials

To Read
To Watch

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 \( P_3(x) \)?
Submit answers through onCourse

Week 10: Due Sunday April 3 @ 11:59 pm

Definite and Indefinite Integrals

To Read
Optional Resources

Pre-Class Questions

Think about these, but no need to submit your answers. Focus on reviewing for Exam 2 this week.

  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.

Week 11: Due Sunday April 10 @ 11:59 pm

Riemann Sums and the Fundamental Theorem of Calculus

To Watch
To Read
Optional Resources

Pre-Class Questions

  1. What is the purpose of a Riemann sum?
  2. Will a Right Hand Rule sum overestimate or underestimate \(\displaystyle\int_0^2 x^2 dx\)? Explain.
  3. Does every continuous function have an antiderivative? Why or why not?
  4. 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.
Submit answers through onCourse

Week 12: Due Sunday April 17 @ 11:59 pm

Antidifferentiation by Substitution

To Read
Optional Resources

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 \(f(x) = 3x^2\cos(x^3)\)
Submit answers through onCourse

Week 13: Due Sunday April 24 @ 11:59 pm

Modeling and applications

To Watch

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.

Week 14: For Sunday May 1

For the last week of the semester we'll be reviewing and looking at the Big Picture of Calculus I. No Pre-Class Assignments for this week.


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