Schedule

Short discussion of the syllabus and this course.

• notation
• evaluation
• domain

We also discussed the interval notation.

1. Read (and keep notes on) our syllabus before the next meeting. If you have any questions, please bring them to that lecture.
2. Read through Chapter 2.1 of the textbook if you want more context about what this course will cover.

• domain calculations
• operations on functions

Review the precalculus. Chapter 1 of the textbook should help.

• intuitive notion
• graphical examples

1. Read (and take notes on) Chapter 2.3 before the next lecture.

2. Complete exercises 46–49 in Chapter 2.2 to turn in at the first in-person lecture.

   Update: see my solutions (printer-friendly).

This was our first in-person lecture following the snow. We spent about ten minutes to have student introductions.

I asked students to take a short diagnostic examination. Here is a bank of practice problems from which I constructed the exam. I know this is an unexciting way to start the semester… However, this is the best way for me to gague student background and thought processes.

In place of our usual meeting, watch the following videos on calculations of limits via algebra.

• Introduction to the Limit Laws
• Limits of Polynomials
• Limits of Rational Functions via Factoring
• Limits via Simplification
• Limits via Conjugates

You can also access the video notes (printer-friendly).

1. The videos have pauses for you to try the problems. Do that…
2. Read (and take notes on) Chapter 2.3 if you haven't yet.

• laws of limits
• polynomial limits
• rational function limits
• piecewise function limits
• examples

1. Solve the Diagnostic Exam again, on clean paper. Let me know if you have issues doing so. This is due at Meeting 08 (Friday, 2 February 2024).
2. Calculate \( \displaystyle \lim_{x \to 1} f(x) \) for \[ f(x) = \begin{cases} 5 + x^3 & \text{if } x \leq -2 \\ x^2 & \text{if } -2 < x < 1 \\ 2 -x & \text{if } 1 \leq x \end{cases}. \]

• practice problems (group work) with solutions on the board
• a method for dealing with infinite limits

Note: The exercises in Chapter 2.3 are your friend. Most of the odd-numbered exercises also come with solutions that you can access by following the link on the problem number!

Solve these exercises from Chapter 2.3: 98, 100, 104, 106, 108, 110, 112 (due at Meeting 09 on Monday, 5 February 2024).

Update: My solutions are here (printer-friendly).

• rules for calculating infinite limits involving infinite pieces
• examples of such calculations

1. Calculate the limits below for \( f(x) = \frac{x^2 + 3x + 2}{x(x+2)^3} \) (due at Meeting 09 on Monday, 5 February 2024 with the homework assigned in meeting 07).
   1. \( \displaystyle \lim_{x \to -2} f(x) \)
   2. \( \displaystyle \lim_{x \to 0} f(x) \)
   3. \( \displaystyle \lim_{x \to -1^{+}} f(x) \)
2. Read about the Squeeze Theorem in Chapter 2.3 for next lecture.

Based on the diagnostic exam results, I decided to take the first half of today's lecture to review some trigonometry. Topics we discussed include…

• terminology
• Pythagorean Theorem
  • In a right triangle with leg lengths \( a, b \) and hypoteneuse \( c \) we have \( a^2 + b^2 = c^2 \).
• trigonometric ratios and their relationships
• unit circle
  • \( \cos^2(\theta) + \sin^2(\theta) = 1 \)
  • \( -1 \leq \cos(\theta) \leq 1 \)
  • \( -1 \leq \sin(\theta) \leq 1 \)
• calculating trigonometric ratios using reference triangles
• Arc–Angle Formula
  • In a circular sector with arc length \( s \), radius \( r \) and angle \( \theta \) we have \( s = r\theta \).
  • this relationship is why we prefer radians to degrees

Read about the Squeeze Theorem in Chapter 2.3 of the textbook. Pay careful attention to the examples involving the trigonometric functions.

• examples

Try exercises 126, 127, and 128 in Chapter 2.3 and read my solutions (printer-friendly).

• Introduction to Continuity
• More Examples of Continuity Calculations
• If you like, you can read the notes from the video (printer-friendly).
• Read Chapter 2.4 by Wednesday, 14 February 2024. For this lecture, focus on the first two subsections (Continuity at a Point and Types of Discontinuities). Make sure to read and take notes on the examples!

Try exercises 139, 141, 144, 131, 132, 133, 134, 138, 145, 147,and 149 in Chapter 2.4.

I've also written solutions (printer-friendly) to these exercises. Let me know if you have questions.

• Continuity of Trigonometric Functions
• Properties of Continuity
• If you like, you can read the notes from the video (printer-friendly).
• Read Chapter 2.4 by Wednesday, 14 February 2024. For this lecture, focus on the last two subsections (Continuity over an Interval). Make sure to read and take notes on the examples!

Try exercises 139, 141, 144, 131, 132, 133, 134, 138, 145, 147,and 149 in Chapter 2.4 (this is the same list of exercises as before, so just finish any you haven't worked through yet).

I've also written solutions (printer-friendly) to these exercises. Let me know if you have questions.

Students asked questions concerning continuity. We solved some problems.

Attempt the following practice problems.

Graphical Limit Computations

Chapter 2.2: 55–67, and see my solutions (printer-friendly).

Algebraic Limit Computations

Chapter 2.3: 83, 85, 87–89, 93, 95, 97, 99, 101, 103, 105, 107–114, 127, and see my solutions (printer-friendly).

Continuity

Chapter 2.4: 133, 137, 139, 141–142, 145, 147, and see my solutions (printer-friendly).

Students should bring questions to this lecture.

During the usual class period. Be sure to arrive a few minutes early to get settled.

• review of lines from precalculus
• average rate of change (slope of the secant line)
• instantaneous rate of change (slope of the tangent line)
• derivation of the slope of the tangent line
• limit definition of the derivative

Read Chapter 3.1 in the textbook.

These videos all include calculations of derivatives using the definition.

• Introductory Calculations
• Alternative Difference Quotient (Square Root Example)
• Alternative Difference Quotient (Polynomial Examples)
• Tangent Line Calculations
• Velocity Given Position

You can also take a look at the notes from the videos (printer-friendly).

• Solve Chapter 3.1 exercises 6, 16, and 28 to turn in on Wednesday, 28 February 2024.

• the derivative as a function
• computations of derivatives via the limit definition
• group practice and quiz

• Read Chapter 3.2 of the textbook. You can skip the section on "graphing derivatives" (unless you want to read it).
• Solve Chapter 3.2 exercises 62, 72, and 76 to turn in on Monday, 4 March 2024.
• Update: You can now read my solutions (printer-friendly).

I gave Exam 1 back at the end of the lecture period.

In an unprecedented act of kindness, I have decided to allow students to write corrections to their exam problems. If you want to receive partial credit for these corrections, we will meet before Wednesday, 6 March 2024 and you will present your corrections to me.

• higher derivatives
• differentiable implies continuous
• group practice

• Solve Chapter 3.2 exercises 62, 72, 76, and 83 to turn in on Monday, 4 March 2024 (the first three were originally assigned on Wednesday, 28 February 2024).
• Read and take notes on Chapter 3.3.
• Update: You can now read my solutions (printer-friendly).

• the absolute value function is continuous but not differentiable at \( x = 0 \)

• Constant Rule
• Sum/Difference Rule
• Constant Multiple Rule
• Power Rule
• many examples, followed by a quiz

• Read and take notes on Chapter 3.3.
• I gave students amnesty on Chapter 3.2 exercise 83 until Wednesday, 6 March 2024.

• derivatives of \( \sin(x) \) and \( \cos(x) \)
• Product Rule

• Read and take notes on Chapter 3.3 (if you haven't already done so).

• Product Rule
• Quotient Rule
• Group practice and quiz

Turn the following problems in on Wednesday, 20 March 2024 at the beginning of lecture.

• Solve Chapter 3.3 exercises 106–116 (evens), and 128.
• Calculate the derivative of \( \tan(x) \).
• Begin studying for the second exam.
• Update: You can now read my solutions (printer-friendly).

Turn the following problems in on Friday, 22 March 2024 at the beginning of lecture.

• Solve Chapter 3.5 exercises 182, 186 (do not graph), and 192.
• Continue studying for the coming exam.
• Update: You can read my solutions (printer-friendly).

• review of function compositions
• recognizing function composition
• derivatives of composite functions
• groups solved these practice problems

Turn the following problems in on Monday, 25 March 2024.

• Solve Chapter 3.6 exercises 228, 232, 234.

  NOTE: I didn't collect these exercises until Wednesday, 27 March 2024.

• Study for the second exam.
• Update: You can read my solutions (printer-friendly).

• reminder of physical interpretations of the derivative
• word problems involving derivatives and their meaning
• groups solved these practice problems

For Wednesday, 27 March 2024…

• Solve Chapter 3.4 exercise 156.
• Continue to study for the second exam.

• word problems involving derivatives and their meaning

• Study for the coming exam.
• Prepare questions for the review session.

This lecture is a student-driven review session for Exam 2.

These are some good practice problems. They are NOT to be turned in, but they are good for studying. I've linked solutions I wrote (I'm human, so sometimes I make mistakes: let me know if you think you find one).

Secant and Tangent Lines

Chapter 3.1: 1–9 (odd), 11–19 (odd), 21–29 (odd), 35, 37, 41, 43. See my solutions (printer-friendly).

Derivative as a Function

Chapter 3.2: 55–63 (odd), 69–73 (odd), 75, 77, 81–83, 85–89 (don't graph). See my solutions (printer-friendly).

Derivative Rules

Chapter 3.3: 106–117, 119, 121, 122–125, 126–129, 133, 135, 137, 139, 141. See my solutions (printer-friendly).

Derivatives as Rates of Change

Chapter 3.4: 151–157 (odd), 161, 163. See my solutions (printer-friendly).

Derivatives of Trigonometric Functions

Chapter 3.5: 175–183 (odd), 185–189 (odd). 191–196, 201. See my solutions (printer-friendly).

Chain Rule

Chapter 3.6: 215–219 (odd), 221–227 (odd), 229–237 (odd), 239, 245–252. See my solutions (printer-friendly).

During the usual class period.

Update: Exams were returned on Wednesday, 3 April 2024.

• Critical points
• Intervals of increase/decrease

Turn the following problems in on Friday, 5 April 2024.

For each of the following functions (A) calculate all critical points of the function, and (B) calculate the intervals of increase and decrease of the function.

1. \( f(x) = 4\sqrt{x} - x^2 \)
2. \( f(x) = \frac{x^2 - 1}{x^2 + 2x - 3} \)

• definitions of extrema, maxima, and minima
• difference between local and absolute extrema
• using the first derivative test to find local extrema
• using the bounded interval method to find absolute extrema

Turn the following problems in on Monday, 8 April 2024.

• Calculate the absolute maxima and minima of \( f(x) = (x^2 - x - 6)^2 \) on \( [-1, 4] \).
• Find and classify all local minima and maxima of the function \( f(x) = 3x^4 + 8x^3 - 18x^2 \).

• meaning of concavity
• second derivative and its connection to concavity
• points of inflection

Turn the following problems in on Wednesday, 10 April 2024.

For \( f(x) = x^3 + x^4 \), calculate…

• the intervals on which \( f \) is increasing/decreasing,
• all local extrema of \( f \),
• the intervals on which \( f \) is concave up/down, and
• all points of inflection of \( f \).

• another example calculating points of inflection and intervals of concavity

• limits at edges of domain

Turn the following problems in on Friday, 12 April 2024.

1. For \( f(x) = \frac{1}{x + x^2} \), calculate…
   • the intervals on which \( f \) is increasing/decreasing,
   • all local extrema of \( f \),
   • the intervals on which \( f \) is concave up/down, and
   • all points of inflection of \( f \).
2. Calculate the asymptotic behaviour of the function \( f(x) = \frac{1}{7-x} \) near its edges of domain.

We again went over a few of the fundamentals of factorization. This includes converting between factors and zeroes, and polynomial division.

• Finding intercepts
• Example: \( f(x) = x^3 - 3x^2 + 4 \) (partially completed)

• first derivative information
• second derivative information
• asymptotic information
• sketching the curve
• Example: \( f(x) = x^3 - 3x^2 + 4 \)
• Example: \( f(x) = \frac{3x + 5}{2x - 1} \)

Turn the following problems in on Friday, 19 April 2024.

Sketch the curves below using the methods from class. Show all work, and ensure your final picture agrees with your work.

• \( y = (x+5) (x-3)^2 (x-2)^3 \)
  • HINT: the second derivative is somewhat nasty for this problem, so I'll provide you with approximations of the zeroes: \( y''(x) = 0 \) when \( x = 2 \), \( x \approx -2.55 \), \( x \approx 2.36 \), and \( x \approx 2.85 \). You should use these to make your sign chart.
• \( y = \frac{x}{x^2-4} \)

• rational function example
• horizontal asymptotes
• vertical asymptotes

• using derivatives to calculate limits of indeterminate types
• only works for \( \frac{\infty}{\infty} \)-type and \( \frac{0}{0} \)-type limits
• other pitfalls (derivatives need to exist, denominator cannot be "too nasty")
• several examples

Turn the following problems in on Friday, 19 April 2024.

• The problems assigned in the prior lecture.
• Calculate the following limits.
  • \( \lim_{x \to 0} \frac{x^2-7x}{x^3+1} \)
  • \( \lim_{x \to \pi} \frac{\cos(x)+1}{x-\pi} \)

Using calculus to optimize real-world quantities.

Turn the following problems in on Friday, 26 April 2024.

• Calculate the maximum area of a rectangle with perimeter \( 10 \) meters.

More using calculus to solve real-world problems.

More complicated examples of real-world optimization.

Some final examples of real-world optimization.

This is a review session for the final exam. Note that the final exam is CUMULATIVE, but there is an emphasis on the topics following exam 2.

Here are the sections we've covered in the textbook. The problems in those sections are nice.

• Chapter 4.3: Maxima and Minima
• Chapter 4.5: Derivatives and the Shape of a Graph
• Chapter 4.6: Limits at Infinity and Asymptotes
• Chapter 4.7: Applied Optimization Problems
• Chapter 4.8: L'Hospital's Rule

This course meets in Canonical Hour A. As such, the final is scheduled by the registrar for Monday, 6 May 2024 at 19:00 in our usual classroom.

This course meets in Canonical Hour B. As such, the final is scheduled by the registrar for Friday, 3 May 2024 at 14:00 in our usual classroom.