Welcome back to class! In our September 13th session, we focused heavily on Section 2-5: Continuity and began our comprehensive review for the upcoming Chapter 1 and 2 test. Understanding continuity is the bridge between limits and derivatives, so mastering these concepts is crucial for the rest of Calculus I.
Understanding Continuity
We started by defining exactly what it means for a function $f$ to be continuous at a number $a$. While intuitively it means you can draw the graph without lifting your pencil, the mathematical definition requires three specific conditions to be met:
- $f(a)$ is defined (that is, $a$ is in the domain of $f$).
- $\lim_{x \to a} f(x)$ exists.
- $\lim_{x \to a} f(x) = f(a)$.
If any of these conditions fail, the function is discontinuous at $a$. We looked at specific examples of discontinuities, such as holes (removable discontinuities) and jumps, particularly in piecewise functions.
The Intermediate Value Theorem (IVT)
One of the most important existence theorems we covered is the Intermediate Value Theorem. It states that if $f$ is continuous on a closed interval $[a, b]$, and $N$ is any number strictly between $f(a)$ and $f(b)$, then there must exist a number $c$ in $(a, b)$ such that $f(c) = N$.
In the notes, we applied this to show that the equation $4x^3 - 6x^2 + 3x - 2 = 0$ has a solution between 1 and 2. This is a powerful tool for locating roots of equations even when we can't solve them algebraically.
Chapter 1 & 2 Test Review
To help you prepare for the upcoming exam, I have attached the Chapter 1 and 2 Test Review Packet. This packet covers the essential topics we have learned so far, including:
- Representing Functions: Practice the "Rule of Four" by representing functions like $f(x) = x^2 + 2x - 4$ verbally, numerically, graphically, and algebraically.
- Evaluating Limits: Be prepared to solve limits analytically using factoring and conjugates (e.g., $\lim_{u \to -2} \sqrt{9 - u^3 + 2u^2}$), as well as reading one-sided limits from graphs.
- Infinite Limits: Review how to handle limits approaching infinity or negative infinity.
- Sketching Graphs: You will be asked to sketch functions based on descriptions of their continuity, such as drawing a graph with a "removable discontinuity at 3 and a jump discontinuity at -2."
Class Materials
Please download the lecture notes and the review packet below to study. I highly recommend working through every problem in the review packet before the test.
Download Section 2-5 & 2-6 Lecture Notes (PDF)
Download Chapter 1-2 Review Test Packet (PDF)
Good luck with your studying! If you get stuck on the algebra for the limits or the logic of the IVT, feel free to reach out or ask questions in the next class.