#AP Calculus AB/BC: Types of Discontinuities - Your Ultimate Guide 🚀

Hey there, future AP Calculus master! Let's dive into the world of discontinuities. This guide is designed to be your go-to resource, especially the night before the exam. We'll make sure everything clicks and you feel confident. Let's get started!

This topic is crucial for understanding function behavior and is frequently tested on both multiple-choice and free-response questions. Understanding discontinuities is key to mastering limits and continuity.

#Introduction to Discontinuities

First, let's nail down what a discontinuity actually is. Remember, for a function to be continuous, three things must be true:

1. It must be defined at all x values in its domain.
2. The limit must exist at all points in its domain.
3. The limit from the left must equal the limit from the right at all points in its domain.

If any of these rules are broken, you've got a discontinuity! Think of it like this: if you have to lift your pencil while drawing a graph, you've hit a discontinuity. ✏️

#Removable Discontinuities (aka "Holes")

These are the simplest type of discontinuity, often called "holes" in the graph. They happen when a single point is out of place. 🕳️

There are two main reasons for removable discontinuities:

1. "Blips" in the function, often caused by piecewise definitions.
2. Common factors in rational functions (fractions).

#Example 1: Piecewise "Blip"

Consider this piecewise function:

$$y = \begin{cases} x^2 & \text{if } x < 1 \\ x - 1 & \text{if } x = 1 \\ -x + 2 & \text{if } x > 1 \end{cases}$$

Here's what it looks like:

• Caption: Notice the "hole" at x=1. The function is not defined at x=1 because the point (1,0) is not on the line. You have to lift your pencil to draw this point.

#Example 2: Common Factor

Let's look at another example:

$$y = \begin{cases} \frac{x^3 - 3x^2 + 2x}{x - 1} & \text{if } 1 > x > 1 \\ x & \text{if } x = 1 \end{cases}$$

Here's the graph:

• Caption: The function simplifies to $y = x^2 - 2x$ after factoring out $(x-1)$, but there's still a hole at $x=1$. Even though the piecewise function defines the value at x=1, the original function has a discontinuity.

#Jump Discontinuities

Imagine a sudden vertical "jump" in the graph. That's a jump discontinuity! It occurs when the left-hand limit does not equal the right-hand limit. Think of it like a staircase or a sudden drop-off. 🦘

#Example of a Jump Discontinuity

Consider this piecewise function:

$$y = \begin{cases} x - 2 & \text{if } x \leq 3 \\ x + 1 & \text{if } x > 3 \end{cases}$$

Here's the graph:

• Caption: Notice the jump at x=3. The left side approaches 1, while the right side approaches 4. The filled circle indicates the function's value at x=3. ## Asymptote Discontinuities

These occur when the function's limits approach infinity (either positive or negative) as x approaches a certain value. The graph has a vertical asymptote, where the function shoots up or down like a rocket! 🚀

#Example of an Asymptote Discontinuity

Let's look at the classic example: $y = \frac{1}{x}$

Here's the graph:

• Caption: Notice the vertical asymptote at x=0. The function approaches infinity as x approaches 0 from both sides.

#Recap: Identifying Discontinuities

Here's a quick summary to help you identify each type of discontinuity:

Removable Discontinuities

• A single point "missing" from the graph.
• Often found in rational functions after factoring.
• The limit at the point does not exist in the original function.

Jump Discontinuities

• The left-hand limit does not equal the right-hand limit.
• The graph has a clear vertical "jump".

Asymptote Discontinuities

• The function approaches infinity (or negative infinity) as x approaches a certain value.
• The graph has a vertical asymptote.

#Final Exam Focus

Okay, let's focus on what's most important for the exam:

• High-Priority Topics: Discontinuities are often combined with limits and continuity, so make sure you understand all three concepts. Pay special attention to piecewise functions and rational functions.

• Common Question Types: Expect questions that ask you to identify the type of discontinuity, find the x-value where a discontinuity occurs, and explain why a function is discontinuous at a given point. You might also be asked to redefine a function to remove a discontinuity.

• Time Management: Practice identifying discontinuities quickly. Look for common factors, piecewise definitions, and potential asymptotes.

#Practice Questions

Let's test your knowledge with some practice questions!

Practice Question

Multiple Choice Questions

1. The function $f(x) = \frac{x^2 - 4}{x - 2}$ has a discontinuity at $x = 2$. Which type of discontinuity is it? (A) Jump discontinuity (B) Asymptote discontinuity (C) Removable discontinuity (D) Continuous

2. Which of the following functions has a jump discontinuity at $x = 0$? (A) $f(x) = \frac{1}{x}$ (B) $g(x) = \begin{cases} x & \text{if } x < 0 \\ x^2 & \text{if } x \geq 0 \end{cases}$ (C) $h(x) = \frac{x^2}{x}$ (D) $k(x) = \begin{cases} x+1 & \text{if } x < 0 \\ x-1 & \text{if } x \geq 0 \end{cases}$

3. The function $f(x)$ is defined as follows: $f(x) = \begin{cases} x^2+1, & x<1 \\ 3, & x=1 \\ 5-x, & x>1 \end{cases}$. Which of the following statements is true about the continuity of $f(x)$ at $x=1$? (A) $f(x)$ is continuous at $x=1$ (B) $f(x)$ has a jump discontinuity at $x=1$ (C) $f(x)$ has a removable discontinuity at $x=1$ (D) $f(x)$ has an infinite discontinuity at $x=1$

Free Response Question

Consider the function:

$$f(x) = \begin{cases} \frac{x^2 - 9}{x - 3} & \text{if } x < 3 \\ 2x + a & \text{if } x \geq 3 \end{cases}$$

(a) Find the limit of $f(x)$ as $x$ approaches 3 from the left. (b) Find the value of $a$ that makes the function continuous at $x = 3$. (c) With the value of $a$ found in part (b), is $f(x)$ differentiable at $x=3$? Justify your answer.

Scoring Breakdown

(a) 2 points * 1 point for factoring and simplifying the expression * 1 point for evaluating the limit (b) 2 points * 1 point for setting the left-hand limit equal to the right-hand limit * 1 point for solving for $a$ (c) 5 points * 1 point for finding the derivative of each piece * 1 point for evaluating the derivative at x=3 from the left * 1 point for evaluating the derivative at x=3 from the right * 1 point for comparing the derivatives from the left and the right * 1 point for correct conclusion and justification

You've got this! Remember, practice makes perfect. Go ace that exam! 🏆