### Confirming Continuity over an Interval: AP Calculus Study Guide

#### Introduction

Hey, Calculus Crusaders! Strap on your mathematical helmets and get ready to dive into the wonderful world of continuity over an interval. Think of it as ensuring our mathematical roller coaster has no crazy jumps, sudden breaks, or those pesky "missing track" moments. 🎢✏️

#### Continuity on an Interval

Let’s start with the basics. According to the College Board (our math overlords), a function is continuous on an interval if it is continuous at every point within that interval. Imagine smoothly tracing a line on your graph without any hiccups or having to lift your pencil—voilà, continuity!

Remember those fancy functions we love so much, like polynomials, rational, power, exponential, logarithmic, and trigonometric functions? They're continuous at every point in their domains. Picture a logarithmic function like ( f(x) = \ln(3x) ). It’s continuous on its domain of ( (0, \infty) ). Easy, right?

#### Continuity for Piecewise Functions

Piecewise functions are like those fancy multi-layered cakes where each layer does its own thing. Calculating the continuity of such functions is crucial. First, check each piece's domain for discontinuities. Then, double-check the points where the function changes its recipe—uh, I mean, its expression.

A piecewise function doesn't use a single equation to define all values. Instead, it uses different formulas for various intervals or subsets of the domain. Imagine it’s like a restaurant menu that changes every hour based on the chef’s whims!

#### Checking Domain Restrictions

Certain functions come with their own set of baggage—restrictions based on their domain. Let’s conquer these one by one.

For square root functions like ( \sqrt{z} ), we can only take the square root of non-negative numbers. So, if a function is ( f(x) = \sqrt{3x + 1} ), the domain is ( x \geq -\frac{1}{3} ). It’s like being invited to a party but only if you’re on the special guest list!

Rational functions involve fractions, and as we all know, dividing by zero is a big no-no. For example, with ( g(x) = \frac{1}{x + 2} ), we need ( x \neq -2 ) to avoid that math disaster. Thus, its domain is ( (-\infty, -2) \cup (-2, \infty) ).

#### Confirming Continuity at a Point

Let’s revisit our friend from Topic 1.11: Confirming Continuity at a Point. We need to ensure the limits from both sides match up. Here’s the formula for you: [ \lim_{{x \to a^{-}}} f(x) = \lim_{{x \to a^{+}}} f(x) = f(a) ] It’s like ensuring both sides of a handshake meet perfectly in the middle. 🤝

#### Practice Problems

**1) Continuity of a Piecewise Function**

Determine if the piecewise function is continuous on the interval ( (-2, 6) ): [ f(x) = \begin{cases} x + 2 & \text{for } x < 3 \ x^2 - 2x + 2 & \text{for } x \geq 3 \end{cases} ]

Step 1: Check domains. Since both expressions are polynomials, they’re continuous within their respective domains.

Step 2: Confirm continuity at ( x = 3 ). Calculate the left-hand limit: [ \lim_{{x \to 3^{-}}} (x + 2) = 5 ] Now, the right-hand limit: [ \lim_{{x \to 3^{+}}} (x^2 - 2x + 2) = 5 ] And finally, the function value: [ f(3) = 5 ] Because all these values match up, the function is continuous at ( x = 3 ).

Step 3: Since we’ve tackled all possible discontinuities, we can confirm the function is continuous on ( (-2, 6) ). 🎉

**2) Continuity of a Rational Function**

Verify if ( f(x) = \frac{3x}{x + 2} ) is continuous on ( (-10, 10) ).

First, find any domain discontinuities. Since ( x = -2 ) would make the denominator zero, the domain is split at ( x = -2 ): [ (-\infty, -2) \cup (-2, \infty) ] Therefore, ( f(x) ) isn’t continuous on ( (-10, 10) ) but instead on intervals that exclude ( x = -2 ). Think of this as the function hitting a “Do Not Enter” sign at -2!

#### Key Terms to Review

**Closed Interval**: A set of real numbers that includes both endpoints, denoted by square brackets [ ].**Continuity**: Describes a function with no breaks, holes, or jumps, like drawing a smooth line without lifting your pencil.**Limit**: The value a function approaches as the input nears a certain point.**Open Interval**: A set of real numbers between two endpoints, where the endpoints are excluded.**Piecewise Functions**: Functions defined by different rules depending on the input intervals.**Rational Functions**: Functions expressed as a ratio of two polynomials.**Simple Polynomials**: Algebraic expressions with non-negative integer powers of variable(s) multiplied by constants.**Trigonometric Functions**: Functions that relate angles to side ratios in a right triangle (e.g., sine, cosine).**Two-sided Limit Test**: Verifies a function's limit at a point by comparing left and right limits.

#### Conclusion

Well done, math warriors! Confirming continuity over an interval is essential for rocking that AP Calculus exam. May your graphs always be smooth, your limits always match, and your pencil never leave the paper! 🏆📚

Now go forth and conquer those continuity questions with confidence and maybe a dash of calculus humor! 🚀