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Defining Average and Instantaneous Rates of Change at a Point

Hannah Hill

Hannah Hill

7 min read

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Study Guide Overview

This study guide covers average and instantaneous rates of change in calculus. It explains how to calculate the average rate of change using the secant line slope and the instantaneous rate of change using the tangent line slope and the limit definition of the derivative. Practice problems and examples are provided for both concepts. A summary table compares the two types of rates of change. Finally, exam tips and high-priority topics are highlighted including the limit definition, calculations, and secant/tangent line relationships.

AP Calculus AB/BC: Differentiation - Rates of Change 🚀

Hey there, future calculus champ! Let's get you prepped for Unit 2 with a super focused review of average and instantaneous rates of change. Think of this as your ultimate cheat sheet for acing the exam. Let's dive in!

2.1: Average vs. Instantaneous Rate of Change

✈️ Average Rate of Change

Think of the average rate of change as the slope of a secant line between two points on a curve. It tells you the overall trend of a function over an interval. It's like calculating your average speed on a road trip – total distance divided by total time.

Quick Fact

Average rate of change is the slope of the secant line.

Average Rate of Change

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For a function f(x)f(x) over an interval [a,b][a, b], the average rate of change is:

f(b)f(a)ba\frac{f(b) - f(a)}{b - a}

Memory Aid

Remember "rise over run"! It's the change in y divided by the change in x.

⛷️ Instantaneous Rate of Change

The instantaneous rate of change is the slope of the tangent line at a single point. It tells you how the function is changing at that exact moment. Imagine looking at your speedometer – that's your instantaneous speed at that precise time. In calculus, we use the derivative to find this.

Quick Fact

Instantaneous rate of change is the slope of the tangent line and is found using the derivative.

The instantaneous rate of change of f(x)f(x) at x=cx = c is denoted as f(c)f'(c) and is given by:

f(c)=limh0f(c+h)f(c)hf'(c) = \lim_{h \to 0} \frac{f(c + h) - f(c)}{h}

Key Concept

This limit definition of the derivative is super important!...

Question 1 of 11

What does the average rate of change represent graphically? 📈

The slope of the tangent line

The area under the curve

The slope of the secant line

The derivative of the function