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Definition of Differentiation

Sarah Miller

Sarah Miller

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Next Topic - Instantaneous Rate of Change

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Study Guide Overview

This study guide covers the average rate of change, including its definition, formula (Δy/Δx), and representation using function notation. It provides a worked example and practice questions to reinforce understanding. Key terms like slope are defined in a glossary, and the guide emphasizes the importance of the average rate of change in calculus.

#Average Rate of Change

#Table of Contents

  1. Introduction
  2. Definition and Formula
  3. Function Notation
  4. Worked Example
  5. Practice Questions
  6. Glossary
  7. Summary and Key Takeaways

#Introduction

Understanding the average rate of change is crucial in analyzing how quantities change over intervals. This concept is foundational in calculus and provides insight into the behavior of functions.

#Definition and Formula

#What is the Average Rate of Change?

Average Rate of Change is the slope of the line segment connecting two points on a graph. It is calculated as:

Average Rate of Change=ΔyΔx=y2−y1x2−x1\text{Average Rate of Change} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}Average Rate of Change=ΔxΔy​=x2​−x1​y2​−y1​​

Where (x1,y1)\left( x_1, y_1 \right)(x1​,y1​) and (x2,y2)\left( x_2, y_2 \right)(x2​,y2​) are the coordinates of the two points.

The average rate of change is undefined if the change in xxx is zero.

#Function Notation

#How Can I Write the Average Rate of Change Using Function Notation?

For a function fff:

  • A point with yyy-coordinate f(x)f(x)f(x)
  • A point with yyy-coordinate f(a)f(a)f(a)

The average rate of change can be written as:

f(x)−f(a)x−a\frac{f(x) - f(a)}{x - a}x−af(x)−f(a)​

If the second point is hhh units to the right of aaa:

f(a+h)−f(a)h\frac{f(a + h) - f(a)}{h}hf(a+h)−f(a)​

Exam Tip

When using function notation, it's important to carefully substitute the values and simplify step by step.

#Worked Example

Let fff be the function defined by f(x)=3x3+2x−8f(x) = 3x^3 + 2x - 8f(x)=3x3+2x−8. Find the average rate of change between the point with an xxx-coordinate of -1 and the point with an xxx-coordinate of 2. **Solution:**

Using the formula:

f(2)−f(−1)2−(−1)\frac{f(2) - f(-1)}{2 - (-1)}2−(−1)f(2)−f(−1)​

Evaluate the function at x=2x = 2x=2 and x=−1x = -1x=−1:

f(2)=3(2)3+2(2)−8=24−4=20f(2) = 3(2)^3 + 2(2) - 8 = 24 - 4 = 20f(2)=3(2)3+2(2)−8=24−4=20

f(−1)=3(−1)3+2(−1)−8=−3−2−8=−13f(-1) = 3(-1)^3 + 2(-1) - 8 = -3 - 2 - 8 = -13f(−1)=3(−1)3+2(−1)−8=−3−2−8=−13

Substitute these values back into the formula:

20−(−13)3=333=11\frac{20 - (-13)}{3} = \frac{33}{3} = 11320−(−13)​=333​=11

The average rate of change is 11.

#Practice Questions

Practice Question

Question 1: Find the average rate of change of the function g(t)=t2−4t+6g(t) = t^2 - 4t + 6g(t)=t2−4t+6 between t=1t = 1t=1 and t=4t = 4t=4.

Question 2: Determine the average rate of change for the function h(x)=xh(x) = \sqrt{x}h(x)=x​ from x=9x = 9x=9 to x=16x = 16x=16.

#Glossary

  • Average Rate of Change: The slope of the line segment connecting two points on a graph.
  • Function Notation: A way to represent functions using symbols like f(x)f(x)f(x).
  • Slope: A measure of how steep a line is, calculated as the ratio of the vertical change to the horizontal change.

#Summary and Key Takeaways

  • The average rate of change is a measure of how a function changes between two points.
  • It is calculated as the difference in yyy-values divided by the difference in xxx-values.
  • Using function notation simplifies calculations and helps in understanding the behavior of functions.

#Key Takeaways

  • Understand the formula: y2−y1x2−x1\frac{y_2 - y_1}{x_2 - x_1}x2​−x1​y2​−y1​​
  • Use function notation: f(x)−f(a)x−a\frac{f(x) - f(a)}{x - a}x−af(x)−f(a)​
  • Practice: Apply these concepts to different functions and intervals.
Key Concept

The average rate of change provides a snapshot of how a function behaves over an interval, and is a precursor to understanding derivatives.

Exam Tip

Always check your calculations and ensure you understand each step when working with function notation.

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Ready to calculate? 🚀 What is the average rate of change of a function between the points (1, 4) and (3, 10)?

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