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AP®MathsCollege BoardCalculus ABStudy GuidesUnit 2: Differentiation Definition & Fundamental PropertiesDefinition of DifferentiationInstantaneous Rate of Change

Instantaneous Rate of Change (College Board AP® Calculus AB)

Study Guide

Jamie Wood

Written by: Jamie Wood

Reviewed by: Dan Finlay

Instantaneous rate of change

What is the instantaneous rate of change?

  • The instantaneous rate of change is the slope of a graph at a specific point, rather than between two points

  • Consider the graph of a function f with two points on the graph, A and B

Two graphs showing a positive gradient curve with two points A and B. The left graph shows distance x-a; the right shows distance h between a and a+h.

  • Using the labeling on the left,

    • the average rate of change from A to B can be written as fraction numerator f open parentheses x close parentheses minus f open parentheses a close parentheses over denominator x minus a end fraction

  • Using the labeling on the right,

    • the average rate of change from A to B can be written as fraction numerator f open parentheses a plus h close parentheses minus f open parentheses a close parentheses over denominator h end fraction

  • To find the instantaneous rate of change, consider finding the slope between A and B as point B moves closer to point A

    • As the distance between the two points becomes smaller, the slope will become a more accurate estimate for the instantaneous rate of change at x equals a

    • Therefore the instantaneous rate of change will be the limit as this distance tends to zero

  • The instantaneous rate of change at x equals a can be written as

    • limit as x rightwards arrow a of fraction numerator f open parentheses x close parentheses minus f open parentheses a close parentheses over denominator x minus a end fraction or

    • limit as h rightwards arrow 0 of fraction numerator f open parentheses a plus h close parentheses minus f open parentheses a close parentheses over denominator h end fraction

  • These only give a valid answer if the relevant limit exists

  • They are both equivalent forms of the definition of the derivative of the function at x equals a, denoted by f to the power of apostrophe open parentheses a close parentheses

Worked Example

A function f open parentheses x close parentheses is defined by f open parentheses x close parentheses equals x squared.

Using the equation below, find the instantaneous rate of change of f open parentheses x close parentheses at the point where x equals 2.

f to the power of apostrophe open parentheses x close parentheses equals limit as h rightwards arrow 0 of fraction numerator f open parentheses x plus h close parentheses minus f open parentheses x close parentheses over denominator h end fraction

Answer:

Substitute x equals 2 into the given formula

f to the power of apostrophe open parentheses 2 close parentheses equals limit as h rightwards arrow 0 of fraction numerator f open parentheses 2 plus h close parentheses minus f open parentheses 2 close parentheses over denominator h end fraction

Evaluate the function, f open parentheses x close parentheses equals x squared, at x equals 2 plus h and x equals 2

f to the power of apostrophe open parentheses 2 close parentheses equals limit as h rightwards arrow 0 of fraction numerator open parentheses 2 plus h close parentheses squared minus open parentheses 2 close parentheses squared over denominator h end fraction

Expand and simplify the numerator

f to the power of apostrophe open parentheses 2 close parentheses equals limit as h rightwards arrow 0 of fraction numerator 4 plus 4 h plus h squared minus 4 over denominator h end fraction f to the power of apostrophe open parentheses 2 close parentheses equals limit as h rightwards arrow 0 of fraction numerator 4 h plus h squared over denominator h end fraction

Simplify the fraction by cancelling terms

f to the power of apostrophe open parentheses 2 close parentheses equals limit as h rightwards arrow 0 of open parentheses 4 plus h close parentheses

Evaluate the limit

f to the power of apostrophe open parentheses 2 close parentheses equals 4

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    Author
    Reviewer
    Jamie Wood

    Author: Jamie Wood

    Expertise: Maths

    Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.

    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.