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Chapter 5 - Applications of Derivatives

5.1 Extreme Values of Functions
5.2 Mean Value Theorem
5.3 Connecting f ' and f'' with the Graph of f
5.4 Modeling and Optimization
5.5 Linearization, Sensitivity, and Differentials
5.6 Related Rates
Chapter 5 Reviews
The derivative is  supremely useful because there are so many ways to think of it.  It is the instantaneous rate of change, so velocity and acceleration can be found when give a position function, it also describes sensitivity of how change in the input variable is reflected in the change of the output variable, and it has geometric meaning as the slope of the tangent to the graph of a function.  This chapter explores how all these understandings of the derivative lead to applications.

The derivative tells us a great deal about the shape of a curve.  The derivative gives us precise information about how the curve bends and exactly where it turns.  Rates of change will be deduced where we cannot measure from rates of change we already know, and how to find a function when we know only its first derivative and its value at a single point.  The key to recovering functions from derivatives is the Mean Value Theorem, the theorem whose corollaries provide the gateway to integral calculus, which begins in chapter 6.

This chapter begins where the early developers of calculus began, with the derivative as the slope of the tangent line and the insights this gives us into finding the greatest and least values of a function.

​
(Taken from an excerpt in the fifth edition of the Finney, Demana, Waits, Kennedy, and Bressoud textbook Calculus - Graphical, Numerical, Algebraic.)

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  • Home
  • Integrated Math 3
    • Homework Calendars
    • Chapter 2 - Linear Equations, Inequalities, and Functions >
      • 2-1/2-2 Solving Equations/Inequalities
      • 2-3 Linear Functions and Slope-Intercept Form
      • 2-4 Families of Functions
      • 2-5 Absolute Value Functions and Graphs
      • 2-6 Two-Variable Inequalities
      • 2-7 Solving Systems of Equations
      • 2-8 Systems of Inequalities
      • 2-9 Systems With Three Variables
      • Chapter 2 Reviews
    • Chapter 3 - Quadratic Functions and Equations >
      • 3-1 Quadratic Functions and Transformations
      • 3-2 Standard Form of a Quadratic Function
      • 3-3 Factoring Quadratic Expressions
      • 3-3b Intercept Form
      • 3-4 Quadratic Equations
      • Graphing Quadratics Handouts
      • 3-5 Completing the Square
      • 3-6 The Quadratic Formula
      • Chapter 3 Reviews
    • Chapter 4 Polynomials and Polynomial Functions >
      • 4-1 Polynomial Functions
      • 4-2 Adding, Subtracting, and Multiplying Polynomials
      • 4-3 Polynomials, Linear Factors, and Zeros
      • 4-4 Solving Polynomial Equations
      • 4-5 Dividing Polynomials
      • 4-6 Theorems About Roots of Polynomial Equations
      • Chapter 4 Handouts
      • Chapter 4 Reviews
    • Chapter 5 - Rational Expressions and Functions >
      • 5-1 Simplifying Rational Expressions
      • 5-2 Multiplying and Dividing Rational Expressions
      • 5-3 Adding and Subtracting Rational Expressions
      • 5-5 The Reciprocal Function Family
      • 5-7 Solving Rational Equations
      • Chapter 5 Handouts
      • Chapter 5 Reviews
    • Chapter 6 - Radical Expressions and Functions >
      • 6-1 Roots and Radical Expressions
      • 6-2 Multiplying and Dividing Radical Expressions
      • 6-3 Binomial Radical Expressions
      • 6-4 Rational Exponents
      • 6-5 Solving Square Root and Other Radical Equations
      • 6-6 Functions Operations
      • 6-7 Inverse Relations and Functions
      • 6-8 Graphing Radical Functions
      • Chapter 6 Handouts
      • Chapter 6 Reviews
    • Chapter 7 - Exponential and Logarithmic Functions >
      • 7-1 Exploring Exponential Models
      • 7-2 Properties of Exponential Functions
      • 7-3 Logarithmic Functions as Inverses
      • 7-4 Properties of Logarithms
      • 7-5 Exponential and Logarithmic Equations
      • 7-6 Natural Logarithms
      • Chapter 7 Handouts
      • Chapter 7 Reviews
    • Chapter 8 - Trigonometric Functions >
      • 8-0 Right Triangle Trigonometry
      • 8-1 Exploring Periodic Data
      • 8-2 Angles and the Unit Circle
      • 8-3 Radian Measure
      • Chapter 8 Handouts
      • Chapter 8 Reviews
    • Spring 2019 Final Project
  • About
  • Contact