2. Applications of Differentiation

Introduction

This unit describes techniques for using differentiation to solve many important problems. Part C of this unit presents the Mean Value Theorem and introduces notation and concepts used in the study of integration, the subject of the next two units.

Part A: Approximation and Curve Sketching

Part B: Optimization, Related Rates and Newton’s Method

Part C: Mean Value Theorem, Antiderivatives and Differential Equations

Exam 2

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Congratulations! You’ve made it through Unit 2. Now it’s time to test your knowledge. Begin with the review, and when you’re ready, take the exam. When you’re done, use the included solutions to check your answers.

» Session 41: Review for Exam 2
» Session 42: Materials for Exam 2

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This section describes how differentiation can be used to simplify complex calculations and graph functions.

» Session 23: Linear Approximation
» Session 24: Examples of Linear Approximation
» Session 25: Introduction to Quadratic Approximation
» Session 26: Using Quadratic Approximations
» Session 27: Sketching Graphs I - Polynomials and Rational Functions
» Session 28: Sketching Graphs II - General Strategies
» Problem Set 3

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The study of the greatest and least outputs of a function – highest profits, least materials used, closest approach – is called optimization. In addition to optimization, this part of the course also describes how to use derivatives to relate two rates of change and to estimate the values for which a function’s output is zero.

» Session 29: Optimization Problems
» Session 30: Optimization Problems II
» Session 31: Related Rates
» Session 32: Ring on a String > » Session 33: Newton’s Method
» Problem Set 4

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The Mean Value Theorem is the key to proving that our abstract definition of a “derivative” faithfully describes our informal notion of a “rate of change.” The second half of this part of the course introduces notation for and discusses the possibility of reversing the process of differentiation.

» Session 34: Introduction to the Mean Value Theorem
» Session 35: Using the Mean Value Theorem
» Session 36: Differentials > » Session 37: Antiderivatives
» Session 38: Integration by Substitution > » Session 39: Introduction to Differential Equations > » Session 40: Separation of Variables > » Problem Set 5

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Single Variable Calculus