Chapter 6: Antiderivatives and Applications
6.4 Undoing the Chain Rule: Integration by Substitution
Calculus for the Life Sciences — University of Rhode Island
Preface
Chapter 1: Functions as Models of Change
1.1 What Makes a Relationship a Function?
1.2 Straight-Line Behavior: Linear Functions
1.3 Measuring Change: Average Rates
1.4 Growth That Builds on Itself: Exponential Functions
1.5 Unpacking Exponentials: The Natural Logarithm
Chapter 2: The Derivative Concept
2.1 Zooming In: Instantaneous Rate of Change
2.2 The Derivative as a Function
2.3 Limits: Making ‘Instantaneous’ Precise
2.4 Defining the Derivative with Limits
2.5 Continuity: When Functions Behave
2.6 Reading the Derivative: Units and Interpretation
2.7 Acceleration and Concavity: The Second Derivative
Chapter 3: Computing Derivatives
3.1 Shortcuts: Derivatives of Powers and Polynomials
3.2 Derivatives of Exponential and Logarithmic Functions
3.3 Combining Functions: The Chain Rule
3.4 Products and Quotients
3.5 Derivatives of Trigonometric Functions
3.6 Synthesis: Choosing and Combining Techniques
Chapter 4: Using Derivatives
4.1 Finding Peaks and Valleys: Local Extrema
4.2 Bending Points: Inflection and Concavity
4.3 The Best Possible: Global Optimization
4.4 Building the Model: Optimization Word Problems
Chapter 5: The Definite Integral
5.1 Accumulating Change: Distance and Area
5.2 The Definite Integral as a Limit
5.3 When Curves Dip Below: Signed Area
5.4 Making Sense of Integrals: Units and Meaning
5.5 The Fundamental Theorem: Connecting Derivatives and Integrals
Chapter 6: Antiderivatives and Applications
6.1 Antiderivatives from Graphs and Tables
6.2 Finding Antiderivatives: The Indefinite Integral
6.3 Evaluating Definite Integrals with the FTC
6.4 Undoing the Chain Rule: Integration by Substitution
Special Topic A: Modeling Boom and Fade
A.1 Logistic Growth: When Growth Has Limits
A.2 The Surge Function: Fast Rise, Slow Fade
Chapter 6: Antiderivatives and Applications
6.4 Undoing the Chain Rule: Integration by Substitution
6.4 Undoing the Chain Rule: Integration by Substitution
This section is under development.
6.3 Evaluating Definite Integrals with the FTC
A.1 Logistic Growth: When Growth Has Limits