Reimagine Calculus

Introduction

1 The Language of Change

1.1 The Nature of Change

1.2 Linear vs. Nonlinear Change

1.3 Functions: The Language of Change

1.4 Linear Function vs. Nonlinear Function

Checkpoint 1

Checkpoint 1 solution

2 The Slope of a Tangent Line 2

2.1 Understanding Slope on a Curve: Two Approaches

2.2 Limit: The Key to the Tangent Line

2.3 Exploring the Limit of a Function

2.4 Using Limits to Find the Slope at a Single Point

2.5 Checkpoint 2

2.6 Checkpoint 2 Solution

3 Limits and Continuity

3.1 Understanding Limits and Continuity

3.2 When Functions Are Continuous

3.3 Continuity and Differentiability

Checkpoint 3

Checkpoint 3 Solution

4 Instantaneous Rate of Change

4.1 Understanding the Rate of Change

4.2 Instantaneous Rate of Change for Linear Functions

4.3 Instantaneous Rate of Change for Nonlinear Functions

4.4 Checkpoint 4

4.5 Checkpoint 4 Solution

5 Derivative

5.1 Derivatives Explained Step by Step

5.2 Standard Definition of the Derivative

5.3 Derivative as a Function

5.4 Common Symbols for Derivatives

5.5 Checkpoint 5

5.6 Checkpoint 5 Solution

6 Differentiation Techniques

6.1 The Power Rule: An Easy Way to Differentiate

6.2 Differentiating Sums and Differences of Functions

6.3 How to Use the Power Rule in Practice

6.4 Checkpoint 6

6.5 Checkpoint 6 Solution

7 Product and Quotient Rules

7.1 Product Rule

7.2 Quotient Rule

7.3 Checkpoint 7

7.4 Checkpoint 7 Solution

8 Chain Rule

8.1 Understanding and Applying the Chain Rule

8.2 Using Substitution with the Chain Rule

8.3 Checkpoint 8

8.4 Checkpoint 8 Solution

9 Understanding Change in Real-World Contexts

9.1 Rate of Change in Business Applications

9.2 Understanding the Motion of Two Moving Objects

9.3 Optimization

9.4 Checkpoint 9

9.5 Checkpoint 9 Solution

Closing Thoughts

Summary

Author's Note on References

Notes

Bibliography