Calculus

4.

Applications of Differentiation

4.1.

4.1.1.

Setting Up Related Rates Problems

4.1.1.1.

Identifying Variables and Relationships

4.1.1.2.

Drawing Diagrams

4.1.2.

Solving Related Rates Problems

4.1.2.1.

Differentiation with Respect to Time

4.1.2.2.

Common Related Rates Scenarios

4.1.3.

4.1.3.1.

Geometric Problems

4.1.3.2.

Physics Problems

4.1.3.3.

Business and Economics

4.2.

Linear Approximation and Differentials

4.2.1.

Linear Approximation

4.2.1.1.

Tangent Line Approximation

4.2.1.2.

Local Linearity

4.2.2.

4.2.2.1.

Definition of Differentials

4.2.2.2.

Relationship to Linear Approximation

4.2.3.

4.2.3.1.

Absolute and Relative Error

4.2.3.2.

Propagation of Error

4.3.

4.3.1.

Absolute Extrema

4.3.1.1.

Definition on Closed Intervals

4.3.1.2.

The Extreme Value Theorem

4.3.2.

4.3.2.1.

Definition of Local Maximum and Minimum

4.3.2.2.

Fermat's Theorem

4.3.3.

Critical Points

4.3.3.1.

Definition and Finding Critical Points

4.3.3.2.

Types of Critical Points

4.3.4.

The Closed Interval Method

4.3.4.1.

Steps for Finding Absolute Extrema

4.4.

The Mean Value Theorem

4.4.1.

Rolle's Theorem

4.4.1.1.

Statement and Geometric Interpretation

4.4.2.

The Mean Value Theorem

4.4.2.1.

Statement and Proof Outline

4.4.2.2.

Geometric and Physical Interpretation

4.4.2.3.

Consequences of the Mean Value Theorem

4.5.

Analyzing Functions with Derivatives

4.5.1.

4.5.1.1.

Increasing and Decreasing Functions

4.5.1.2.

The First Derivative Test for Monotonicity

4.5.1.3.

Finding Intervals of Increase and Decrease

4.5.2.

4.5.2.1.

Definition of Concave Up and Concave Down

4.5.2.2.

The Second Derivative Test for Concavity

4.5.2.3.

Inflection Points

4.5.3.

The First Derivative Test

4.5.3.1.

Using f' to Classify Critical Points

4.5.3.2.

Applications to Optimization

4.5.4.

The Second Derivative Test

4.5.4.1.

Using f'' to Classify Critical Points

4.5.4.2.

When the Test Fails

4.6.

Curve Sketching

4.6.1.

Systematic Approach to Graphing

4.6.1.1.

Domain and Range

4.6.1.2.

4.6.1.3.

4.6.1.4.

4.6.1.5.

Critical Points and Extrema

4.6.1.6.

Inflection Points and Concavity

4.6.2.

Putting It All Together

4.6.2.1.

Step-by-Step Graphing Process

4.6.2.2.

Common Function Families

4.7.

Optimization Problems

4.7.1.

Setting Up Optimization Problems

4.7.1.1.

Identifying the Objective Function

4.7.1.2.

Identifying Constraints

4.7.1.3.

Expressing as Single-Variable Problems

4.7.2.

Solving Optimization Problems

4.7.2.1.

Finding Critical Points

4.7.2.2.

Checking Endpoints and Boundaries

4.7.2.3.

Verifying Solutions

4.7.3.

4.7.3.1.

Geometric Optimization

4.7.3.2.

Business and Economics

4.7.3.3.

Physics and Engineering

4.8.

Indeterminate Forms and L'Hôpital's Rule

4.8.1.

Indeterminate Forms

4.8.1.1.

4.8.1.2.

4.8.1.3.

Other Indeterminate Forms

4.8.2.

L'Hôpital's Rule

4.8.2.1.

Statement and Conditions

4.8.2.2.

Repeated Applications

4.8.2.3.

When L'Hôpital's Rule Doesn't Apply

4.9.

Newton's Method

4.9.1.

4.9.1.1.

Derivation and Formula

4.9.1.2.

Geometric Interpretation

4.9.2.

4.9.2.1.

Choosing Initial Approximations

4.9.2.2.

Iteration Process

4.9.3.

Convergence and Limitations

4.9.3.1.

When Newton's Method Fails

4.9.3.2.

Rate of Convergence

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5. Antiderivatives and Indefinite Integration