Calculus

The Limit Concept

1. Intuitive Limits Using Graphs

2. Estimating Limits Using Numbers

Evaluating Limits Algebraically

1. Algebraic Limits: Piecewise-Defined Functions

2. Algebraic Limits: Rational Functions

3. Algebraic Limits: Complex Fractions

4. Algebraic Limits: Absolute Value

5. Algebraic Limits: Rationalizing Square Roots

The Squeeze Theorem

1. The Squeeze Theorem

The Epsilon-Delta Definition of a Limit

1. The Epsilon-Delta Definition of a Limit

2. Finding a Delta Given an Epsilon

3. Proof of a Limit Value Using Epsilon and Delta

Continuity

1. Continuity of a Function

2. Continuity on an Interval

3. Continuity of Well-Known Functions

4. Example: Showing Where a Function is Continuous

5. The Intermediate Value Theorem

6. A Basic Proof Using the Intermediate Value Theorem

Limits at Infinity

1. Limits at Infinity: Ratios

2. Limits at Infinity: Roots

3. Evaluating a Limit at Infinity by Using the Squeeze Theorem

4. Evaluating a Limit at Infinity for Arctan

The Limit Definition of Derivative

1. The Limit Definition of a Derivative

2. Example: Derivative & Tangent Line Equation

3. Examples: Using Limits to Find Derivative Formulas

4. The Derivative as a Function

5. Differentiation

6. Differentiable Functions

7. Derivative Values from a Function Graph

8. Graphing a Derivative

The Power Rule and Polynomial Derivatives

1. Derivatives of Constant Multiples, Sums, and Differences

2. The Derivative Power Rule & Polynomials

The Natural Exponential Derivative

1. The Derivative of the Natural Exponential Function

The Product Rule

1. The Product Rule

The Quotient Rule

1. The Quotient Rule

Trigonometric Derivatives

1. Derivatives of Sine and Cosine

2. Trigonometric Derivatives

The Chain Rule

1. The Chain Rule

Exponential Derivatives

1. Derivatives of Exponential Functions

Implicit Differentiation

1. Implicit Differentiation

2. Implicit Differentiation Examples

Inverse Trigonometric Derivatives

1. Derivatives of the Inverse Trigonometric Functions

Logarithmic Derivatives

1. Derivatives of Logarithmic Functions

2. Logarithmic Differentiation

Related Rates

1. Introduction to Related Rates with an Example Using Pythagorean Theorem

2. Related Rates Example Using Area of a Circle and Radius

3. Related Rates Example Using Similar Triangles

4. Related Rates - Tip of Shadow

5. Related Rates Example Using Trigonometry

Critical Numbers; Maximum and Minimum

1. Maximum and Minimum Values of a Function

2. Critical Numbers

3. Absolute Maximum and Minimum Values of a Function

Rolle's and the Mean Value Theorem

1. Rolle's Theorem

2. The Mean Value Theorem

Derivatives and Graphing

1. What the Derivative Tells Us About the Graph of a Function

2. Using Derivatives to Graph a Function

L'Hospital's Rule

1. L'Hospital's Rule

2. L'Hospital's Rule: Products, Sums, and Differences

Optimization

1. Optimization Explained Using a Basic Example

2. Optimization Example: Maximum and Minimum Vertical Distances Between Functions

3. Optimization Example: Minimizing Fencing

4. Optimization Example: Cylinder

5. Optimization Example: Two-Dimensional Distance

6. Optimization Example: Inscribing a Rectangle Inside a Triangle

Linear Approximation

1. Linear Approximations

Newton's Method

1. The Concept of Newton's Method

2. Newton's Method Examples

Antiderivatives

1. Introduction to Antiderivatives Using Constant and Power Functions

2. Antiderivatives of Exponential Functions

3. Basic Antiderivatives with Trigonometric Functions

4. Antiderivatives with Inverse Trigonometric Functions

5. Finding a Function Given One of its Derivatives

6. Finding a Particular Antiderivative Given Initial Conditions

Antiderivatives

1. Introduction to Antiderivatives Using Constant and Power Functions

2. Antiderivatives of Exponential Functions

3. Basic Antiderivatives with Trigonometric Functions

4. Antiderivatives with Inverse Trigonometric Functions

5. Finding a Function Given One of its Derivatives

6. Finding a Particular Antiderivative Given Initial Conditions

Area Under a Curve and the Definite Integral

1. Approximating the Area Under a Curve Using Rectangles

2. The Definite Integral and the Riemann Sum

3. Examples: Definite Integrals and Sums

4. Evaluating a Definite Integral Using Geometry

The Fundamental Theorem of Calculus

1. Basic Properties of Definite Integrals

2. The Fundamental Theorem of Calculus: Part 1 of 2

3. Exploring an "Area so Far" Function from FTC: Part 1

4. The Fundamental Theorem of Calculus: Part 2 of 2

Definite vs Indefinite Integrals

1. Integrals: Definite vs Indefinite

Integral Applications

1. Integrals and Net Change

Integral Techniques

1. Evaluating Indefinite Integrals Using Substitution

2. Evaluating Definite Integrals Using Substitution

Areas and Volumes

1. Concept: Area Between Curves

2. Examples: Area Between Curves

3. Concept: Volume by Disks

4. Examples: Volume by Disks

5. Concept: Volume by Washers

6. Examples: Volume by Washers

7. Concept: Volume by Cylindrical Shells

8. Examples: Volume by Cylindrical Shells

The Mean Value Theorem for Integrals

1. The Mean Value Theorem for Integrals and the Average Value of a Function

2. Example: Average Value of a Function (Integrate by Substitution)

Functions of Multiple Variables