Calculus

UNIT 1

Foundations for Calculus

Build the algebra, functions and graphical thinking needed for calculus.

1. What Calculus Is
2. Functions and Their Graphs
3. Function Notation
4. Domain and Range
5. Rates of Change
6. Secant Lines
7. Preparing for Limits

UNIT 2

Limits

Understand how functions behave as inputs approach a value.

1. The Idea of a Limit
2. Limits from Tables
3. Limits from Graphs
4. One-Sided Limits
5. Limits Algebraically
6. Limits at Infinity
7. Infinite Limits

UNIT 3

Continuity

Study when functions behave without breaks, jumps or holes.

1. What Continuity Means
2. Continuity at a Point
3. Continuity on an Interval
4. Removable Discontinuities
5. Jump Discontinuities
6. Infinite Discontinuities
7. The Intermediate Value Theorem

UNIT 4

The Derivative

Develop the derivative from instantaneous rate of change.

1. Average vs Instantaneous Change
2. Tangent Lines
3. Derivative from First Principles
4. Derivative Notation
5. Differentiability
6. Estimating Derivatives
7. Meaning of the Derivative

UNIT 5

Basic Differentiation Rules

Differentiate common algebraic functions efficiently.

1. Constant Rule
2. Power Rule
3. Constant Multiple Rule
4. Sum and Difference Rules
5. Polynomial Derivatives
6. Higher-Order Derivatives
7. Derivative Practice

UNIT 6

Product, Quotient and Chain Rules

Differentiate products, quotients and compositions of functions.

1. Product Rule
2. Quotient Rule
3. Function Composition Review
4. Chain Rule
5. Multiple Chain Rules
6. Combining Differentiation Rules
7. Advanced Derivative Practice

UNIT 7

Trigonometric Derivatives

Differentiate sine, cosine and the other trigonometric functions.

1. Derivative of Sine
2. Derivative of Cosine
3. Derivative of Tangent
4. Derivatives of Secant and Cosecant
5. Derivative of Cotangent
6. Trig Functions with the Chain Rule
7. Trig Derivative Applications

UNIT 8

Exponential and Logarithmic Derivatives

Differentiate exponential and logarithmic functions.

1. Derivative of e^x
2. General Exponential Derivatives
3. Derivative of ln(x)
4. General Logarithmic Derivatives
5. Logarithmic Differentiation
6. Exponential Chain Rule
7. Growth and Decay Rates

UNIT 9

Implicit and Related Rates

Differentiate relationships that are not solved explicitly for one variable.

1. Implicit Differentiation
2. Implicit Curves
3. Tangent Lines to Implicit Curves
4. What Related Rates Are
5. Geometry Related Rates
6. Motion Related Rates
7. Applied Related Rates

UNIT 10

Applications of Derivatives

Use derivatives to understand motion, shape and optimization.

1. Critical Points
2. Increasing and Decreasing Functions
3. Local Extrema
4. Concavity
5. Inflection Points
6. Optimization
7. Curve Sketching

UNIT 11

Antiderivatives and Indefinite Integrals

Reverse differentiation and build the idea of accumulation.

1. What an Antiderivative Is
2. Basic Antiderivative Rules
3. Power Rule for Integration
4. Constant of Integration
5. Integrating Polynomials
6. Basic Trig Antiderivatives
7. Indefinite Integral Practice

UNIT 12

Definite Integrals

Measure accumulated change and signed area.

1. Area Under a Curve
2. Riemann Sums
3. Left and Right Sums
4. Midpoint Sums
5. Definite Integral Notation
6. Properties of Definite Integrals
7. Numerical Integration

UNIT 13

Fundamental Theorem of Calculus

Connect derivatives and integrals as inverse processes.

1. Why Derivatives and Integrals Connect
2. Fundamental Theorem Part I
3. Fundamental Theorem Part II
4. Evaluating Definite Integrals
5. Accumulation Functions
6. Net Change
7. Fundamental Theorem Applications

UNIT 14

Techniques and Applications of Integration

Use integration to solve geometric and physical problems.

1. Substitution
2. Area Between Curves
3. Volumes by Disks
4. Volumes by Washers
5. Average Value of a Function
6. Position from Velocity
7. Applied Integration

UNIT 15

Advanced Calculus Applications

Combine calculus ideas in science, engineering, economics and modeling.

1. Motion and Kinematics
2. Optimization in Engineering
3. Population Growth
4. Economics and Marginal Analysis
5. Physics and Accumulation
6. Differential Equations Introduction
7. Comprehensive Calculus Problem Solving