Calculus: Course Contents

1. Functions and Their Graphs

1.1 Function basics

• 1.1.1 Definition of functions, domain, range (14 pages)
• 1.1.2 Addition, subtraction, multiplication and divisions of functions (7 pages)
• 1.1.3 Composition of functions (12 pages)
• 1.1.4 Inverses of functions (18 pages)
• 1.1.5 Graphs of combined functions (20 pages)

1.2 Powers, roots and polynomials

• 1.2.1 Properties of exponents
• 1.2.2 Power functions and their graphs
• 1.2.3 Root functions and their graphs
• 1.2.4 Polynomials

1.3 Trigonometric functions

• 1.3.1 Basic definitions and properties
• 1.3.2 Trigonometric identities
• 1.3.3 Graphs of trigonometric functions
• 1.3.4 Inverse trigonometric functions

1.4 Exponential and logarithm functions

• 1.4.1 Exponential functions
• 1.4.2 Logarithm functions
• 1.4.3 Natural exponential and logarithm functions
• 1.4.4 Graphs of exponential and logarithm functions
• 1.4.5 Hyperbolic and inverse hyperbolic functions

1.5 Polar and parametric curves

• 1.5.1 Definition of polar coordinates
• 1.5.2 Equations in polar coordinates
• 1.5.3 Special polar curves
• 1.5.4 Definition of parametric curves
• 1.5.5 Special parametric curves
• 1.5.6 Polar curves as parametric curves

1.6 Conic sections

• 1.6.1 General form of a conic section
• 1.6.2 Parabolas
• 1.6.3 Ellipses
• 1.6.4 Hyperbolas
• 1.6.5 Degenerate conics
• 1.6.6 Conics in polar coordinates, eccentricity

2. Limits and Continuity

2.1 Definition and calculation of a limit

• 2.1.1 Intuitive definition of a limit
• 2.1.2 Limits at infinity and infinite limits
• 2.1.3 Rules for calculating limits
• 2.1.4 One-sided limits
• 2.1.5 The squeeze law and the basic trigonometric limits
• 2.1.6 The epsilon-delta definition of a limit

2.2 Definition of continuity

• 2.2.1 Intuitive definition of continuity
• 2.2.2 Formal definition of continuity
• 2.2.3 Continuity of the basic functions
• 2.2.4 Combinations of continuous functions
• 2.2.5 Intermediate value theorem
• 2.2.6 One-sided continuity

3. Definition and Interpretation of Derivatives

3.1 Derivative at a point

• 3.1.1 Definition of a derivative at a point
• 3.1.2 Calculation of derivatives at a point

3.2 Derivative of a function

• 3.2.1 Definition of the derivative of a function
• 3.2.2 Calculation of the derivative of a function
• 3.2.3 Higher order derivatives
• 3.2.4 Differentiability implies continuity

3.3 Interpretations of derivatives

• 3.3.1 Tangent lines
• 3.3.2 Velocity and acceleration
• 3.3.3 Rates of change

4. Calculation Rules for Derivatives

4.1 Rules for differentiating basic functions

• 4.1.1 Derivatives of constants, powers and roots
• 4.1.2 Derivatives of trigonometric functions
• 4.1.3 Derivatives of exponential functions
• 4.1.4 Derivatives of logarithm functions
• 4.1.5 Derivatives of inverse trigonometric functions
• 4.1.6 Derivatives of hyperbolic and inverse hyperbolic functions

4.2 Rules for differentiating combinations of functions

• 4.2.1 Rules for sums, differences and constant multiples
• 4.2.2 The product rule and the quotient rule
• 4.2.3 The chain rule

4.3 Rules for differentiating implicitly defined functions

• 4.3.1 Implicit differentiation
• 4.3.1 Related rates problems
• 4.3.3 Logarithmic differentiation
• 4.3.4 Derivatives of inverse functions
• 4.3.5 Slopes of parametric curves

5. Using Limits and Derivatives to Analyze Curves

5.1 Asymptotes

• 5.1.1 Horizontal asymptotes
• 5.1.2 Vertical asymptotes
• 5.1.3 Oblique asymptotes

5.2 Monotonicity

• 5.2.1 Increasing and decreasing functions
• 5.2.2 Critical points
• 5.2.3 Relative extrema

5.3 Concavity

• 5.3.1 Concave up and concave down functions
• 5.3.2 Points of inflection
• 5.3.3 Second derivative test

5.4 Graphing curves

• 5.4.1 Domain and range
• 5.4.2 Symmetry
• 5.4.3 Intercepts and asymptotes, behaviour at infinity
• 5.4.4 Regions of monotonicity and extrema
• 5.4.5 Sign of concavity and inflection points
• 5.4.6 Curve sketching strategy

6. Applications of Derivatives

6.1 Optimizing Functions

• 6.1.1 Absolute extrema
• 6.1.2 Second derivative test
• 6.1.3 Word problems

6.2 Mean value theorem

• 6.2.1 Rolle's theorem
• 6.2.2 The mean value theorem
• 6.2.3 Consequences of the mean value theorem

6.3 Newton's method

6.4 Differentials and linear approximations

6.5 Taylor polynomials

6.6 L'Hopital's rule

• 6.6.1 Basic 0/0 and ∞/∞ forms
• 6.6.2 0.∞ form
• 6.6.3 00, ∞0 and 1∞ forms