Mathematics 3208 (Calculus)

Unit 1: Pre-Calculus

Combining and composing functions, rational functions and their asymptotes, exponential and logarithmic functions, and inverse trigonometric functions - the function theory calculus builds on.

• Combining functions
• Composition of functions

• Rational functions, asymptotes and holes
• Exponential and logarithmic functions

• Inverse trigonometric functions

Unit 2: Limits and Continuity

The concept of a limit, one-sided and infinite limits, evaluating limits algebraically, limits at infinity and asymptotes, continuity, and trigonometric and exponential limits.

• The idea and notation of a limit
• Limits from graphs and tables

• The properties of limits
• Evaluating limits algebraically
• Infinite limits and vertical asymptotes

• Limits at infinity and end behaviour
• Continuity and discontinuities
• Trigonometric and exponential limits

Unit 3: Derivatives and Their Applications

The derivative as a limit and a rate of change, differentiability, the differentiation rules, curve sketching, related rates and optimization, and the derivatives of the trigonometric, inverse-trigonometric, exponential, logarithmic, and hyperbolic functions.

• Average and instantaneous rate of change
• The definition of the derivative
• Tangent lines, notation, and differentiability
• Graphs of a function and its derivative

• The basic differentiation rules
• The chain rule
• Implicit differentiation

• First-derivative analysis of curves
• Concavity and sketching
• Sketching rational functions

• Motion and rates of change
• Related rates
• Optimization

• Derivatives of trigonometric functions
• Derivatives of inverse trigonometric functions
• Derivatives of exponential and logarithmic functions
• Hyperbolic functions

Unit 4: Antidifferentiation and Integration

Antiderivatives and indefinite integrals, applications to motion, area by Riemann sums, and the definite integral and area between curves.

• Antiderivatives and notation
• Particular antiderivatives and motion

• Area by Riemann sums
• Definite integrals and area between curves