MCV4U, Grade 12 Calculus and Vectors

Introduction to CalculusWhat is Calculus?

Reviews of Functions

Basic Classes of Functions

Trigonometric Functions

Inverse Functions

Exponential and Logarithmic Functions

Review Exercises

LimitsIntroduction

Preview of Calculus

The Limit of a Function

The Limit Laws

Continuity

DerivativesIntroduction

The Derivative as a Function

Differentiation Rules

Derivatives as Rates of Change

Derivatives of Trigonometric Functions

The Chain Rule

Derivatives of Inverse Functions

Implicit Differentiation

Derivatives of Exponential and Logarithmic Functions

Derivatives of Exponential, Logarithmic, and Trigonometric FunctionsIntegrals Involving Exponential and Logarithmic Functions

Integrals Resulting in Inverse Trigonometric Functions

Video
Introduction
Science fiction writers often imagine spaceships that can travel to faroff planets in distant galaxies. However, back in 1905, Albert Einstein showed that a limit exists to how fast any object can travel. The problem is that the faster an object moves, the more mass it attains (in the form of energy), according to the equation,
where is the object’s mass at rest, is its speed, and is the speed of light. What is this speed limit? (We explore this problem further in (Figure).)
The idea of a limit is central to all of calculus. We begin this chapter by examining why limits are so important. Then, we go on to describe how to find the limit of a function at a given point. Not all functions have limits at all points, and we discuss what this means and how we can tell if a function does or does not have a limit at a particular value. This chapter has been created in an informal, intuitive fashion, but this is not always enough if we need to prove a mathematical statement involving limits. The last section of this chapter presents the more precise definition of a limit and shows how to prove whether a function has a limit.