Unit 2: Functions, Graphs, Limits, and Continuity
The concepts of continuity and the meaning of a limit form the foundation for all of calculus. Not only must you understand both of these concepts individually, but you must understand how they relate to each other. They are "twins" in calculus problems: they usually show up together.
A student taking a calculus course during a winter term came up with a great analogy for tying these concepts together: "The weather was raining ice – the kind of weather where no one should be driving a car. The student stepped out on his front porch to see whether the ice rain had stopped, and he couldn't believe his eyes: he saw headlights heading down the road that dead-ended at a brick house. When the car hit the brakes, the student intuited that, at the rate the car's velocity was decreasing (the continuity), there was no way it could stop in time without hitting the house (the limiting value). Oops! However, the student forgot that there was a gravel stretch at the end of the road, so the car stopped before hitting the brick house. The gravel represented a discontinuity in the student's calculations, so his limiting value was not correct."
Completing this unit should take you approximately 7 hours.
2.1: Tangent Lines, Velocities, and Growth
In this section, you will start with finding the slope of a line tangent to a function at a point. This method will not require you to use a graph.
2.2: The Limit of a Function
Understanding limits in calculus is essential. The idea of limit is the basis of learning calculus. In this section, you will learn what the limit of a function is, how to evaluate limits, and familiarize yourself with limit laws.
2.3: Properties of Limits
This section expands on limits. In this section, you will learn how to apply the properties of limits. You will be able to compute limits directly by knowing these properties.
2.4: Continuous Functions
Now that you are familiar with the behavior of functions, you will learn about continuous functions. It is important to be able to identify a continuous function since you will find them in a variety of applications.
2.5: Definition of a Limit
Now that you have used limits, you will learn how to define a limit of a function. Defining the limit of a function will help you verify the limits of some functions.
Unit 2 Assessment
- Receive a grade