• 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.

    • Upon successful completion of this unit, you should be able to:

      • explain the limit of a function;
      • determine the slope of the line tangent to the graph of a function at a point;
      • determine the values of one- or two-sided limits for a function given by a graph;
      • use algebraic methods to determine the values of one- and two-sided limits for a function given by a formula or state that the limit "does not exist";
      • state whether a given function is continuous at a point, and use the properties of continuity to find limits and values of related functions;
      • use the Intermediate Value Theorem to determine the number of times a function has a given value;
      • approximate the roots of functions using the Bisection Algorithm;
      • state the epsilon-delta definition of limit; and
      • find the required delta graphically and algebraically for linear and quadratic functions for a given epsilon.

    • 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.

      • Tangent Lines, Velocities, and Growth Book

        Read this section for an introduction to connecting derivatives to quantities we can see in the real world. Work through practice problems 1-4.

      • Slope of a Function at a Point Page
        Watch this video on how to find the slope of the line tangent to the graph of a function at a point.
      • Practice Problems Book

        Work through the odd-numbered problems 1-9. Once you have completed the problem set, check your answers for the odd-numbered questions.

    • 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.

      • Properties of Limits Book
        Read this section to learn about the properties of limits. Work through practice problems 1-6.
      • Solving Limits (Rationalization) Page

        Watch this video on finding limits algebraically. Be warned that removing x-4 from the numerator and denominator in Step 4 of this video is only legal inside this limit. The function \frac{x - 4}{x - 4} is not defined at x = 4 ; however, when x is not 4, it simplifies to 1. Because the limit as x approaches 4 depends only on values of x different from 4, inside that limit \frac{x - 4}{x - 4} and 1 are interchangeable. Outside that limit, they are not! However, this kind of cancellation is a key technique for finding limits of algebraically complicated functions.
      • Calculating Slope of Tangent Line Using the Definition of a Derivative Page

        Watch this video on limits as the slopes of tangent lines.

        Before you watch, know that, for this problem, the limit that gives the slope of the tangent line to a curve is y = f(x) at a point x = a , which is the derivative of f(x) at a . We will talk about this more in Unit 3.
      • Practice Problems Book

        Work through the odd-numbered problems 1-21. Once you have completed the problem set, check your answers.

    • 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.

      • Definition of a Limit Book
        Read this section to learn how a limit is defined. Work through practice problems 1-6.
      • Epsilon-Delta Limits Page
        Watch this video to learn the epsilon-delta definition of a limit.
      • Practice Problems Book

        Work through the odd-numbered problems 1-23. Once you have completed the problem set, check your answers.

    • Unit 2 Assessment

      • Unit 2 Assessment Quiz
        Receive a grade

        Take this assessment to see how well you understood these concepts.

        • This assessment does not count towards your grade. It is just for practice!
        • You will see the correct answers when you submit your answers. Use this to help you study for the final exam!
        • You can take this assessment as many times as you want, whenever you want.