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Mathematics LibreTexts

3: The Graphical Behavior of Functions


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Our study of limits led to continuous functions, which is a certain class of functions that behave in a particularly nice way. Limits then gave us an even nicer class of functions, functions that are differentiable. This chapter explores many of the ways we can take advantage of the information that continuous and differentiable functions provide.

  • 3.1: Extreme Values
    Given any quantity described by a function, we are often interested in the largest and/or smallest values that quantity attains. For instance, if a function describes the speed of an object, it seems reasonable to want to know the fastest/slowest the object traveled. If a function describes the value of a stock, we might want to know how the highest/lowest values the stock attained over the past year. We call such values extreme values.
  • 3.2: The Mean Value Theorem
    The mean value theorem states that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. This theorem is used to prove statements about a function on an interval starting from local hypotheses about derivatives at points of the interval.
  • 3.3: Increasing and Decreasing Functions
    In this section we begin to study how functions behave between special points; we begin studying in more detail the shape of their graphs. The first derivative of a function helps determine when the function is going "up" or "down."
  • 3.4: Concavity and the Second Derivative
    We have been learning how the first and second derivatives of a function relate information about the graph of that function. We have found intervals of increasing and decreasing, intervals where the graph is concave up and down, along with the locations of relative extrema and inflection points.
  • 3.5: Curve Sketching
    We have been learning how we can understand the behavior of a function based on its first and second derivatives. While we have been treating the properties of a function separately (increasing and decreasing, concave up and concave down, etc.), we combine them here to produce an accurate graph of the function without plotting lots of extraneous points.
  • 3.E: Applications of Integration (Exercises)


  • Gregory Hartman (Virginia Military Institute). Contributions were made by Troy Siemers and Dimplekumar Chalishajar of VMI and Brian Heinold of Mount Saint Mary's University. This content is copyrighted by a Creative Commons Attribution - Noncommercial (BY-NC) License.