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Chapter Three: Section Three Increasing and Decreasing Functions and the First Derivative Test

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Chapter Three: Section Three Increasing and Decreasing Functions and the First Derivative Test Chapter Three: Section Three What do we mean if we say that a function ... – PowerPoint PPT presentation

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Title: Chapter Three: Section Three Increasing and Decreasing Functions and the First Derivative Test


1
Chapter Three Section ThreeIncreasing and
Decreasing Functions and the First Derivative
Test
2
Chapter Three Section Three
  • What do we mean if we say that a function is
    increasing? Informally, I think that we can all
    agree that this means the function has a positive
    slope. Formally, we say the following
  • A function f is increasing on a region if
    whenever a lt b, then f(a) lt f(b) on that region

3
Chapter Three Section Three
  • Similarly, we can discuss when a function is
    decreasing. Informally, we would say that the
    slope of the curve is negative. Formally, we say
    the following
  • A function is decreasing on an interval if
    whenever a lt b, then f(b) lt f(a).

4
Chapter Three Section Three
  • So, how does the derivative of a function tell us
    when a function is increasing or decreasing?
  • We will use a tool known as the sign chart to
    analyze a function and see how the derivative
    tells us about the physical behavior of the
    function.

5
Chapter Three Section Three
  • Lets look at this function
  • To discuss when this function is increasing or
    decreasing, we must first know the derivative of
    the function.

6
Chapter Three Section Three
  • The derivative of the function is
  • The zeros of this function are x 1 and x -1.
  • What we want to know is whether the function is
    positive or negative over the regions on the
    number line formed by these two critical values.

7
Chapter Three Section Three

8
Chapter Three Section Three
  • What you see on the preceding slide is a helpful
    (at least I think it is helpful) visualization of
    the graphical behavior of the function.
  • From the left-hand extreme of the function until
    the critical value of x -1, the function is
    consistently increasing. Between the critical
    values of x -1 and x 1, the function is
    decreasing. After that point, the function rises
    on the rest of its domain.
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