# Chapter Three: Section Three Increasing and Decreasing Functions and the First Derivative Test - PowerPoint PPT Presentation

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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.