INCREASING AND DECREASING FUNCTIONS AND THE FIRST DERIVATIVE TEST

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INCREASING AND DECREASING FUNCTIONS AND THE FIRST
DERIVATIVE TEST

• Section 3.3

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When you are done with your homework, you should
be able to

• Determine intervals on which a function is
  increasing or decreasing
• Apply the first derivative test to find relative
  extrema of a function

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Ptolemy lived in 150AD. He devised the 1st
accurate description of the solar system. What
branch of mathematics did he create?

• Calculus
• Number Theory
• Trigonometry
• Statistics

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Definitions of Increasing and Decreasing Functions

• A function f is increasing on an interval if for
  any two numbers and in the interval,
  implies .
• A function is decreasing on an interval if for
  any two numbers and in the interval,
  implies .
• How does this relate to the derivative?!

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Theorem Test for Increasing and Decreasing
Functions

• Let f be a function that is continuous on the
  closed interval and differentiable on the
  open interval
• If for all , then f is
  increasing on .
• If for all , then f is
  decreasing on .
• If for all , then f is
  constant on .

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Guidelines for Finding Intervals on which a
Function is Increasing or Decreasing

• Let f be continuous on the interval To
  find the open intervals on which f is increasing
  or decreasing, use the following steps.
• Locate the critical numbers of in
  and use these numbers to determine the test
  intervals.
• Determine the sign of at one test value
  in each of the intervals.
• Use the previous theorem to determine whether f
  is increasing or decreasing on each interval.
• These guidelines are also valid if the interval
  
• is replaced by an interval of the form

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The weight (in pounds) of a newborn infant during
its 1st three months of life can be modeled by
the equation below, where t is measured in
months. Determine when the infant was gaining
weight and losing weight.

• The infant lost weight for approximately the
  first month and then gained for the 2nd and 3rd.
• The infant lost weight for approximately .56
  month and then gained thereafter.
• The infant gained weight for approximately the
  first month and then lost weight during the 2nd
  and 3rd months.
• The infant lost weight for approximately .56
  month and then gained weight during the 2nd and
  3rd months.

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Fibonacci lived in 1200. Why was he famous?

• He introduced the Hindu-Arabic system of
  numeration (aka base 10)
• He developed the study of numbers in the form of
  1, 1, 2, 3, 5, 8, 13, 21, . . .
• He found all integer solutions to
• All of the above.

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Theorem The First Derivative Test

• Let c be a critical number of a function f that
  is continuous on an open interval I containing c.
  If f is differentiable on the interval, except
  possibly at c, then can be classified as
  follows
•  If changes from negative to positive at
  c, then f has a relative minimum at
• If changes from positive to negative at
  c, then f has a relative maximum at
• If is positive on both sides of c or
  negative on both sides of c, then is
  neither a relative minimum or relative maximum.

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Find the relative extrema of the function and the
intervals on which it is increasing or decreasing.

• Relative min _at_ (5, 0), relative max _at_
  (5, -75/16), increasing on
  , decreasing on .
• Relative max _at_ (5, 0), relative min _at_
  (5, -75/16), increasing on ,
  decreasing on .
• None of the above