First Derivative Test

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First Derivative Test

• Lesson 4.3

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Increasing/Decreasing Functions

• Consider the following function
• For all x lt a we note that x1ltx2 guarantees that
  f(x1) lt f(x2)

The function is said to be strictly increasing
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Increasing/Decreasing Functions

• Similarly -- For all x gt a we note that x1ltx2
  guarantees that f(x1) gt f(x2)
• If a function is either strictly decreasing or
  strictly increasing on an interval, it is said to
  be monotonic

The function is said to be strictly decreasing
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Monotone Function Theorem

• If a function is differentiable and f (x) gt 0
  for all x on an interval, then it is strictly
  increasing
• If a function is differentiable and f (x) lt 0
  for all x on an interval, then it is strictly
  decreasing
• Consider how to find the intervals where the
  derivative is either negative or positive

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Monotone Function Theorem

• Finding intervals where the derivative is
  negative or positive
• Find f (x)
• Determine where
• Try for
• Where is f(x) strictly increasing / decreasing

• f (x) 0
• f (x) gt 0
• f (x) lt 0
• f (x) does not exist

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Monotone Function Theorem

• Determine f (x)
• Note graphof f(x)
• Where is it pos, neg
• What does this tell us about f(x)

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First Derivative Test

• Given that f (x) 0 at x 3, x -2, and x
  5.25
• How could we find whether these points are
  relative max or min?
• Check f (x) close to (left and right) the point
  in question
• Thus, relative min

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First Derivative Test

• Similarly, if f (x) gt 0 on left, f (x) lt 0 on
  right,
• We have a relative maximum

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First Derivative Test

• What if they are positive on both sides of the
  point in question?
• This is called aninflection point

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Examples

• Consider the following function
• Determine f (x)
• Set f (x) 0, solve
• Find intervals

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Algorithm for Curve Sketching

• Determine critical points
• Places where f (x) 0
• Plot these points on f(x)
• Determine where function is
• increasing
• decreasing
• resulting relative maximums, minimums
• Sketch

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Assignment

• Lesson 4.3
• Page 244
• Exercises 3, 5, 7, 9, 15, 19, 23,
  29, 33, 41, 45, 47