4'2 The Mean Value Theorem

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4.2 The Mean Value Theorem

• p. 186-194

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Theorem 3 The Mean Value Theorem for
Derivatives

• If y?(x) is continuous at every point of the
  closed interval a,b and differentiable at every
  point of its interior (a,b), then there is at
  least one point c in (a,b) at which

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Example 1

• Show that the function (x)x2/3 satisfies the
  hypotheses of the Mean Value Theorem on the
  interval 0,1. Then find a solution c to the
  equation

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What is the Mean Value Theorem really Saying???

• Lets say Ms. Bartley buys a brand new red sports
  car. If she starts accelerating from zero, it
  takes 8 seconds for her to go 352 ft, her average
  velocity for the 8-sec interval is
  352ft/8sec44ft/sec, or 30 mph. This means that
  at some point during the acceleration, the
  theorem says, her speedometer must read exactly
  30 mph.

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Increasing and Decreasing Functions

• Let be continuous on a,b and differentiable
  on (a,b).
• If ?gt0 at each point of (a,b), then increases
  on a,b.
• If ?lt0 at each point of (a,b), then decreases
  on a,b.

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Example 2
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Example 3
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Example 4
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A function that is always increasing on an
interval or always decreasing on an interval is
said to be Monotonic.
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Assignment

• p. 192-193
• 1-9 odd, 15,18, 20, 36bd, 39, 51a