Chapter 3 Applications of Derivatives

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Chapter 3 Applications of Derivatives

• 3.1 Extrema

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Extrema

• p154 Packages/Containers
• Goal ___________ the Surface Area
• (and thus the cost) for a given Volume.
• Notice the low point on the graph.

MINIMIZE
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Extrema

• p210 16
• Goal __________ Area for animals
• Love your dog and/or cat!

MAXIMIZE
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Extrema

• p233 24
• Goal What path will __________ the cost of the
  pipeline?
• Avoid higher gas prices!

MINIMIZE
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Extrema

• p212 41
• Goal Find the angle to __________ the range
  for a projectile.
• such as a Patriot missile aimed at Saddam
  Hussein

MAXIMIZE
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Extrema

• p210 18
• Goal Next week we will each make an open box
  attempting to the VOLUME.

MAXIMIZE
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Extrema Concept

• A maximum point is the
• ________ point on a graph,
• and
• a minimum point is the
• ________ point on a graph.

HIGHEST
LOWEST
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Extrema

• There are Two kinds of Extrema
• Relative Extrema
• vs.
• Absolute Extrema

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Extrema

• Refer to page 156 top left graph
• Relative Extrema are
• mountain peaks and valleys.
• (A function may or may not have them.)

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Extrema

• Refer to the 3 graphs on page 155.

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Extrema

• Absolute Extrema
• may be the same as the
• Relative Extrema
• or
• they may occur at endpoints of an interval.
• (A function may or may not have them.)

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Extrema

• Indicate where a minimum or maximum occurs IN
  TERMS OF X.
• The minimum value or maximum value
• of a function is the y value.

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Extrema

• KNOW THIS DEFINITION
• Critical Number
• Let f be ________ at x c.
• If _____ or if _________ ,
• then c is a critical number of f.

defined
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Extrema

• RELATIVE EXTREMA OCCUR ONLY AT CRITICAL
  NUMBERS.
• .BUT, NOT ______ CRITICAL NUMBERS LEAD TO
  RELATIVE EXTREMA.

ALL
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PROCEDURETo find Absolute Extrema of a
function f on an interval a, b

• 1) Find the critical numbers of f on a,b.
• Differentiate f.
• Set the derivative equal to zero and solve.
• Set the denominator (if there is one) equal
  to zero, and solve.

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PROCEDURETo find Absolute Extrema of a
function f on an interval a, b

• 2) Evaluate f at each critical number in a,
  b.

3) Evaluate f at each endpoint.
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PROCEDURETo find Absolute Extrema of a
function f on an interval a, b

• The smallest value is the absolute min and
• the largest value is the absolute max.

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Example 1

• Determine the absolute extrema
• of on -3,3 .

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Example 2

• Determine the absolute extrema
• of on 0,4 .

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Example 3

• Determine the absolute extrema
• of on -5,5 .