2.1 Rates of Change and Limits

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2.1 Rates of Change and Limits

• Average and Instantaneous Speed
• A moving bodys average speed during an interval
  of time is found by dividing the distance covered
  by the elapsed time.
• The unit of measure is length per unit time ex.
  Miles per hour, etc.

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Finding an Average Speed

• A rock breaks loose from the top of a tall cliff.
  What is its average speed during the first 2
  seconds of fall?
• Experiments show that objects dropped from rest
  to free fall will fall y 16t² feet in the first
  t seconds.
• For the first 2 seconds of, we change t 0 to t
  2.

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Finding an Instantaneous Speed

• Find the speed of the rock in example 1 at the
  instant t 2.
• Since we cannot use h 0 because it will give us
  an undefined answer, evaluate the formula at
  values close to 0.
• See the table 2.1 on p. 60 in your textbook.
• Notice, the average speed approaches the limiting
  value of 64 ft/sec.

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Average Speeds over Short Time Intervals Starting
at t 2
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Finding an Instantaneous Speed

• Confirm algebraically
• So, we can see why the average speed has the
  limiting value of 64 16(0) 64 ft/sec as h
  approaches 0.

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Limits

• Most limits of interest in the real world can be
  viewed as numerical limits of values of
  functions.
• A calculator can suggest the limits, and calculus
  can give the mathematics for confirming the
  limits analytically.

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Properties of Limits
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Properties of Limits
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Using Properties of Limits

• Use the observations and and
  the properties of limits to find the following
  limits.
• a. b.

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Polynomial and Rational Functions
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Using Theorem 2

• a.
• b.

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Using the Product Rule

• Determine

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Exploring a Nonexistent Limit

• Use a graph to show that does not exist.
• Notice that the denominatoris 0 when x is
  replaced by2, so we cannot usesubstitution.
• The graph suggests that asx approaches 2 from
  eitherside, the absolute values getvery large.
  This suggests thatthe limit does not exist.

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One-Sided and Two-Sided Limits

• Limits can approach a function from opposite
  sides.
• Right-hand limit limit approaches from the
  right side.
• Left-hand limit limit approaches from the left
  side.

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One-sided and Two-sided Limits
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Exploring Right- and Left-Hand Limits
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Sandwich Theorem
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Using the Sandwich Theorem

• Show that

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Homework!!!!!

• Textbook p. 66 67 1, 2, 5, 6, 7 14, 20 28
  even, 37, 40 44 even.