2.1 rates of change and limits average and instantaneous speed –a moving body’s average speed...
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2.1 Rates of Change and Limits• Average and Instantaneous Speed
– A moving body’s 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.
2 216(2) 16(0) .32
2 0 sec
y ft
t
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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.
2 216(2 ) 16(2)y h
t h
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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.
2 216(2 ) 16(2)y h
t h
216(4 4 ) 64h h
h
264 16h h
h
64 16h
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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.3 2lim( 4 3)
x cx x
4 2
2
1lim
5x c
x x
x
limx c k k limx c x c
3 2lim lim4 lim3x c x c x cx x
3 24 3c c
4 2
2
lim( 1)
lim( 5)x c
x c
x x
x
4 2
2
lim lim lim1
lim lim5x c x c x c
x c x c
x x
x
4 2
2
1
5
c c
c
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Polynomial and Rational Functions
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Using Theorem 2a.
b.
2
3lim[ (2 )]x
x x
2(3) (2 3)
92
2
2 4lim
2x
x x
x
2(2) 2(2) 4
2 2
12
34
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Using the Product Rule
• Determine
0
tanlimx
x
x
0
tanlim .x
x
x
0
sin 1lim
cosx
x
x x
0 0
sin 1lim lim
cosx x
x
x x
11cos0
1
1 11
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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.
3
2
1lim
2x
x
x
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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.
lim ( )x c
f x
lim ( )x c
f x
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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 2
0lim[ sin(1 )] 0.x
x x
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Homework!!!!!
• Textbook p. 66 – 67 #1, 2, 5, 6, 7 – 14, 20 – 28 even, 37, 40 – 44 even.