1.1 a preview of calculus and 1.2 finding limits graphically and numerically
DESCRIPTION
1.1 A Preview of Calculus and 1.2 Finding Limits Graphically and Numerically. Objectives. Understand what calculus is and how it compares to precalculus. Estimate a limit using a numerical or graphical approach. Learn different ways that a limit can fail to exist. Swimming Speed. - PowerPoint PPT PresentationTRANSCRIPT
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1.1 A Preview of Calculus 1.1 A Preview of Calculus and and
1.2 Finding Limits 1.2 Finding Limits Graphically and Graphically and
NumericallyNumerically
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ObjectivesObjectives
Understand what calculus is and how it Understand what calculus is and how it compares to precalculus.compares to precalculus.
Estimate a limit using a numerical or Estimate a limit using a numerical or graphical approach.graphical approach.
Learn different ways that a limit can fail to Learn different ways that a limit can fail to exist.exist.
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Swimming SpeedSwimming Speed
Swimming Speed: Taking it to the LimitSwimming Speed: Taking it to the Limit
Questions 1-5Questions 1-5
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Preview of CalculusPreview of Calculus
Diagrams on pages 43 and 44Diagrams on pages 43 and 44
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Two Areas of Calculus: Two Areas of Calculus: DifferentiationDifferentiation
Animation of Differentiation Animation of Differentiation
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Two Areas of Calculus: IntegrationTwo Areas of Calculus: Integration
Animation of IntegrationAnimation of Integration
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LimitsLimits
Both branches of calculus were originally Both branches of calculus were originally explored using limits.explored using limits.
Limits help define calculus.Limits help define calculus.
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1.2 Finding Limits Graphically and 1.2 Finding Limits Graphically and NumericallyNumerically
2Graph ( ) 1 with a hole at x=1.
f(x) is not defined at x=1, but it has a limit at 1.
What value does f approach as x gets closer to 1?
f x x x
3 1( )
1
xf x
x
2( 1)( 1)
( )1
x x xf x
x
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Find the Limit Find the Limit 3 1
( )1
xf x
x
xx .75.75 .9.9 .99.99 .999.999 11 1.0011.001 1.011.01 1.11.1 1.251.25
f(x)f(x) 2.3132.313 2.7102.710 2.9702.970 2.9972.997 ?? 3.0033.003 3.033.03 3.3103.310 3.8133.813
x approaches 1 from the left
x approaches 1 from the right
1lim ( ) 3xf x
Limits are independent
of single points.
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Exploration (p. 48)Exploration (p. 48)
From the graph, it looks like f(2) is defined.From the graph, it looks like f(2) is defined.Look at the table.Look at the table.On the calculator: tblstart 1.8 and On the calculator: tblstart 1.8 and ∆Tbl=0.1.∆Tbl=0.1.Look at the table again.Look at the table again.What does f approach as x gets closer to 2 What does f approach as x gets closer to 2
from both sides?from both sides?
2
2
3 2lim
2x
x x
x
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ExampleExample
Look at the graph and the table.Look at the graph and the table.
0lim
1 1x
x
x
0lim 2
1 1x
x
x
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ExampleExample
Limits are NOT affected by single points!Limits are NOT affected by single points!
1, 2( )
0, 2
xf x
x
2(2) 0, but lim ( ) 1.
xf f x
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Three Examples of Limits that Fail Three Examples of Limits that Fail to Existto Exist
If the left-hand limit doesn't equal right-hand If the left-hand limit doesn't equal right-hand limit, the two-sided limit limit, the two-sided limit does not existdoes not exist..
( )
1, x>0
1, x<0
xf x
x
0 0lim 1 but lim 1 x x
x x
x x
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Three Examples of Limits that Fail Three Examples of Limits that Fail to Existto Exist
If the graph approaches If the graph approaches ∞ or -∞ from one ∞ or -∞ from one or both sides, the limit or both sides, the limit does not existdoes not exist..
2
1( )f x
x
From left, as x approaches 0, f(x) approaches .
From right, as x approaches 0, f(x) approaches .
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Three Examples of Limits that Fail Three Examples of Limits that Fail to Existto Exist
Look at the graph and table.Look at the graph and table.As x gets close to 0, f(x) doesn't approach As x gets close to 0, f(x) doesn't approach
a number, but oscillates back and forth.a number, but oscillates back and forth.If the graph has an oscillating behaviorIf the graph has an oscillating behavior, ,
the limit the limit does not existdoes not exist..
1( ) sinf x
x
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Limits that Fail to ExistLimits that Fail to Exist
f(x) approaches a different number from f(x) approaches a different number from the right side of c than it approaches from the right side of c than it approaches from the left side.the left side.
f(x) increases or decreases without bound f(x) increases or decreases without bound as x approaches c.as x approaches c.
f(x) oscillates as x approaches c.f(x) oscillates as x approaches c.
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HomeworkHomework
1.2 (page 54)1.2 (page 54)
#5, 7, #5, 7,
15-23 odd15-23 odd