Post on 01-Jan-2016
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
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.
Swimming SpeedSwimming Speed
Swimming Speed: Taking it to the LimitSwimming Speed: Taking it to the Limit
Questions 1-5Questions 1-5
Preview of CalculusPreview of Calculus
Diagrams on pages 43 and 44Diagrams on pages 43 and 44
Two Areas of Calculus: Two Areas of Calculus: DifferentiationDifferentiation
Animation of Differentiation Animation of Differentiation
Two Areas of Calculus: IntegrationTwo Areas of Calculus: Integration
Animation of IntegrationAnimation of Integration
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.
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
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.
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
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
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
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
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 .
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
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.
HomeworkHomework
1.2 (page 54)1.2 (page 54)
#5, 7, #5, 7,
15-23 odd15-23 odd