express the repeating decimal 0.5757... as the ratio of two integers without your calculator....
TRANSCRIPT
Express the repeating decimal 0.5757... as the ratio of two integers without your calculator.
Warm-Up
Express the repeating decimal 0.5757... as the ratio of two integers without your calculator.
Warm-=Up
x = 0.57
100x = 57.57
99x = 57 x = 57/99
𝑥=0.57
What is Calculus?
There are only 3 main concepts in calculus.
1) The Limit
2) The Derivative
3) The Integral
What is Calculus?
There are only 3 main concepts in calculus.
1) The Limit
2) The Derivative
3) The Integral
4) You will need a graphing calculator.
1-2:Finding Limits Graphically and NumericallyObjectives:•Understand the concept of a limit
•Calculate limits
Important Ideas•Limits are what make calculus different from algebra and trigonometry•Limits are fundamental to the study of calculus•Limits are related to rate of change•Rate of change is important in engineering & technology
Analysis•Slope is a rate of change•Rate of change is constant at every value on a linear f(x)
m=2
f(x)
x
m=3m=2m=1m=-
1
Analysisf(x)
x
•Rate of change is different at every value on a non-linear f(x)• Rate of change is the slope of the tangent line at a point
Important Ideas•The slope of a secant line is an average rate of change•The slope of a tangent line is an instantaneous rate of change at a point
•We know how to calculate average rate of change
Analysis
•The tangent line problem…
Go to Sketchpad
•We don’t know how to calculate instantaneous rate of change
,therefore,
Warm-Up-You need a graphing calculator. I’m using a TI-84.
Put your signature pages in the box
Important IdeaInstantaneousRate of change is different at every point on f(x)
f(x)
x
Limits are used to calculate the slopes of the tangents
Example1. Graph: 2( )f x x
2. Trace to x=2.
3. Zoom in at least 4 times.4. Describe the graph.
Example
Consider
3 1( ) , 1
1
xf x x
x
What happens at x=1?
x .75 .9 .99 .999
f(x)
Let x get close to 1 from the left:
Try This
Consider
3 1( ) , 1
1
xf x x
x
x 1.25 1.1 1.01
1.001
f(x)
Let x get close to 1 from the right:
Try ThisWhat number does f(x) approach as x approaches 1 from the left and from the right?
3
1
1lim 3
1x
x
x
Try This
Graph and
3
1
1
1
xY
x
2
2 1Y x x on the same axes. What is the
difference between these graphs?
3 1( )
1
xf x
x
Why is there a “hole” in the graph at x=1?
Analysis
ExampleConsider3 1
( )1
xf x
x
for ( ,1) (1, ) and
( ) 4f x
for x=1
3
1
1lim
1x
x
x
=?
Try ThisFind: if
1lim ( )x
f x
2( ) 2, 1f x x x
( ) 1, 1f x x
1lim ( ) 3x
f x
Important Idea
The existence or non-existence of f(x) as x approaches c has no bearing on the existence of the limit of f(x) as x approaches c.
Important Idea
What matters is…what value does f(x) get very, very close to as x gets very,very close to c. This value is the limit.
Try This
Find:
f(0)is undefined; 2 is the limit
2( )
1 1
xf x
x
0lim ( )x
f x
Find:
( ) 1, 0f x x
Try This
( ) , 01 1
xf x x
x
f(0) is defined; 2 is the limit
21
0lim ( )x
f x
Warm-Up
Try ThisFind the limit of f(x) as x approaches 3 where f is defined by:
2 , 3( )
3 , 3
xf x
x
3lim ( ) 2x
f x
ExampleGraph and find the limit (if it exists):
3
3lim
3x x
Important IdeaSome limits do not exist. If f(x) approaches as x approaches c, we say that the limit does not exist at c or, sometimes we say the AP Exam says the limit approaches infinity at c.
ExampleFind the limit if it exists:
0limx
x
x
Important Idea
0lim 1
x
x
x
0lim 1
x
x
x
But…
0limx
x
x
Does not exist
Definition
If a function has a limit, the limit from the right must equal the limit from the left.
Example
1.Graph using a
friendly window:
1sin
x
2. Zoom at x=0
3. Wassup at x=0?
Important Idea
If f(x) bounces from one value to another (oscillates) as x approachs c, the limit of f(x) does not exist at c:
Lesson Close
Name 3 ways a limit may fail to exist.
Assignment
Page 54 Problems 1 - 7 odd, 8 – 24 all In class, we will not cover the formal definition of a limit, sometimes called epsilon-delta definition. I’ll talk about it in NMSI tutoring.