1002 Limits 1 Local Behavior REVIEW ALGEBRA is a machine that a function a point CALCULUS is a machine that a function a point ID: 296735
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Slide1
AP CALCULUS
1002
- Limits 1: Local BehaviorSlide2
REVIEW:
ALGEBRA is a ________________________ machine that ___________________ a function ___________ a point.
CALCULUS is a ________________________ machine that ___________________________ a function ___________ a pointSlide3
Limits Review:
PART 1: LOCAL BEHAVIOR
(1).
General Idea: Behavior of a function very near the point where (2). Layman’s Description of Limit (Local Behavior) L a (3). Notation (4). Mantra Slide4
G N A W
Graphically
“We Don’t Care” Postulate”:Slide5
G N A W
NumericallySlide6
The Formal DefinitionSlide7
(5). Formal Definition
( Equation Part)
Graphically:
Find a
If
3 2 1 1 2 3 4Slide8
Analytically
Find a if
given and
for
------------------------------------------
Slide9
Find a for any
Slide10
Day 2Slide11
FINDING LIMITSSlide12
G N A W
-.1
-.01
-.001
0
.001
.01
.1
X
Mantra:
Numerically
Words
Verify these also:Slide13
(6). FINDING LIMITS
“We Don’t Care” Postulate…..
The existence or non-existence of
f(x)
at
x = 2
has no bearing on the limit as GraphicallySlide14
FINDING LIMITS
Analytically
“a”
in
the Domain
Use _______________________________
“a”
not in the Domain This produces ______ called the _____________________ Rem: Always start with Direct SubstitutionSlide15
Rem: Always start with Direct Substitution
Method 1: Algebraic - Factorization
Method 2: Algebraic - Rationalization
Method 3: Numeric – Chart (last resort!)
Method 4: Calculus
To be Learned Later !Slide16
Do All Functions have Limits?
Where LIMITS fail to exist.
Why?Slide17Slide18
Review :
1) Write the Layman’s description of a Limit.
2) Write the formal definition. ( equation part)
3) Find each limit.
4) Does f(x) reach L at either point in #3?Slide19
Homework Problems
From the figure,
determine a
such thatSlide20
Review:
(5). The graph of the function displays the graph of a function with
Estimate how close x must be to 2 in order to insure that f(x) is within 0.5 of 4.
(6). Find a such that Slide21
Last Update: 08/12/10Slide22
Using Direct Substitution
BASIC (
k
is a constant.
x
is a variable)1)2) 3) 4)IMPORTANT: Goes BOTH ways!
Properties of LimitsSlide23
Properties of Limits: cont.
POLYNOMIAL, RADICAL, and RATIONAL FUNCTIONS
all us
Direct Substitution
as long as
a
is in the domainOPERATIONS Take the limits of each part and then perform the operations.EX:Slide24
Composite Functions
REM: Notation
THEOREM:
and
Use Direct Substitution.
EX: EX:Slide25
Limits of TRIG Functions
Squeeze Theorem: if f(x) ≤
g(x) ≤
h(x) for
x
in the interval about
a
, except possibly at a and theThen exists and also equals L
f
g
h
a
This theorem allow us to use
DIRECT SUBSTIUTION
with
Trig Functions
.Slide26
Limits of TRIG Functions:cont.
In a
UNIT CIRCLE
measured in
RADIANS
:
THEREFORE:
Defn. of radians!Slide27
Exponential and Logarithmic Limits
Use DIRECT SUBSTITUTION. REM: the Domain
of the functions
REM: Special Exponential Limit
For a > 0