lesson 1
Continuity
5 lessons + test day
2008-2009 9/16-9/25
2009-2010 9/15-9/22
5 lessons + test day-
2010 5 lessons + test 9/21-9/28
2014- 9/16
Contents
lesson 1
Continuous Functions http://www.calculus-help.com/the-intermediate-value-theorem/ p. 84 #1-6, 11-18, p. 85 #19-24, p. 86# 56-59 sketch 𝑓(𝑥) = ( 𝑥
2
−4, (0,∞ 𝑥 (−∞,0]
) continuous in its domain-square root new book uses infinite discontinuity as well as non removable
Definition of Continuity:
Functions are continuous at c if
1 f(c) is defined
2 lim
𝑥→𝑐
3 lim
𝑥→𝑐 𝑓(𝑥) 𝑒𝑥𝑖𝑠𝑡𝑠 𝑓(𝑥) = 𝑓(𝑐)
Informally you might say that a function is continuous on an open interval if its graph can be drawn with a pencil without lifting the pencil from the paper. Use a graphing calculator; graph each of the following function on the indicated interval. From the graphs, which functions would you say are continuous on the interval? Do you think you can trust the results you obtained graphically?
Y= x 2 +1 (-3,3) 𝑦 =
1
𝑥−2
(-3,3)
Y=
𝑥
2
− 4 𝑥+2
(−3,3)
Point of Discontinuity –
If a function is discontinuous at a point, the discontinuity may be Removable or Nonremovable depending upon whether the limit of the function exists at the point of discontinuity.
5. Test for continuity at the indicated points-
I must clearly see the three things that make a function continuous at a point.
Definition of Continuity :
Functions are continuous at c if
1 f(c) is defined
2 lim
𝑥→𝑐 𝑓(𝑥) 𝑒𝑥𝑖𝑠𝑡𝑠
3 lim
𝑥→𝑐 𝑓(𝑥) = 𝑓(𝑐)
These functions are not continuous at c on the interval (a,b). –
for which reason above
There are two categories of discontinuity: Removable and Non-Removable.
Removable because you can change the function to cover the hole such as:
Here are a few more examples of removable discontinuity:
Non Removable
Here are some examples of non removable discontinuous functions:
Find the x values at which the function is not continuous. Is it removable or non removable?
𝑓(𝑥) = 𝑥
1 this function has a non removable discontinuity at x=0 it has an asymptote g(x)= 𝑥
2
−1 𝑥−1
can be rewritten as g(x)=
(𝑥+1)(𝑥−1) 𝑥−1 it is discontinuous at x=1 it is removable discontinuity if used reverse classroom after limits test, then this is the
For each of the following, Find the vertical asymptote the removable discontinuity the non removable discontinuity the horizontal asymptote
f
(
x
)
2
x
1
2
F(x) =
x
2 x
2
2 x
4
8 f ( x )
x x
2
2
1
1 f ( x )
x
2 x x
2
2
Lesson 2
McClearys website- making a function continuous
Making functions continuous p. 85 #47-50 and
WS - IVT problems and asymptotes from old book p. 78 # 75-77, 83-85
Answers- 83) f(3)=11 84) f(2)=0 85) f(2)=4 http://tutorial.math.lamar.edu/Classes/CalcI/Continuity.aspx
Example 1 - Given the graph of f(x) , shown below, determine if f(x) is continuous at & if not give the reason why it’s not f(-2) f(0) f(3)
Solution
To answer the question for each point we’ll need to get both the limit at that point and the function value at that point. If they are equal the function is continuous at that point and if they aren’t equal the function isn’t continuous at that point.
What are the 3 conditions for continuity
?
Functions are continuous at c if
1 f(c) is defined
2 lim
𝑥→𝑐 𝑓(𝑥) 𝑒𝑥𝑖𝑠𝑡𝑠
3 lim
𝑥→𝑐 𝑓(𝑥) = 𝑓(𝑐)
Ask- limit exists when it is the same from both sides
This is a piecewise function so we need to worry about what is happening where the 2 pieces meet
In a piecewise function-to be continuous- the two functions must have the same y-value at the breaking point-or if the function is undefined at a point, we need to fill in the open circle
What value of B would make this function continuous f(x) =
3 x x
2
B
1
,
, x x
5
5
what IVT means
If you are 5 ft on your 13
th
birthday and on your 14
th
you are 5’6” at some pt you had to be 5’4”
http://www.calculus-help.com/the-intermediate-value-theorem/
Intermediate Value Theorem
-( mcclearys website)
Look at IVT on http://www.calculus-help.com/funstuff/phobe.html
Class Work-Worksheet-continuity problems
lesson 3
HW –2.3 -Worksheet-
Go over hw- AP style test on ________________
Multiple choice- Free response
½ with calculator, ½ without calculator
Asymptote power point http://online.math.uh.edu/HoustonACT/ 2.2
1.
Use the Intermediate Value theorem to explain why the function f(x) = x 2 +2x-1
Must have at least one root on the interval [-1,1]
2 .
What is the value of k to make this a continuous function?
4. If f ( x )
x
2
(A) ln 2 ln ln x x for for
(B)
0
2 ln 8
x x
2
4
then lim
(C) x
2 f ( x ) is ln16
(D)4 E)nonexistent
3
Lesson 4
1
Give back limits test –do a conjugate problem and –infinity square root problem
HW-Worksheet –next 2 pages of continuity packet
CW- below Review questions-from mcclearys website,
2.
Find the value of a and b that makes this function continuous
Functions are continuous at c if
1 f(c) is defined
2 lim
𝑥→𝑐 𝑓(𝑥) 𝑒𝑥𝑖𝑠𝑡𝑠
3 lim
𝑥→𝑐 𝑓(𝑥) = 𝑓(𝑐)
Lesson 5
Review problems p. 88-89
Go over HW- give answer books
AP Style test ________________ cw- Review Problems
How do you find the limits- analytically, graphically, and table
Direct Substitution put the limit (c ) into the equation
If #/# it is the limit
If 0/0 it often has a limit, you must do some work- Factor, conjugates…
If #/0 limit never exists DNE
If 0/# limit is 0
Vertical asymptotes occur where the denominator=0 ie- set the denominator = 0, solve for x
Horizontal asymptotes occur-
Highest degree in denominator getting closer to 0, horizontal asymptote at y=0
Equal degree in numerator and denominator-horizontal asymptote is the leading coefficients
Highest degree in numerator-no horizontal asymptote
Whenever there is a vertical asymptote you have a limit of
+∞ or -∞
Test CH 1
