Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... ·...
Transcript of Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... ·...
![Page 1: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/1.jpg)
Slide 1 / 102
Limits and Continuity
![Page 2: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/2.jpg)
Slide 2 / 102
Table of ContentsLimitsLimits of End BehaviorThe Indeterminable Form of 0/0Trig LimitsContinuityDifference Quotient
![Page 4: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/4.jpg)
Slide 4 / 102
A rubber ball is dropped and bounces back up to half the height it was on the previous bounce. Given this scenario, does the ball ever come to rest? What is the approximate height as time goes to ∞?
![Page 5: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/5.jpg)
Slide 5 / 102
A limit allows a function, f(x), to have a value as the function approaches x.
In the previous example, the balls height was zero as time went to ∞, even if the object never stopped moving.
![Page 6: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/6.jpg)
Slide 6 / 102
Consider the functions as x approaches 3.
![Page 7: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/7.jpg)
Slide 7 / 102
As the function f(x) approaches x=3 what is the value f(x)?
As the function g(x) approaches x=3 what is the value g(x)?
A limit describes what happens to the function as it gets closer and closer to a certain value of x. The function doesn't have to have a value at that x for the limit to exist.
![Page 8: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/8.jpg)
Slide 8 / 102
We can find a limit by substituting into the function the value of x.
So in the previous example f(x)= x-3, f(3)=0, which the graph shows.
What about ?
Substitution doesn't work. But since a limit is defined by getting close to that value we can look at the graph and see that the function has a value approaching -3 as x approaches 0 from both the left and right. Algebraically, we can factor and reduce and we also will get a limit of 0.
![Page 9: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/9.jpg)
Slide 9 / 102
LimitsLimits describe what happens to a function as x approaches a value.
is read "The limit of f of x, as x approaches c, is L.
For the previous example of f(x)= x-3:
![Page 10: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/10.jpg)
Slide 10 / 102
Limits of a Composite Function can be found by finding the limits of the individual terms.
![Page 11: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/11.jpg)
Slide 11 / 102
Ex:
Approaches 1 from the right only.
Approaches 1 from the left only.
![Page 12: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/12.jpg)
Slide 12 / 102
1 Find the indicated limit, if it exists. If it doesn't exist enter 0.0
![Page 13: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/13.jpg)
Slide 13 / 102
2 Find the indicated limit, if it exists. If it doesn't exist enter 0.0
![Page 14: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/14.jpg)
Slide 14 / 102
![Page 15: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/15.jpg)
Slide 15 / 102
![Page 16: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/16.jpg)
Slide 16 / 102
![Page 17: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/17.jpg)
Slide 17 / 102
6 Find the indicated limit, if it exists. If it doesn't exist enter 0.0
![Page 18: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/18.jpg)
Slide 18 / 102
![Page 19: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/19.jpg)
Slide 19 / 102
Use graphing calculator to determine:
Using the second function, find the value of x=2.What does this demonstrate?
![Page 20: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/20.jpg)
Slide 20 / 102
Using a graphing calculator with
Go to 2nd -> TBLSET TblSTART= -2 Auto #Tbl=.5 Auto
Now use 2nd -> TABLE What do you observe? What happens at x = 2?
![Page 21: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/21.jpg)
Slide 21 / 102
![Page 22: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/22.jpg)
Slide 22 / 102
8 Find the indicated limit, if it exists. If it doesn't exist enter 0.0
![Page 23: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/23.jpg)
Slide 23 / 102
9 Find the indicated limit, if it exists. If it doesn't exist enter 0.0
![Page 24: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/24.jpg)
Slide 24 / 102
10 Find the indicated limit, if it exists. If it doesn't exist enter 0.0
![Page 26: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/26.jpg)
Slide 26 / 102
When a limit goes to infinity we are looking for any horizontal asymptotes and a number of things can happen.
The first example isn't a problem, as was seen in the last section.
But what about the second and third examples?
Not all # are the same.
![Page 27: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/27.jpg)
Slide 27 / 102
Consider is the limit 1?
Reduce the rational before substituting in # . The limit is # .
In calculus, divide each term by the highest power of x, then take the limit.
Remember that #/# is 0.
![Page 28: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/28.jpg)
Slide 28 / 102
![Page 29: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/29.jpg)
Slide 29 / 102
Be aware that some of the limits are going to -# .
![Page 30: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/30.jpg)
Slide 30 / 102
Short cut: Compare the highest power in the numerator and denominator.
Same power, divide the coefficents
If numerator is greater, #
If the denominator is greater, 0
What is the highest power in the numerator and denominator?
![Page 31: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/31.jpg)
Slide 31 / 102
by the short cut, the highest powers are the same. But be careful before dividing the coefficents?
reduces to 2. But the limit is going to -# . Since the denominator is always positive, only the numerator is negative.
![Page 32: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/32.jpg)
Slide 32 / 102
11
A
B
C
D
![Page 33: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/33.jpg)
Slide 33 / 102
12
A
B
C
D
![Page 34: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/34.jpg)
Slide 34 / 102
13
A
B
C
D
![Page 35: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/35.jpg)
Slide 35 / 102
Why? Which is getting bigger faster?
![Page 36: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/36.jpg)
Slide 36 / 102
Use conjugates to rewrite the expression as a fraction and then solve like # /# .
![Page 37: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/37.jpg)
Slide 37 / 102
![Page 38: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/38.jpg)
Slide 38 / 102
Try it.
Ans: 5/2
![Page 40: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/40.jpg)
Slide 40 / 102
Limits that go to # /# or # -# are called indeterminate forms. The third type we study are 0/0.
Think back to the example
When we graphed the function and looked at the table of values the limit was 4, but subbing in 2 was 0/0.
So what gives?
![Page 41: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/41.jpg)
Slide 41 / 102
Think back to Algebra, when you plugged a number into an equation and got zero we said that the number was a root. Since we are getting 0/0, that means the numerator and denominator share a root. So this means we should be able to factor and reduce.
Remember a limit is what the function is doing as it approaches a value from the left and right. x# 2 but the function can approach it.
![Page 42: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/42.jpg)
Slide 42 / 102
Try:
![Page 43: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/43.jpg)
Slide 43 / 102
Try: Find:
a DNE
b -5
c 4
answers
![Page 44: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/44.jpg)
Slide 44 / 102
14 Find the limit of
A
B 4
C 0
D Does not exist
![Page 45: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/45.jpg)
Slide 45 / 102
15 Find the limit of
A
B 4
C 0
D Does not exist
![Page 46: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/46.jpg)
Slide 46 / 102
16 Find the limit of
A
B -1
C 0
D Does not exist
![Page 47: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/47.jpg)
Slide 47 / 102
17 Find the limit of
A
B -1/2
C 0
D Does not exist
![Page 48: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/48.jpg)
Slide 48 / 102
18
A
B
C
D Does not exist
![Page 49: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/49.jpg)
Slide 49 / 102
Review of Limits1) plug in a for x, if it equals a number, there's your limit.
2) If when a is plugged gives:
A) :Factor and reduce. Try plugging in again.
B) i) Divide each term by the highest exponent term ii)Short cut: compare highest exponent of the numerator with the highest exponent of the denominator. a) numerator greater -># b) denominator greater -> 0 c) degrees are the same->lead coefficient of numerator lead coefficient of denominator (Note: watch your signs when using -# ) C) :Multiply by conjugate and then try plugging in again D) : This is # but check to see that
![Page 51: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/51.jpg)
Slide 51 / 102
If you can plug in and get a number, good. But what if you get an indeterminable form?
3 Trig Limits to Know
![Page 52: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/52.jpg)
Slide 52 / 102
![Page 53: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/53.jpg)
Slide 53 / 102
![Page 54: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/54.jpg)
Slide 54 / 102
19 Find the limit of
(Answer in fraction form if not an integer.)
![Page 55: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/55.jpg)
Slide 55 / 102
20 Find the limit of
![Page 56: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/56.jpg)
Slide 56 / 102
21 Find the limit of
![Page 57: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/57.jpg)
Slide 57 / 102
22 Find the limit of
![Page 58: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/58.jpg)
Slide 58 / 102
23 Find the limit of
![Page 59: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/59.jpg)
Slide 59 / 102
![Page 61: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/61.jpg)
Slide 61 / 102
a b c de f g h
At what points do you think the graph is continuous?At what points do you think the graph is discontinuous?
What should the definition of continuous be?
![Page 62: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/62.jpg)
Slide 62 / 102
AP Calculus Definition of Continuous
1) f(a) exis ts
2) exis ts
3)
This definition shows continuity at a point on the interior of a function.
For a function to be continuous, every point in its domain must be continuous.
![Page 63: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/63.jpg)
Slide 63 / 102
Continuity at an EndpointReplace step 3 in the previous definition with:
Left Endpoint:
Right Endpoint
![Page 64: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/64.jpg)
Slide 64 / 102
Types of Discontinuity
Infinite Jump Removable Essential
![Page 65: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/65.jpg)
Slide 65 / 102
24 Given the function decided if it is continuous or not. If it is not state the reason it is not.
A continuousB
C
D
f(a) does not exis t for all a
does not exis t for all a
is not true for all a
![Page 66: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/66.jpg)
Slide 66 / 102
25 Given the function decided if it is continuous or not. If it is not state the reason it is not.
A continuousB
C
D
f(a) does not exis t for all a
does not exis t for all a
is not true for all a
![Page 67: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/67.jpg)
Slide 67 / 102
26 Given the function decided if it is continuous or not. If it is not state the reason it is not.
A continuousB
C
D
f(a) does not exis t for all a
does not exis t for all a
is not true for all a
![Page 68: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/68.jpg)
Slide 68 / 102
27 Given the function decided if it is continuous or not. If it is not state the reason it is not.
A continuousB
C
D
f(a) does not exis t for all a
does not exis t for all a
is not true for all a
![Page 69: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/69.jpg)
Slide 69 / 102
Removable discontinuities come from rational functions.
A piecewise function can be used to fill the "hole".
What should k be so that g(x) is continuous?
![Page 70: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/70.jpg)
Slide 70 / 102
28What value(s) would remove the discontinuity(s) of the given function?
A -3
B -2
C -1
D -1/2
E 0
F 1/2
G 1
H 2
I 3
J DNE
![Page 71: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/71.jpg)
Slide 71 / 102
29What value(s) would remove the discontinuity(s) of the given function?
A -3
B -2
C -1
D -1/2
E 0
F 1/2
G 1
H 2
I 3
J DNE
![Page 72: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/72.jpg)
Slide 72 / 102
30What value(s) would remove the discontinuity(s) of the given function?
A -3
B -2
C -1
D -1/2
E 0
F 1/2
G 1
H 2
I 3
J DNE
![Page 73: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/73.jpg)
Slide 73 / 102
31What value(s) would remove the discontinuity(s) of the given function?
A -3
B -2
C -1
D -1/2
E 0
F 1/2
G 1
H 2
I 3
J DNE
![Page 74: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/74.jpg)
Slide 74 / 102
Making a Function Continuous
, find a so that f(x) is continuous .
Both 'halves' of the function are continuous. The concern is making
Solution:
![Page 75: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/75.jpg)
Slide 75 / 102
32What value of k will make the function continuous?
![Page 76: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/76.jpg)
Slide 76 / 102
33What value of k will make the function continuous?
![Page 77: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/77.jpg)
Slide 77 / 102
Characteristics of a function on closed continuous interval.
1) Somewhere on the interval there will be a maximum and a minimum.
i ii iii iv
![Page 78: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/78.jpg)
Slide 78 / 102
When discussing Maximum and Minimum, there is a difference between Absolute Maximum and Relative Maximum.
Think of dropping a ball.The starting point is absolute max.The starting point and each bounce are relative max.
![Page 79: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/79.jpg)
Slide 79 / 102
Characteristics of a function on closed continuous interval.
*2) Intermediate Value Theorem- if f(x) is a closed continuous interval on [a,b], then f(x) takes on every value between f(a) and f(b).
* IVT is a theorem that can be used as a justification on the AP exam.
af(a)
b
f(b)
This comes in handy when looking for zeros.
![Page 80: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/80.jpg)
Slide 80 / 102
Can you use the Intermediate Value Theorem to find the zeros of function?
X Y
-2 -6
-1 -2
0 3
1 5
2 -1
3 -4
4 -2
5 2
![Page 81: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/81.jpg)
Slide 81 / 102
Can you use the Intermediate Value Theorem to find the zeros of function?
X Y
-3 4
-2 1
-1 -1
0 1
1 2
2 3
3 -2
4 2
![Page 82: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/82.jpg)
Slide 82 / 102
34Give the letter that lies in the same interval as a zero of this continuous function.
A B C DX 1 2 3 4 5
Y -2 -1 0.5 2 3
![Page 83: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/83.jpg)
Slide 83 / 102
35Give the letter that lies in the same interval as a zero of this continuous function.
A B C DX 1 2 3 4 5
Y 4 3 2 1 -1
![Page 84: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/84.jpg)
Slide 84 / 102
36Give the letter that lies in the same interval as a zero of this continuous function.
A B C DX 1 2 3 4 5
Y -2 -1 -0.5 2 3
![Page 86: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/86.jpg)
Slide 86 / 102
Draw a possible graph of traveling 100 miles in 2 hours.
Distance
Time
100
t
d
2
![Page 87: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/87.jpg)
Slide 87 / 102
Using the graph on the previous slide: What is the average rate of change for the trip?
Is this constant for the entire trip?
What formula could be used to find the average rate of change between 45 minutes and 1 hour?
![Page 88: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/88.jpg)
Slide 88 / 102
The slope formula of represents the Velocity or
Average Rate of Change. This is the slope of the secant line from (t1,d1) to (t2,d2).
Suppose we were looking for Instantaneous Velocity at 45 minutes, what values of (t1,d1) and (t2,d2) should be used?
Is there a better approximation?
![Page 89: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/89.jpg)
Slide 89 / 102
![Page 90: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/90.jpg)
Slide 90 / 102
The Difference Quotient gives the instantaneous velocity, which is the slope of tangent line at a point.
A Derivative is used to find the slope of a tangent line. So Difference Quotient can be used find a derivative algebraically.
![Page 91: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/91.jpg)
Slide 91 / 102
Example of the Difference QuotientFind the slope of the tangent line to the functionat x=3.
![Page 92: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/92.jpg)
Slide 92 / 102
Example of the Difference QuotientFind an equation that can be used to find the slope of the tangent line at any point on the function
To find the slope at x=2, sub 2 into 9x2. So the slope at x=2 is 36.
![Page 93: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/93.jpg)
Slide 93 / 102
![Page 94: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/94.jpg)
Slide 94 / 102
![Page 95: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/95.jpg)
Slide 95 / 102
![Page 96: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/96.jpg)
Slide 96 / 102
40What expression can used to find the slope of any tangent line to
A
B
C
D
![Page 97: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/97.jpg)
Slide 97 / 102
Now try finding the slope of a tangent of a graph.
y1=sin(x)Find the slope at x=# / 4
Try ZOOM IN at the point till it looks like a line in the viewing window. (To get calculator to zoom at x=# /4, use 2nd CALC 1:Value, then zoom once the cursor is on π/4.)
![Page 98: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/98.jpg)
Slide 98 / 102
The cursor is on the function at x=# /4 . We don't want round on the AP exam until the end but who wants to copy down all those numbers?
2nd QUIT will take you to the computation screen with x and y still equal to the point on the graph. STORE x->A and y->B using the STORE key and ALPHA.
Go back to graph and "bump" the cursor over to a point close by.2nd QUIT again and this time STORE x->C and y->D.
Calculate the slope (D-B)/(C-A)= .707 (now you can round)
![Page 99: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/99.jpg)
Slide 99 / 102
The calculator does have a built in derivative key for graphs.
It is dy/dx. When the cursor is where you want it, use 2nd CALC 6: dy/dx (You may need to zoom in a few times to take the derivative where you want it)
These both stand for the derivative of a function.
![Page 100: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/100.jpg)
Slide 100 / 102
![Page 101: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/101.jpg)
Slide 101 / 102
![Page 102: Limits and Continuity - content.njctl.orgcontent.njctl.org/courses/math/archived-courses... · Substitution doesn't work. But since a limit is defined by getting close to that value](https://reader033.fdocuments.net/reader033/viewer/2022053015/5f13a304a34f6100383e7956/html5/thumbnails/102.jpg)
Slide 102 / 102