1. Warm up!…you know you love ‘em!
1. For the graph shown, which of these
statements is FALSE?
(A) f(x) is continuous at x=2
(B) (C)
(D)
(E) f(x) is continuous everywhere from 2.

2. Warm up!…you know you love ‘em!
1. For the graph shown, which of these
statements is FALSE?
(A) f(x) is continuous at x=2
(B) (C)
(D)
(E) f(x) is continuous everywhere from 2.

3. HW questions:

4. Day 6: Unit Wrap-up

5. 3 common ways a function can be discontinuous
Hole when factors divide out
Removable discontinuity
Asymptotes when denominator = 0
Infinite discontinuity, nonremovable
Piecewise Graph or Greatest Integer Function
Jump Discontinuity, nonremovable

6. Do the functions have any discontinuities? If yes, where and what type?

7. What could be done to the following function to make it continuous?

8. What would the value of “k” need to be to make the following function continous?

9. f(x) is continuous at x=c if . . .
Official Definition
Which means . . .

10. Is f(x) continuous at x = 4? Justify your answer
therefore,
f(x) is not continuous at x = 4
This is what your test paper should show.

11. Is f(x) continuous at x = 3? Justify your answer.
therefore,
f(x) is continuous at x = 3
This is what your test paper should show.

12. Continuity if “c” happens to be the endpoint of the domain [a,b]. . . .
Continuity at an endpoint –
Can’t examine the other limits

13. Composition of functions. . . Is f(g(x)) continuous at x = 4?

14. We must evaluate
Therefore the composition is continuous

15. Find the limit of the composition

16. Find the limit of the composition

17. Find the limit of the composition

18. A couple more things to wrap up!
Continuity
A couple more things to wrap up!

19. Intermediate Value Thm.
A continuous function takes on all y values between f(a) and f(b).
In other words…
If k is between f(a) & f(b), then k = f(c) for some c in [a,b]

20. Graphically: f(b) Any k value in here will be “hit” at least once f(a)

21. Intermediate Value Theorem in Action
Given f(x) is continuous on the interval [0,4].
What is the minimum number of times f(x) = 8? Justify. x 1 2 3 4
f(x) 10 6 7 9
Answer: at least twice. f(x) = 8 in the interval (0,1) because f(0) = 10 and f(1) = 6 since the Intermediate Value Theorem says that if a function is continuous on an interval, the function takes on all values between f(0) and f(1).
f(x) = 8 in the interval (3,4) for the same reason.

22. Other things to review for the test

23. Find the Domain and Zeroes
x = -5 is not a zero because -5 is not in the domain

24. One last way to find a limit…
Squeeze Thm. (aka Sandwich Thm.) If
and
then

25. Sandwich Thm Example

26. Test format
One Calculator Inactive Free Response Question with multiple parts
One Calculator Active Free Response Question with multiple parts
11 Multiple Choice Calculator Inactive
7 Multiple Choice Calculator Active
Best to NOT leave a multiple choice answer blank. You are earning points for the number you answer correctly. SO, if you guess correctly you earn the point.

27. QUIZ common mistakes: