Warm – Up: NO CALCULATOR!

Homework: pg. 65 (23 – 35 odd)

23. a. 4

b. 64

c. 64

25. a. 3

b. 2

c. 2

27. 1

29. -.5

31. 1

33. ½

35. -1

Homework: Packet pg. 4

1.0

2. 2

3. 5/3 4. 8/95. 26. -2/77. -3/28. 2a9. 2710. -1 11. 2

12. -1/a2

13. -1/9

14. 3a2

15. 1/27

16. 2

17. -1

18. -1

19. 1

20. 10

21. 3

22. -1

23. 1

24. DNE

25. 14

26. 18

27. DNE

28. 0

29. 1

Quiz

Good Luck Show lots of work You may use an extra sheet of paper!

Video: Segment 1

www.calculus-help.com/continuity/

Summary : Types Discontinuities

3 main types: 1) Point discontinuity

Type(s) of Function: _______________

2) Infinite discontinuity

Type(s) of Function: _______________

3) Jump Discontinuity  Type(s) of Function: _______________.

Discontinuity can either be

REMOVALBE or NONREMOVABLE.

Points are Removable.

Infinite and Jump are Not

Continuity at a Point

Discuss the discontinuity (if any) of the functions below:

Video: Segment 2

Slide 2- 9

Continuity at a Point

Function f is continuous at x = c if and only if

1. f(c) exists

2.

3.

existsxfcx

)(lim

)()(lim cfxfcx

Slide 2- 10

Continuity at a Point

If a function f is not continuous at a point c , we say that f is discontinuous at c and c is a point of discontinuity of f.

Note that c need not be in the domain of f.

Slide 2- 11

Example Continuity at a Point

There is an infinite discontinuity at 1.x

2

3Find and identify the points of discontinuity of

1y

x

[-5,5] by [-5,10]

Where are these discontinuous?And what type?

1 51. ( ) 2. ( )

1 2cos( ) 2

13. ( ) cot( ) 4.

5 2sin( )

xf x f x

x x

f x xx

Video: Intermediate Value Theorem

http://www.calculus-help.com/the-intermediate-value-theorem/

Intermediate Value Thm.

A continuous functions on [a,b]

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]

Graphically:

f(b)

f(a)

a b

Any k value in here will be “hit” at least once

Example 1: Make the function continuous

Steps: 1. Determine if

discontinuity is removable

2. Find values that are causing discontinuity

3. Find the limit at the found value(s)

4. Write a piecewise function that includes found value

Example 2: Make the following continuous

Example 3: Make the following continuous

Example 4: Make the following continuous

Example 5: Determine the value of k that makes the following continuous

Packet pg. 5