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Warm – Up: NO CALCULATOR!. Homework: pg. 65 (23 – 35 odd). 31. 1 33. ½ 35. -1. 23. a. 4 b. 64 c. 64 25. a. 3 b. 2 c. 2 27. 1 29. -.5. Homework: Packet pg. 4. 1.0 2. 2 3. 5/3 4. 8/9 5. 2 6. -2/7 7. -3/2 8. 2a 9. 27 10. -1 11. 2 12. -1/a 2 -1/9 3a 2 - PowerPoint PPT Presentation
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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
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
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]
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
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