1. Continuity and Intermediate Value Theorem
Section 1.4A
Calculus AB & BC AP/Dual, Revised ©2013
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1.4A - Continunity and Intermediate Value Theorem

2. Continuity on a Closed Interval
A function is continuous on the closed interval [a, b] if it is continuous on the open interval (a, b) and if lim 𝑥→ 𝑎 + 𝑓(𝑎) and lim 𝑥→ 𝑏 – 𝑓(𝑏) .
The function f is continuous from the right at a and continuous from the left at b.
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1.4A - Continunity and Intermediate Value Theorem

3. Continuity on a Closed Interval
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1.4A - Continunity and Intermediate Value Theorem

4. 1.4A - Continunity and Intermediate Value Theorem
Example 1
Determine the continuity of 𝑓 𝑥 = 1− 𝑥 2 from [–1, 0] –1 1
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1.4A - Continunity and Intermediate Value Theorem

5. 1.4A - Continunity and Intermediate Value Theorem
Example 2
Determine the continuity of 𝑓 𝑥 = 𝑥+1,𝑥≤0 𝑥 2 +1,𝑥>0 from [–1, 1]
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1.4A - Continunity and Intermediate Value Theorem

6. 1.4A - Continunity and Intermediate Value Theorem
Your Turn
Determine the continuity of 𝑓 𝑥 =4− 16− 𝑥 2 from [–4, 4]
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1.4A - Continunity and Intermediate Value Theorem

7. Properties of Continuity
If the functions f and g are continuous at x = c, then the following are also continuous at c (just at a certain point, not everywhere).
Types:
Scalar Multiple: b • f
Sum and Difference: f + g
Product: fg
Quotient: f/g if g(c) ≠ 0
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1.4A - Continunity and Intermediate Value Theorem

8. 1.4A - Continunity and Intermediate Value Theorem
Example 3
If f(x) = 3x is continuous at x = 2 and g(x) = 1/(x – 1) is continuous at x = 2, would f(x) • g(x) be continuous at 2? Show all work.
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1.4A - Continunity and Intermediate Value Theorem

9. 1.4A - Continunity and Intermediate Value Theorem
Example 3
If f(x) = 3x is continuous at x = 2 and g(x) = 1/(x – 1) is continuous at x = 2, would f(x) • g(x) be continuous at 2? Show all work.
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1.4A - Continunity and Intermediate Value Theorem

10. 1.4A - Continunity and Intermediate Value Theorem
Your Turn
If f(x) = 3x is continuous at x =1 and g(x) = 1/(x – 1) is discontinuous at x = 1, would f(x) • g(x) be continuous at 1? Show all work.
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1.4A - Continunity and Intermediate Value Theorem

11. Intermediate Value Theorem
If f(x) is continuous on the closed interval [a, b]
f(a) ≠ f(b)
If k is between f(a) and f(b) then there exists a number c between a and b for f(c) = k
What is the purpose?
If the function is continuous and function has positive and negative values of f(c), then somewhere f(c) = 0
Examples include temperature or growth
This is one of the theorems that show up on the AP Test. It is an existence theorem
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1.4A - Continunity and Intermediate Value Theorem

12. Proof of Intermediate Value Theorem
Can you prove that at one time, you were exactly feet tall?
If k is between f(a) and f(b) then there exists a number c between a and b for f(c) = k
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1.4A - Continunity and Intermediate Value Theorem

13. Intermediate Value Theorem
To guarantee the Intermediate Value Theorem, function must be continuous.
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1.4A - Continunity and Intermediate Value Theorem

14. 1.4A - Continunity and Intermediate Value Theorem
Example 4
Use the IVT to prove that the function f(x) = x2 is 7 on the interval between [2, 5].
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1.4A - Continunity and Intermediate Value Theorem

15. 1.4A - Continunity and Intermediate Value Theorem
Example 5
If 𝒇 𝒙 =𝐥𝐧 𝒙 , prove by the IVT that there is a root on the interval of [1/2, 3].
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1.4A - Continunity and Intermediate Value Theorem

16. Example 6
If f(x) = x2 + x – 1, prove the IVT holds through the indicated interval of [0, 5]. If the IVT applies, find the value of c for f(c) = 11.
What is the extremes? (other words f(a) and f(b))?
These are the two extremes.
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1.4A - Continunity and Intermediate Value Theorem

17. 1.4A - Continunity and Intermediate Value Theorem
Example 6
If f(x) = x2 + x – 1, prove the IVT holds through the indicated interval of [0, 5]. If the IVT applies, find the value of c for f(c) = 11.
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1.4A - Continunity and Intermediate Value Theorem

18. 1.4A - Continunity and Intermediate Value Theorem
Example 7
If 𝑓 𝑥 = 𝑥 2 +𝑥 𝑥−1 , prove the IVT holds through the indicated interval of [5/2, 4] if f(c) = 6. If the IVT applies, find the value of c for f(c) = 6.
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1.4A - Continunity and Intermediate Value Theorem

19. 1.4A - Continunity and Intermediate Value Theorem
Example 7 (extension)
Would the IVT hold for 𝑓 𝑥 = 𝑥 2 +𝑥 𝑥−1 , through the indicated interval of [–3, 7]? Explain why.
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1.4A - Continunity and Intermediate Value Theorem

20. 1.4A - Continunity and Intermediate Value Theorem
Your Turn
If f(x) = 𝟏 𝒙−𝟐 , use the Intermediate Value Theorem to prove there is zero on the interval [5/2, 7] if f(c) = 1/4.
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1.4A - Continunity and Intermediate Value Theorem

21. 1.4A - Continunity and Intermediate Value Theorem
Piecewise Functions
For a piecewise function to be continuous each function must be continuous on its specified interval and the limit of the endpoints of each interval must be equal.
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1.4A - Continunity and Intermediate Value Theorem

22. 1.4A - Continunity and Intermediate Value Theorem
Example 8
What value of a will make the given piecewise function f(x) continuous at x = –3 of 𝑓 𝑥 = 2 𝑥 2 +5𝑥−3 𝑥 2 −9 ,𝑥≠−3 𝑎, 𝑥=−3
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1.4A - Continunity and Intermediate Value Theorem

23. 1.4A - Continunity and Intermediate Value Theorem
Example 8
What value of a will make the given piecewise function f(x) continuous at x = –3 of 𝑓 𝑥 = 2 𝑥 2 +5𝑥−3 𝑥 2 −9 ,𝑥≠−3 𝑎, 𝑥=−3
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1.4A - Continunity and Intermediate Value Theorem

24. 1.4A - Continunity and Intermediate Value Theorem
In Conclusion…
A function exists when:
Point Exists
Limit Exists
Limit = Point
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1.4A - Continunity and Intermediate Value Theorem

25. AP Multiple Choice Practice Question (non-calculator)
Let f be a continuous function on the closed interval [–3, 6]. If f(–3) = –1 and f(6) = 3, then the Intermediate Value Theorem guarantees that:
[A] f(0) = 0
[B] f ’(c) = 4/9 for at least one c between –3 and 6
[C] –1 ≤ f(x) ≤ 3 for all x between –3 and 6
[D] f(c) = 1 for at least one c between –3 and 6
[E] f(c) = 0 for at least one c between –1 and 3
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1.4A - Continunity and Intermediate Value Theorem

26. 1.4A - Continunity and Intermediate Value Theorem
Assignment
Page odd, odd, 59, odd
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1.4A - Continunity and Intermediate Value Theorem