1. Intermediate Value Theorem
Section 1.4B
Calculus AP/Dual, Revised ©2017
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§1.4B: Intermediate Value Theorem

2. Definition of Continuity
A function is continuous at the point 𝒙=𝒄 if and only if:
𝒇(𝒄) is continuous
2) 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 exists
3) 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 = 𝒇(𝒄)
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§1.4B: Intermediate Value Theorem

3. Examples of Discontinuous
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§1.4B: Intermediate Value Theorem

4. Proof of Intermediate Value Theorem
Can you prove that at one time, you were exactly feet tall?
If 𝒇 is continuous on 𝒂,𝒃 and 𝒌 is between 𝒇(𝒂) and 𝒇(𝒃) then there exists a number 𝒄 between 𝒂 and 𝒃 such that 𝒇(𝒄)=𝒌
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§1.4B: Intermediate Value Theorem

5. 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
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§1.4B: Intermediate Value Theorem

6. §1.4B: Intermediate Value Theorem
Example 1
Use the IVT to prove that the function 𝒇 𝒙 = 𝒙 𝟐 is 7 on the interval between 𝟐,𝟓 .
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§1.4B: Intermediate Value Theorem

7. §1.4B: Intermediate Value Theorem
Example 2
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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§1.4B: Intermediate Value Theorem

8. These are the two extremes. §1.4B: Intermediate Value Theorem
Example 3
If 𝒇 𝒙 = 𝒙 𝟐 +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 are the extremes? (other words 𝒇 𝒂 and 𝒇 𝒃 )?
These are the two extremes.
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§1.4B: Intermediate Value Theorem

9. §1.4B: Intermediate Value Theorem
Example 3
If 𝒇 𝒙 = 𝒙 𝟐 +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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§1.4B: Intermediate Value Theorem

10. §1.4B: Intermediate Value Theorem
Example 4
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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§1.4B: Intermediate Value Theorem

11. §1.4B: Intermediate Value Theorem
Example 4 (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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§1.4B: Intermediate Value Theorem

12. §1.4B: Intermediate Value Theorem
Example 5
If 𝒇 𝒙 = 𝒙 𝟐 −𝟔𝒙+𝟖, prove the IVT holds through the indicated interval of 𝟎,𝟑 . If the IVT applies, find the value of 𝒄 for 𝒇 𝒄 =𝟑.
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§1.4B: Intermediate Value Theorem

13. §1.4B: Intermediate Value Theorem
Your Turn
If 𝒇 𝒙 = 𝟏 𝒙−𝟐 , use the Intermediate Value Theorem to prove for 𝒄 on the interval 𝟓 𝟐 ,𝟕 if 𝒇 𝒄 = 𝟏 𝟒 .
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§1.4B: Intermediate Value Theorem

14. §1.4B: Intermediate Value Theorem
To Earn Full Credit:
The function, 𝒇 𝒙 (or whatever they give) is identified, and stated to be CONTINUOUS.
Include the function is continuous in 𝒂, 𝒃 where 𝒂 and 𝒃 are defined
State the value of 𝒄, if asked to be defined.
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§1.4B: Intermediate Value Theorem

15. §1.4B: 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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§1.4B: Intermediate Value Theorem

16. §1.4B: Intermediate Value Theorem
Example 6
What value of 𝒌 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 𝒌, 𝒙=−𝟑 ?
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§1.4B: Intermediate Value Theorem

17. §1.4B: Intermediate Value Theorem
Example 6
What value of 𝒌 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 𝒌, 𝒙=−𝟑 ?
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§1.4B: Intermediate Value Theorem

18. §1.4B: Intermediate Value Theorem
Example 6
What value of 𝒌 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 𝒌, 𝒙=−𝟑 ?
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§1.4B: Intermediate Value Theorem

19. §1.4B: Intermediate Value Theorem
In Conclusion…
A function exists when:
Point Exists
Limit Exists
Limit = Point
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§1.4B: Intermediate Value Theorem

20. AP Multiple Choice Practice Question 1 (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 ′ 𝒄 = 𝟒 𝟗 for at least one 𝒄 between −𝟑 and 𝟔 (B) −𝟏≤ 𝒇(𝒙)≤𝟑 for all 𝒙 between −𝟑 and 𝟔 (C) 𝒇(𝒄)=𝟏 for at least one 𝒄 between −𝟑 and 𝟔 (D) 𝒇(𝒄)=𝟎 for at least one 𝒄 between −𝟏 and 𝟑
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§1.4B: Intermediate Value Theorem

21. AP Multiple Choice Practice Question 1 (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:
Vocabulary
Connections and Process
Answer and Justifications
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§1.4B: Intermediate Value Theorem

22. §1.4B: Intermediate Value Theorem
Assignment
Worksheet
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§1.4B: Intermediate Value Theorem