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Intermediate Value Theorem

Section 1.4B Calculus AP/Dual, Revised ©2017 11/9/2018 3:50 PM §1.4B: Intermediate Value Theorem

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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) 11/9/2018 3:50 PM §1.4B: Intermediate Value Theorem

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Examples of Discontinuous

11/9/2018 3:50 PM §1.4B: Intermediate Value Theorem

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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 11/9/2018 3:50 PM §1.4B: Intermediate Value Theorem

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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 11/9/2018 3:50 PM §1.4B: Intermediate Value Theorem

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§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 . 11/9/2018 3:50 PM §1.4B: Intermediate Value Theorem

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§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 . 11/9/2018 3:50 PM §1.4B: Intermediate Value Theorem

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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. 11/9/2018 3:50 PM §1.4B: Intermediate Value Theorem

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§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. 11/9/2018 3:50 PM §1.4B: Intermediate Value Theorem

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§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. 11/9/2018 3:50 PM §1.4B: Intermediate Value Theorem

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§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. 11/9/2018 3:50 PM §1.4B: Intermediate Value Theorem

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§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. 11/9/2018 3:50 PM §1.4B: Intermediate Value Theorem

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§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 . 11/9/2018 3:50 PM §1.4B: Intermediate Value Theorem

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§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. 11/9/2018 3:50 PM §1.4B: Intermediate Value Theorem

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§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. 11/9/2018 3:50 PM §1.4B: Intermediate Value Theorem

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§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 ? 11/9/2018 3:50 PM §1.4B: Intermediate Value Theorem

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§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 ? 11/9/2018 3:50 PM §1.4B: Intermediate Value Theorem

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§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 ? 11/9/2018 3:50 PM §1.4B: Intermediate Value Theorem

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§1.4B: Intermediate Value Theorem

In Conclusion… A function exists when: Point Exists Limit Exists Limit = Point 11/9/2018 3:50 PM §1.4B: Intermediate Value Theorem

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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 11/9/2018 3:50 PM §1.4B: Intermediate Value Theorem

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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 11/9/2018 3:50 PM §1.4B: Intermediate Value Theorem

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§1.4B: Intermediate Value Theorem

Assignment Worksheet 11/9/2018 3:50 PM §1.4B: Intermediate Value Theorem

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