Continuity and Intermediate Value Theorem Section 1.4A Calculus AB & BC AP/Dual, Revised ©2013 viet.dang@humble.k12.tx.us 2/17/2019 7:28 PM 1.4A - Continunity and Intermediate Value Theorem

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. 2/17/2019 7:28 PM 1.4A - Continunity and Intermediate Value Theorem

Continuity on a Closed Interval 2/17/2019 7:28 PM 1.4A - Continunity and Intermediate Value Theorem

1.4A - Continunity and Intermediate Value Theorem Example 1 Determine the continuity of 𝑓 𝑥 = 1− 𝑥 2 from [–1, 0] –1 1 2/17/2019 7:28 PM 1.4A - Continunity and Intermediate Value Theorem

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] 2/17/2019 7:28 PM 1.4A - Continunity and Intermediate Value Theorem

1.4A - Continunity and Intermediate Value Theorem Your Turn Determine the continuity of 𝑓 𝑥 =4− 16− 𝑥 2 from [–4, 4] 2/17/2019 7:28 PM 1.4A - Continunity and Intermediate Value Theorem

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 2/17/2019 7:28 PM 1.4A - Continunity and Intermediate Value Theorem

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. 2/17/2019 7:28 PM 1.4A - Continunity and Intermediate Value Theorem

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. 2/17/2019 7:28 PM 1.4A - Continunity and Intermediate Value Theorem

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. 2/17/2019 7:28 PM 1.4A - Continunity and Intermediate Value Theorem

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 2/17/2019 7:28 PM 1.4A - Continunity and Intermediate Value Theorem

Proof of Intermediate Value Theorem Can you prove that at one time, you were exactly 5.2578 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 2/17/2019 7:28 PM 1.4A - Continunity and Intermediate Value Theorem

Intermediate Value Theorem To guarantee the Intermediate Value Theorem, function must be continuous. 2/17/2019 7:28 PM 1.4A - Continunity and Intermediate Value Theorem

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]. 2/17/2019 7:28 PM 1.4A - Continunity and Intermediate Value Theorem

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]. 2/17/2019 7:28 PM 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. What is the extremes? (other words f(a) and f(b))? These are the two extremes. 2/17/2019 7:28 PM 1.4A - Continunity and Intermediate Value Theorem

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. 2/17/2019 7:28 PM 1.4A - Continunity and Intermediate Value Theorem

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. 2/17/2019 7:28 PM 1.4A - Continunity and Intermediate Value Theorem

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. 2/17/2019 7:28 PM 1.4A - Continunity and Intermediate Value Theorem

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. 2/17/2019 7:28 PM 1.4A - Continunity and Intermediate Value Theorem

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. 2/17/2019 7:28 PM 1.4A - Continunity and Intermediate Value Theorem

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 2/17/2019 7:28 PM 1.4A - Continunity and Intermediate Value Theorem

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 2/17/2019 7:28 PM 1.4A - Continunity and Intermediate Value Theorem

1.4A - Continunity and Intermediate Value Theorem In Conclusion… A function exists when: Point Exists Limit Exists Limit = Point 2/17/2019 7:28 PM 1.4A - Continunity and Intermediate Value Theorem

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 2/17/2019 7:28 PM 1.4A - Continunity and Intermediate Value Theorem

1.4A - Continunity and Intermediate Value Theorem Assignment Page 79-80 7-21 odd, 25-41 odd, 59, 75-83 odd 2/17/2019 7:28 PM 1.4A - Continunity and Intermediate Value Theorem