Section 3.2 Day 1 Mean Value Theorem AP Calculus AB

Learning Targets Define & apply the Mean Value Theorem

Mean Value Theorem: Definition If 𝑓 is continuous on the closed interval [𝑎, 𝑏] and differentiable on the open interval (𝑎, 𝑏), then there exists a number 𝑐 in (𝑎, 𝑏) such that 𝑓 ′ 𝑐 = 𝑓 𝑏 −𝑓 𝑎 𝑏−𝑎

Mean Value Theorem: Conceptual Geometrically: The tangent line is parallel to the secant line at some point Application: The IROC is equal to the AROC at some point

Example 1 Determine all the numbers 𝑐 which will satisfy the conclusions of the Mean Value Theorem for the following function: 𝑓 𝑥 = 𝑥 3 +2 𝑥 2 −𝑥 on [−1, 2] 1. Confirm that 𝑓(𝑥) is continuous and differentiable on the interval 2. 𝑓 ′ 𝑐 = 𝑓 2 −𝑓 −1 2− −1 3. 3 𝑐 2 +4𝑐−1=4 4. Use calculator: 𝑐=0.7863

Example 2 Show that the function 𝑓 𝑥 = 𝑥 2 satisfies the hypotheses of the Mean Value Theorem on the interval 0, 2 . Then, find the value of 𝑐 that satisfies the Mean Value Theorem 1. Confirm 𝑓(𝑥) is continuous and differentiable on the interval 2. 𝑓 ′ 𝑐 = 𝑓 2 −𝑓 0 2−0 3. 2𝑐=2 4. 𝑐=1

Example 3 The functions 𝑓 and 𝑔 are differentiable for all real numbers. Also, ℎ 𝑥 =𝑓 𝑔 𝑥 −6 Explain why there must be a value 𝑐 for 1<𝑐<3 such that ℎ ′ 𝑐 =−5.

Example 3: Answer ℎ 3 −ℎ 1 3−1 = −7−3 3−1 =−5 ℎ 3 −ℎ 1 3−1 = −7−3 3−1 =−5 Since ℎ is continuous and differentiable, by the Mean Value Theorem, there exists a value 𝑐, 1 <𝑐<3, such that ℎ ′ 𝑐 =−5

Example 4 Find a value 𝑐, that satisfies the Mean Value Theorem for the function 𝑓 𝑥 =4 𝑥 2 −𝑥−6 on the interval [1, 3] A) 15 B) 2 C) -6 D) 8 B

Example 5 A trucker handed in a ticket at a toll both showing that in 2 hours she had covered 159 miles on a toll road with speed limit 65 mph. Does the trucker deserve a ticket? A) Yes B) No C) Cannot be determined

Example 6 Determine the value of 𝑐 that satisfies the Mean Value Theorem for 𝑓 𝑥 = 1 𝑥−1 on the interval 0, 3 A) 1/2 B) 0 C) 1 D) Cannot be determined

Exit Ticket for Feedback Let 𝑓 𝑥 = 𝑥 4 −2 𝑥 2 . Find all the values of x where 𝑓 ′ 𝑥 =0 on the interval (−2, 2)