Section 3.2 Calculus AP/Dual, Revised ©2017 viet.dang@humbleisd.net Mean Value Theorem Section 3.2 Calculus AP/Dual, Revised ©2017 viet.dang@humbleisd.net 2/19/2019 12:04 AM §3.2: Mean Value Theorem

Activity Draw a curve 𝒇(𝒙) on a separate sheet of paper within a defined closed interval 𝒂, 𝒃 Make sure the graph is continuous and differentiable Pick two points on the curve and connect them with a straight edge through a secant line Then, pick a point that will have the same slope line 2/19/2019 12:04 AM §3.2: Mean Value Theorem

Mean Value Theorem 2/19/2019 12:04 AM §3.2: Mean Value Theorem

Determine Where the Same Slope Exists Given these graphs, determine where the same slope exists from the points 𝒂 to 𝒃. 2/19/2019 12:04 AM §3.2: Mean Value Theorem

Determine Where the Same Slope Exists Given these graphs, determine where the same slope exists from the points 𝒂 to 𝒃. 2/19/2019 12:04 AM §3.2: Mean Value Theorem

Mean Value Theorem A. Mean Value Theorem states that if 𝒇 𝒙 is defined and continuous on the interval 𝒂,𝒃 and differentiable on 𝒂,𝒃 , then there is at least one number 𝒄 in the interval 𝒂,𝒃 (that is 𝒂<𝒄<𝒃). 2/19/2019 12:04 AM §3.2: Mean Value Theorem

Mean Value Theorem Equation If 𝒇 is continuous on the closed interval 𝒂, 𝒃 If 𝒇 is differentiable on the open interval 𝒂, 𝒃 then there exists a number 𝒄 in 𝒂, 𝒃 such that 𝒇′ 𝒄 is instantaneous rate of change 𝒇 𝒃 −𝒇 𝒂 𝒃−𝒂 is average rate of change 𝒇 ′ 𝒄 is the slope of tangent line 𝒇 𝒃 −𝒇 𝒂 𝒃−𝒂 is the slope of the secant line 2/19/2019 12:04 AM §3.2: Mean Value Theorem

Mean Value Theorem Equation 2/19/2019 12:04 AM §3.2: Mean Value Theorem

Mean Value Theorem (𝒄, 𝒇(𝒄)) 𝒃, 𝒇 𝒃 𝒂, 𝒇 𝒂 Tangent line Secant line 2/19/2019 12:04 AM §3.2: Mean Value Theorem

Cop Story Video Start at 1:21 to 3:27 2/19/2019 12:04 AM §3.2: Mean Value Theorem

Steps of Mean Value Theorem Prove that the functions are continuous and differentiable Identify what is given and apply the slope equation, 𝒇 ′ 𝒄 = 𝒇 𝒃 −𝒇 𝒂 𝒃−𝒂 Take the derivative of the original function and equal it to the slope Identify 𝒄 Special Case: ‘Rolle’s Theorem’ 𝒇 ′ 𝒄 =𝟎 There must be at least one 𝒙-value between 𝒂 and 𝒃 at which the graph of 𝒇 has a horizontal tangent 2/19/2019 12:04 AM §3.2: Mean Value Theorem

Example 1 Given the function 𝒇 𝒙 =𝒙 𝒙 𝟐 −𝒙−𝟐 on the interval −𝟏, 𝟏 . Show that the Mean Value Theorem applies and find the 𝒄 that the theorem guarantees. 2/19/2019 12:04 AM §3.2: Mean Value Theorem

Example 1 Given the function 𝒇 𝒙 =𝒙 𝒙 𝟐 −𝒙−𝟐 on the interval −𝟏, 𝟏 . Show that the Mean Value Theorem applies and find the 𝒄 that the theorem guarantees. 2/19/2019 12:04 AM §3.2: Mean Value Theorem

Example 1 Given the function 𝒇 𝒙 =𝒙 𝒙 𝟐 −𝒙−𝟐 on the interval −𝟏, 𝟏 . Show that the Mean Value Theorem applies and find the 𝒄 that the theorem guarantees. 2/19/2019 12:04 AM §3.2: Mean Value Theorem

Example 1 Given the function 𝒇 𝒙 =𝒙 𝒙 𝟐 −𝒙−𝟐 on the interval −𝟏, 𝟏 . Show that the Mean Value Theorem applies and find the 𝒄 that the theorem guarantees. 2/19/2019 12:04 AM §3.2: Mean Value Theorem

Example 2 Given the function 𝒇 𝒙 = 𝒙 𝟐 −𝒙−𝟔 on the interval −𝟐, 𝟑 . Show that the Mean Value Theorem applies and find the 𝒄 that the theorem guarantees. 2/19/2019 12:04 AM §3.2: Mean Value Theorem

Example 2 Given the function 𝒇 𝒙 = 𝒙 𝟐 −𝒙−𝟔 on the interval −𝟐, 𝟑 . Show that the Mean Value Theorem applies and find the 𝒄 that the theorem guarantees. 2/19/2019 12:04 AM §3.2: Mean Value Theorem

Example 2 Given the function 𝒇 𝒙 = 𝒙 𝟐 −𝒙−𝟔 on the interval −𝟐, 𝟑 . Show that the Mean Value Theorem applies and find the 𝒄 that the theorem guarantees. 2/19/2019 12:04 AM §3.2: Mean Value Theorem

Example 2 Given the function 𝒇 𝒙 = 𝒙 𝟐 −𝒙−𝟔 on the interval −𝟐, 𝟑 . Show that the Mean Value Theorem applies and find the 𝒄 that the theorem guarantees. 2/19/2019 12:04 AM §3.2: Mean Value Theorem

Example 3 Given the function 𝒇 𝒙 = 𝟔 𝒙 𝟐 on the interval −𝟐, 𝟑 . Show that the Mean Value Theorem applies and find the 𝒄 that the theorem guarantees. 2/19/2019 12:04 AM §3.2: Mean Value Theorem

Example 3 Given the function 𝒇 𝒙 = 𝟔 𝒙 𝟐 on the interval −𝟐, 𝟑 . Show that the Mean Value Theorem applies and find the 𝒄 that the theorem guarantees. 2/19/2019 12:04 AM §3.2: Mean Value Theorem

Your Turn Given the function 𝒇 𝒙 =𝟑− 𝟓 𝒙 on the interval 𝟏, 𝟓 . The graph of 𝒇(𝒙) is given. Estimate the point 𝒄 where the MVT applies. Show that the Mean Value Theorem applies and find the 𝒄 that the theorem guarantees. 2/19/2019 12:04 AM §3.2: Mean Value Theorem

Your Turn A Given the function 𝒇 𝒙 =𝟑− 𝟓 𝒙 on the interval 𝟏, 𝟓 . A) The graph of 𝒇(𝒙) is given. Estimate the point 𝒄 where the MVT applies. 𝒄, 𝒇 𝒄 2/19/2019 12:04 AM §3.2: Mean Value Theorem

Your Turn B Given the function 𝒇 𝒙 =𝟑− 𝟓 𝒙 on the interval 𝟏, 𝟓 . B) Show that the Mean Value Theorem applies and find the 𝒄 that the theorem guarantees. 2/19/2019 12:04 AM §3.2: Mean Value Theorem

Your Turn B Given the function 𝒇 𝒙 =𝟑− 𝟓 𝒙 on the interval 𝟏, 𝟓 . B) Show that the Mean Value Theorem applies and find the 𝒄 that the theorem guarantees. 2/19/2019 12:04 AM §3.2: Mean Value Theorem

Your Turn Given the function 𝒇 𝒙 =𝟑− 𝟓 𝒙 on the interval 𝟏, 𝟓 . ( 𝟓 , 𝒇 𝒄 ) 2/19/2019 12:04 AM §3.2: Mean Value Theorem

INTERMEDIATE VALUE THEOREM Recap MEAN VALUE THEOREM EXTREME VALUE THEOREM INTERMEDIATE VALUE THEOREM Continuous and Differentiable; Slope of Tangent Line = Slope of the Secant Line 𝒇′ 𝒄 = 𝒇 𝒃 −𝒇 𝒂 𝒃−𝒂 where 𝒂<𝒄<𝒃 𝒇 is in a closed interval; there is an absolute max ( 𝒇(𝒄)≥𝒇(𝒙) ) and an absolute min ( 𝒇(𝒄)≤ 𝒇(𝒙) ) 𝒇 is continuous; 𝒄 is between 𝒂 and 𝒃, 𝒇 𝒂 ≠𝒇 𝒃 ; and 𝒇 𝒄 is between 𝒇 𝒂 and 𝒇 𝒃 where 𝒂<𝒄<𝒃 Slope of the Tangent Line = Slope of the Secant Line 𝒇 ′ 𝒄 = 𝒇 𝒃 −𝒇 𝒂 𝒃−𝒂 Slope of the Tangent Line = Slope of the Secant Line 𝒇 ′ 𝒄 = 𝒇 𝒃 −𝒇 𝒂 𝒃−𝒂 𝒇 is continuous and in a closed interval; there is a max ( 𝒇(𝒄)≥𝒇(𝒙) ) and a min ( 𝒇(𝒄)≤ 𝒇(𝒙) ) 2/19/2019 12:04 AM §3.2: Mean Value Theorem

Review CONDITIONS THEOREMS EXTREME VALUE THEOREM (EVT) CONTINUOUS INTERMEDIATE VALUE THEOREM (IVT) CONTINUOUS ON 𝒂,𝒃 DIFFFERENTIABLE ON 𝒂,𝒃 MEAN VALUE THEOREM (MVT) 2/19/2019 12:04 AM §3.2: Mean Value Theorem

To Earn Full Credit using MVT: The difference quotient 𝒇 ′ 𝒄 = 𝒇 𝒃 −𝒇 𝒂 𝒃−𝒂 and the plugging into the numbers will give you credit. If the MVT exists, state so but also include continuous in 𝒂,𝒃 and differentiable 𝒂,𝒃 with 𝒂 and 𝒃 defined. 2/19/2019 12:04 AM §3.2: Mean Value Theorem

AP Multiple Choice Practice Question 1 (non-calculator) Let 𝒇 be a function that is differentiable on the interval 𝟏, 𝟏𝟎 . If 𝒇 𝟐 =−𝟓, 𝒇 𝟓 =𝟓 and 𝒇 𝟗 =−𝟓, which of the following must be true? 𝒇 has at least two zeros The graph of 𝒇 has at least one horizontal tangent line. For some 𝒄, 𝟐<𝒄<𝟓, 𝒇 𝒄 =𝟑. I only. I and III only. II and III only. I, II, and III. 2/19/2019 12:04 AM §3.2: Mean Value Theorem

AP Multiple Choice Practice Question 1 (non-calculator) Let 𝒇 be a function that is differentiable on the interval 𝟏, 𝟏𝟎 . If 𝒇 𝟐 =−𝟓, 𝒇 𝟓 =𝟓 and 𝒇 𝟗 =−𝟓, which of the following must be true? 𝒇 has at least two zeros The graph of 𝒇 has at least one horizontal tangent line. For some 𝒄, 𝟐<𝒄<𝟓, 𝒇 𝒄 =𝟑. Vocabulary Connections and Process Answer and Justifications 2/19/2019 12:04 AM §3.2: Mean Value Theorem

Assignment Worksheet 2/19/2019 12:04 AM §3.2: Mean Value Theorem