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Section 3.2 Calculus AP/Dual, Revised Β©2017 viet.dang@humbleisd.net
Mean Value Theorem Section 3.2 Calculus AP/Dual, Revised Β©2017 2/19/ :04 AM Β§3.2: Mean Value Theorem
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Activity Draw a curve π(π) on a separate sheet of paper within a defined closed interval π, π 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/ :04 AM Β§3.2: Mean Value Theorem
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Mean Value Theorem 2/19/ :04 AM Β§3.2: Mean Value Theorem
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Determine Where the Same Slope Exists
Given these graphs, determine where the same slope exists from the points π to π. 2/19/ :04 AM Β§3.2: Mean Value Theorem
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Determine Where the Same Slope Exists
Given these graphs, determine where the same slope exists from the points π to π. 2/19/ :04 AM Β§3.2: Mean Value Theorem
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Mean Value Theorem A. Mean Value Theorem states that if π π is defined and continuous on the interval π,π and differentiable on π,π , then there is at least one number π in the interval π,π (that is π<π<π). 2/19/ :04 AM Β§3.2: Mean Value Theorem
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Mean Value Theorem Equation
If π is continuous on the closed interval π, π If π is differentiable on the open interval π, π then there exists a number π in π, π such that πβ² π is instantaneous rate of change π π βπ π πβπ is average rate of change π β² π is the slope of tangent line π π βπ π πβπ is the slope of the secant line 2/19/ :04 AM Β§3.2: Mean Value Theorem
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Mean Value Theorem Equation
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Mean Value Theorem (π, π(π)) π, π π π, π π Tangent line Secant line
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Cop Story Video Start at 1:21 to 3:27 2/19/2019 12:04 AM
Β§3.2: Mean Value Theorem
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Steps of Mean Value Theorem
Prove that the functions are continuous and differentiable Identify what is given and apply the slope equation, π β² π = π π βπ π πβπ Take the derivative of the original function and equal it to the slope Identify π Special Case: βRolleβs Theoremβ π β² π =π There must be at least one π-value between π and π at which the graph of π has a horizontal tangent 2/19/ :04 AM Β§3.2: Mean Value Theorem
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Example 1 Given the function π π =π π π βπβπ on the interval βπ, π . Show that the Mean Value Theorem applies and find the π that the theorem guarantees. 2/19/ :04 AM Β§3.2: Mean Value Theorem
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Example 1 Given the function π π =π π π βπβπ on the interval βπ, π . Show that the Mean Value Theorem applies and find the π that the theorem guarantees. 2/19/ :04 AM Β§3.2: Mean Value Theorem
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Example 1 Given the function π π =π π π βπβπ on the interval βπ, π . Show that the Mean Value Theorem applies and find the π that the theorem guarantees. 2/19/ :04 AM Β§3.2: Mean Value Theorem
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Example 1 Given the function π π =π π π βπβπ on the interval βπ, π . Show that the Mean Value Theorem applies and find the π that the theorem guarantees. 2/19/ :04 AM Β§3.2: Mean Value Theorem
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Example 2 Given the function π π = π π βπβπ on the interval βπ, π . Show that the Mean Value Theorem applies and find the π that the theorem guarantees. 2/19/ :04 AM Β§3.2: Mean Value Theorem
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Example 2 Given the function π π = π π βπβπ on the interval βπ, π . Show that the Mean Value Theorem applies and find the π that the theorem guarantees. 2/19/ :04 AM Β§3.2: Mean Value Theorem
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Example 2 Given the function π π = π π βπβπ on the interval βπ, π . Show that the Mean Value Theorem applies and find the π that the theorem guarantees. 2/19/ :04 AM Β§3.2: Mean Value Theorem
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Example 2 Given the function π π = π π βπβπ on the interval βπ, π . Show that the Mean Value Theorem applies and find the π that the theorem guarantees. 2/19/ :04 AM Β§3.2: Mean Value Theorem
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Example 3 Given the function π π = π π π on the interval βπ, π . Show that the Mean Value Theorem applies and find the π that the theorem guarantees. 2/19/ :04 AM Β§3.2: Mean Value Theorem
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Example 3 Given the function π π = π π π on the interval βπ, π . Show that the Mean Value Theorem applies and find the π that the theorem guarantees. 2/19/ :04 AM Β§3.2: Mean Value Theorem
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Your Turn Given the function π π =πβ π π on the interval π, π .
The graph of π(π) is given. Estimate the point π where the MVT applies. Show that the Mean Value Theorem applies and find the π that the theorem guarantees. 2/19/ :04 AM Β§3.2: Mean Value Theorem
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Your Turn A Given the function π π =πβ π π on the interval π, π . A) The graph of π(π) is given. Estimate the point π where the MVT applies. π, π π 2/19/ :04 AM Β§3.2: Mean Value Theorem
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Your Turn B Given the function π π =πβ π π on the interval π, π . B) Show that the Mean Value Theorem applies and find the π that the theorem guarantees. 2/19/ :04 AM Β§3.2: Mean Value Theorem
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Your Turn B Given the function π π =πβ π π on the interval π, π . B) Show that the Mean Value Theorem applies and find the π that the theorem guarantees. 2/19/ :04 AM Β§3.2: Mean Value Theorem
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Your Turn Given the function π π =πβ π π on the interval π, π .
( π , π π ) 2/19/ :04 AM Β§3.2: Mean Value Theorem
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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 πβ² π = π π βπ π πβπ where π<π<π π is in a closed interval; there is an absolute max ( π(π)β₯π(π) ) and an absolute min ( π(π)β€ π(π) ) π is continuous; π is between π and π, π π β π π ; and π π is between π π and π π where π<π<π Slope of the Tangent Line = Slope of the Secant Line π β² π = π π βπ π πβπ Slope of the Tangent Line = Slope of the Secant Line π β² π = π π βπ π πβπ π is continuous and in a closed interval; there is a max ( π(π)β₯π(π) ) and a min ( π(π)β€ π(π) ) 2/19/ :04 AM Β§3.2: Mean Value Theorem
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Review CONDITIONS THEOREMS EXTREME VALUE THEOREM (EVT) CONTINUOUS
INTERMEDIATE VALUE THEOREM (IVT) CONTINUOUS ON π,π DIFFFERENTIABLE ON π,π MEAN VALUE THEOREM (MVT) 2/19/ :04 AM Β§3.2: Mean Value Theorem
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To Earn Full Credit using MVT:
The difference quotient π β² π = π π βπ π πβπ and the plugging into the numbers will give you credit. If the MVT exists, state so but also include continuous in π,π and differentiable π,π with π and π defined. 2/19/ :04 AM Β§3.2: Mean Value Theorem
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AP Multiple Choice Practice Question 1 (non-calculator)
Let π be a function that is differentiable on the interval π, ππ . If π π =βπ, π π =π and π π =βπ, which of the following must be true? π has at least two zeros The graph of π has at least one horizontal tangent line. For some π, π<π<π, π π =π. I only. I and III only. II and III only. I, II, and III. 2/19/ :04 AM Β§3.2: Mean Value Theorem
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AP Multiple Choice Practice Question 1 (non-calculator)
Let π be a function that is differentiable on the interval π, ππ . If π π =βπ, π π =π and π π =βπ, which of the following must be true? π has at least two zeros The graph of π has at least one horizontal tangent line. For some π, π<π<π, π π =π. Vocabulary Connections and Process Answer and Justifications 2/19/ :04 AM Β§3.2: Mean Value Theorem
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Assignment Worksheet 2/19/ :04 AM Β§3.2: Mean Value Theorem
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