MEAN VALUE THEOREM Section 3. Calculus AP/Dual, Revised 017 viet.dang@humbleisd.net 7/30/018 1:16 AM 3.: Mean Value Theorem 1
ACTIVITY A. Draw a curve (x) on a separate sheet o paper within a deined closed interval a, b B. Make sure the graph is continuous and dierentiable C. Pick two points on the curve and connect them with a straight edge through a secant line D. Then, pick a point that will have the same slope line 7/30/018 1:16 AM 3.: Mean Value Theorem
MEAN VALUE THEOREM c ' c b b a a x ' c b a [ a c ] b 7/30/018 1:16 AM 3.: Mean Value Theorem 3
DETERMINE WHERE THE SAME SLOPE EXISTS Given these graphs, determine where the same slope exists rom the points a to b. 0 a C1 C b 0 a C b 7/30/018 1:16 AM 3.: Mean Value Theorem 4
DETERMINE WHERE THE SAME SLOPE EXISTS Given these graphs, determine where the same slope exists rom the points a to b. 0 a C1 C b 0 a C b 7/30/018 1:16 AM 3.: Mean Value Theorem 5
MEAN VALUE THEOREM A. Mean Value Theorem states that i x is deined and continuous on the interval a, b and dierentiable on a, b, then there is at least one number c in the interval a, b (that is a < c < b). 7/30/018 1:16 AM 3.: Mean Value Theorem 6
MEAN VALUE THEOREM EQUATION b a ' c b a b a A. Equation: c = b a B. I is continuous on the closed interval a, b C. I is dierentiable on the open interval a, b then there exists a number c in a, b such that c is instantaneous rate o change D. b a b a is average rate o change E. c is the slope o tangent line F. b a b a is the slope o the secant line 7/30/018 1:16 AM 3.: Mean Value Theorem 7
MEAN VALUE THEOREM EQUATION b a ' c b a Slope o Tangent Line Slope o Secant Line Instantaneous ROC Average ROC 7/30/018 1:16 AM 3.: Mean Value Theorem 8
MEAN VALUE THEOREM (c, (c)) Tangent line Secant line b, b a, a 7/30/018 1:16 AM 3.: Mean Value Theorem 9
COP STORY Video Start at 1:1 to 3:7 7/30/018 1:16 AM 3.: Mean Value Theorem 10
STEPS OF MEAN VALUE THEOREM A. Prove that the unctions are continuous and dierentiable b a B. Identiy what is given and apply the slope equation, c = b a C. Take the derivative o the original unction and equal it to the slope D. Identiy c E. Special Case: Rolle s Theorem 1. c = 0. There must be at least one x-value between a and b at which the graph o has a horizontal tangent 7/30/018 1:16 AM 3.: Mean Value Theorem 11
EXAMPLE 1 Given the unction x = x x x on the interval 1, 1. Show that the Mean Value Theorem applies and ind the c that the theorem guarantees. x x x x 3 1 x x x 1 1 1 1 1 0 1 0 1 x is MVT Applies 3 Continuous and Dierentiable 3 1 x x x 7/30/018 1:16 AM 3.: Mean Value Theorem 1 1 1 1 1 1
EXAMPLE 1 Given the unction x = x x x on the interval 1, 1. Show that the Mean Value Theorem applies and ind the c that the theorem guarantees. c c c c Slope o Tangent Line c c c c 3 ' c 3c c 3 Slope o Secant Line 1 1 1 1 7/30/018 1:16 AM 3.: Mean Value Theorem 13 0 1
EXAMPLE 1 Given the unction x = x x x on the interval 1, 1. Show that the Mean Value Theorem applies and ind the c that the theorem guarantees. Slope o Tangent Line ' c 3c c 3c c 1 3c c 1 0 Slope o Secant Line 0 1 TS : 3, TP : 3c 3c 1c 1 0 c 3c c 1 1 1 0 7/30/018 1:16 AM 3.: Mean Value Theorem 14
EXAMPLE 1 Given the unction x = x x x on the interval 1, 1. Show that the Mean Value Theorem applies and ind the c that the theorem guarantees. 3c c 1 1 c 1 0 3c1 c1 0 c 1, c 1 3 Is x continuous between 1,1? Is x dierentiable between 1,1? x Since is continuous on 1,1 and dierentiable on 1,1, 1 the Mean Value Theorem exists or which there is one value o c 3 7/30/018 1:16 AM 3.: Mean Value Theorem 15 YES YES
EXAMPLE Given the unction x = x x 6 on the interval, 3. Show that the Mean Value Theorem applies and ind the c that the theorem guarantees. x x x 3 0 7/30/018 1:16 AM 3.: Mean Value Theorem 16 6 Continuous and Dierentiable ' c MVT applies b b a a 0 3 0
EXAMPLE Given the unction x = x x 6 on the interval, 3. Show that the Mean Value Theorem applies and ind the c that the theorem guarantees. c 1 ' c c 1 b a 7/30/018 1:16 AM 3.: Mean Value Theorem 17 b a 3 3 3 3 6 6 3 c 1 0 0 5
EXAMPLE Given the unction x = x x 6 on the interval, 3. Show that the Mean Value Theorem applies and ind the c that the theorem guarantees. c 1 0 c Is x continuous between,3? Is x dierentiable between,3? 7/30/018 1:16 AM 3.: Mean Value Theorem 18 1 YES YES x Since is continuous on,3 and dierentiable on,3, the Mean Value Theorem exists or which there is one value o c 1
EXAMPLE Given the unction x = x x 6 on the interval, 3. Show that the Mean Value Theorem applies and ind the c that the theorem guarantees. x Since is continuous on,3 and dierentiable on,3, 1 the Mean Value Theorem exists or which there is one value o c 7/30/018 1:16 AM 3.: Mean Value Theorem 19
EXAMPLE 3 Given the unction x = 6 on the interval, 3. Show that the x Mean Value Theorem applies and ind the c that the theorem guarantees. 6 x 6 6 3 4 3 3 3 6 x 6 3 6 9 3 3 3 3 7/30/018 1:16 AM 3.: Mean Value Theorem 0
EXAMPLE 3 Given the unction x = 6 on the interval, 3. Show that the x Mean Value Theorem applies and ind the c that the theorem guarantees. 3 3 3 Since x is not continuous on,3 and not dierentiable on,3, thereore the Mean Value Theorem does not exist. 7/30/018 1:16 AM 3.: Mean Value Theorem 1
YOUR TURN Given the unction x = 3 5 x on the interval 1, 5. (a) The graph o (x) is given. Estimate the point c where the MVT applies. (b) Show that the Mean Value Theorem applies and ind the c that the theorem guarantees. 7/30/018 1:16 AM 3.: Mean Value Theorem
YOUR TURN A Given the unction x = 3 5 on the interval 1, 5. A) The graph o x (x) is given. Estimate the point c where the MVT applies. c, c 7/30/018 1:16 AM 3.: Mean Value Theorem 3
YOUR TURN B Given the unction x = 3 5 on the interval 1, 5. B) Show that the x Mean Value Theorem applies and ind the c that the theorem guarantees. 5 1 5 ' c 1 3 51 1 5 5 3 3 5 5 1 ' c 5 3 5 51 31 3 5 Continuous and Dierentiable ' c 51 MVT Applies ' c 1 51 7/30/018 1:16 AM 3.: Mean Value Theorem 4
YOUR TURN B Given the unction x = 3 5 on the interval 1, 5. B) Show that the x Mean Value Theorem applies and ind the c that the theorem guarantees. x 35x 1 ' c 5x 5 ' c c 5 c 1 c 5 c 7/30/018 1:16 AM 3.: Mean Value Theorem 5 c 5 5 on I 1,5
YOUR TURN Given the unction x = 3 5 x on the interval 1, 5. ( 5, c ) 7/30/018 1:16 AM 3.: Mean Value Theorem 6
RECAP MEAN VALUE THEOREM EXTREME VALUE THEOREM INTERMEDIATE VALUE THEOREM Continuous Slope o the and Tangent Dierentiable; Line = Slope Slope o the o Tangent Secant Line = Slope o the Secant Line b a c = b a c = where a < c < b b a b a is continuous a closed interval; and a there closed is interval; an absolute there is a max (( (c) (x) )) and an a min ( absolute (c) (x) min () (c) (x) ) is continuous; c is between a and b, a b ; and c is between a and b where a < c < b 7/30/018 1:16 AM 3.: Mean Value Theorem 7
REVIEW CONDITIONS CONTINUOUS ON a, b CONTINUOUS ON a, b DIFFFERENTIABLE ON a, b THEOREMS EXTREME VALUE THEOREM (EVT) INTERMEDIATE VALUE THEOREM (IVT) MEAN VALUE THEOREM (MVT) 7/30/018 1:16 AM 3.: Mean Value Theorem 8
TO EARN FULL CREDIT USING MVT: A. The dierence quotient c = numbers will give you credit. b a b a and the plugging into the B. I the MVT exists, state so but also include continuous in a, b and dierentiable a, b with a and b deined. 7/30/018 1:16 AM 3.: Mean Value Theorem 9
AP MULTIPLE CHOICE PRACTICE QUESTION 1 (NON-CALCULATOR) Let be a unction that is dierentiable on the interval 1, 10. I = 5, 5 = 5 and 9 = 5, which o the ollowing must be true? I. has at least two zeros II. III. The graph o has at least one horizontal tangent line. For some c, < c < 5, c = 3. (A) I only. (B) I and III only. (C) II and III only. (D) I, II, and III. 7/30/018 1:16 AM 3.: Mean Value Theorem 30
AP MULTIPLE CHOICE PRACTICE QUESTION 1 (NON-CALCULATOR) Let be a unction that is dierentiable on the interval 1, 10. I = 5, 5 = 5 and 9 = 5, which o the ollowing must be true? I. has at least two zeros II. The graph o has at least one horizontal tangent line. III. For some c, < c < 5, c = 3. Vocabulary Connections and Process Answer and Justiications IVT MVT Continuous and Dierentiable is dierentiable and thereore Continuous. D I and III is the deinition o the IVT because is continuous. II is the deinition o MVT when is continuous 7/30/018 1:16 AM 3.: Mean Value Theorem 31 and dierentiable and ' c 0.
ASSIGNMENT Worksheet 7/30/018 1:16 AM 3.: Mean Value Theorem 3