4.2 - Rolle & Mean Theorems.pdf - 4.2 Rolle\u2019s Theorem The Mean Value Theorem SWBAT \u221e Understand use Rolle\u2019s Theorem \u221e Understand use the

4.2 - Rolle & Mean Theorems.pdf - 4.2 Rolleu2019s Theorem...

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Rolle’s Theorem & The Mean Value Theorem 4.2 SWBAT: Understand & use Rolle’s Theorem. Understand & use the Mean Value Theorem
Find the relative extrema on the following interval: ࠵? ࠵? = ! " ࠵? # + 5࠵? \$ − 4࠵? " + 7 ࠵? % ࠵? = 2࠵? \$ + 15࠵? " − 8࠵? = 0 ࠵? 2࠵? " + 15࠵? − 8 = 0 ࠵? 2࠵? − 1 ࠵? + 8 = 0 ࠵? = 0 2࠵? − 1 = 0 ࠵? + 8 = 0 ࠵? = ! " ࠵? = −8 0.5, 6.66 0, 7 −8, −761 Do Now 4.2a 2࠵? = 1 ࠵? 0 = ! " 0 # + 5 0 \$ − 4 0 " + 7 = 7 ࠵? ! " = ! " ! " # + 5 ! " \$ − 4 ! " " + 7 = 6.65625 ࠵? −8 = ! " −8 # + 5 −8 \$ − 4 −8 " + 7 = −761
Proves Between the _____________ There is at least one ________ 1) 2) 3) 4) Steps Rolle’s Theorem Let ࠵? be continuous on the closed interval [ ࠵? , ࠵? ] and differentiable on the open interval ࠵? , ࠵? , if: Then there is at least one number ࠵? in ࠵? , ࠵? such that ࠵? ! ࠵? = 0 ࠵? = ࠵? ࠵? ࠵? Factor Solve for ࠵? Find critical points Set ࠵? % ࠵? = 0 ࠵? ! ࠵? = 0 ࠵? ࠵? ࠵? ࠵? − ࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵? ࠵?࠵?࠵?࠵?࠵?࠵?࠵? Theorem 4.3
5) Graph Illustrating Rolle’s Theorem 1. Find the ࠵? − ࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵? & critical points of ࠵? ࠵? = ࠵? " − 3࠵? + 2 1) Factor 2) Solve for ࠵? 4) Find critical points 3) Set ࠵? ! ࠵? = 0 ࠵? − ࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵? ࠵? " − 3࠵? + 2 = 0 ࠵? − 2 ࠵? − 1 = 0 ࠵? − 2 = 0 ࠵? = 2 ࠵? − 1 = 0 ࠵? = 1 ࠵? % ࠵? = 2࠵? − 3 = 0 2࠵? = 3 ࠵? = \$ " ࠵? \$ " = \$ " " − 3 \$ " + 2 ! # \$ " , − ! # 1.5, −0.25 = & # & " + 2 =
Illustrating Rolle’s Theorem 2. Find the ࠵? − ࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵? & critical points of ࠵? ࠵? = ࠵? " − 5࠵? + 4 5) Graph 1) Factor 2) Solve for ࠵? 4) Find critical points 3) Set ࠵?′ ࠵? = 0 ࠵? " − 5࠵? + 4 = 0 ࠵? − 4 ࠵? − 1 = 0 ࠵? − 4 = 0 ࠵? = 4 ࠵? − 1 = 0 ࠵? = 1 ࠵? % ࠵? = 2࠵? − 5 = 0 2࠵? = 5 ࠵? = " ࠵? " = " " − 5 " + 4 = 9 4 5 2 , − 9 4 2.5, −2.25
Illustrating Rolle’s Theorem 1) Find the ࠵? − intercepts 2) Find the endpoints 3) Find the critical points 3. Let ࠵? ࠵? = ࠵? # − 2࠵? " Find all values of ࠵? in the interval −2, 2 such that ࠵?′ ࠵? = 0 ࠵? # − 2࠵? " = 0 ࠵? " ࠵? " − 2 = 0 ࠵? " = 0 ࠵? = 0 ࠵? " − 2 = 0 ࠵? = ± 2 ࠵? ! ࠵? = 4࠵? \$ − 4࠵? = 0 4࠵? ࠵? " − 1 = 0 ࠵? " = 2 ࠵? −2 = −2 # − 2 −2 " = 8 ࠵? 2 = 2 # − 2 2 " = 8 −2,8 2,8 4࠵? = 0 ࠵? " − 1 = 0 ࠵? = 0 ࠵? " = 1 ࠵? = ±1 −1, −1 1, −1 0,0 0 # − 2 0 " = 0 −1 # − 2 −1 " = −1 1 # − 2 1 " = −1
4. Find the the ࠵? − ࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵? & critical points of ࠵? ࠵? = ࠵? − 1 ࠵? − 2 ࠵? − 3 1) Find the ࠵? − intercepts 2) Set ࠵?’ ࠵? = 0 3) Find the critical points ࠵? − 1 = 0 ࠵? = 1 ࠵? − 2 = 0 ࠵? = 2 ࠵? ! ࠵? = 3࠵? " − 12࠵? + 11 = 0 ࠵? ࠵? − 3 = 0 ࠵? = 3 = 0 ࠵? ࠵? = ࠵? " − 3࠵? + 2 ࠵? − 3 ࠵? ࠵? = ࠵? \$ − 6࠵? " + 11࠵? − 6 ࠵? ࠵? Quadratic Equation −࠵? ± ࠵? " − 4࠵?࠵? 2࠵? ࠵?࠵? " + ࠵?࠵? + ࠵? = 0 −12 ± −12 " − 4 3 11 2 3 12 ± 12 6 = 1.42 ࠵?࠵? 2.58 1.42, 0.38 2.58, −0.38 1 1 −1 −1
࠵? ࠵? = 3࠵? " + 17࠵? + 20 1࠵? 3 ࠵? Factor 1 20 1 3 1 3 3 20 = 61 ࠵? + 3࠵? + 32 2 10 1 3 2 3 3 10 = 19 4 5 1 3 4 3 3 5 = 17 5 4 1 3 5 3 3 4 = 26 10 2 1 3 10 3 3 2 = 23 20 1 1 3 20 3 3 1 = 5 4 3࠵? + + 5࠵? + 12࠵? + 20 3࠵? + + 17࠵? + 20 Do Now 4.2b ࠵? + 4 3࠵? + 5
1) Find ࠵? − intercepts 2) Find the critical points 3) Graph 3) 5.