Section 4 2 Rolles Thm Mean Value Thm

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Section 4. 2 Rolle’s Th’m & Mean Value Th’m AP Calculus October 26, 2009

Section 4. 2 Rolle’s Th’m & Mean Value Th’m AP Calculus October 26, 2009 Berkley High School, D 2 B 2 todd 1@toddfadoir. com Calculus, Section 4. 2

Rolle’s Th’m (or “What goes up must come down”) n IF (condition) ¨f is

Rolle’s Th’m (or “What goes up must come down”) n IF (condition) ¨f is continuous on [a, b] ¨ f is differentiable on (a, b) ¨ f(a)=f(b) n THEN (conclusion) ¨ There exist a number c in (a, b) such that f’(c)=0 Calculus, Section 4. 2 2

Rolle’s Th’m (or “What goes up must come down”) n IF ¨f is continuous

Rolle’s Th’m (or “What goes up must come down”) n IF ¨f is continuous on [a, b] ¨ f is differentiable on (a, b) ¨ f(a)=f(b) n a c b THEN ¨ There exist a number c in (a, b) such that f’(c)=0 Calculus, Section 4. 2 3

Rolle’s Th’m (or “What goes up must come down”) n Since we know such

Rolle’s Th’m (or “What goes up must come down”) n Since we know such a c exists, we now can solve from c with confidence. a Calculus, Section 4. 2 c b 4

Using Rolle’s Th’m n n Prove that the equation x 3+x-1=0 has exactly one

Using Rolle’s Th’m n n Prove that the equation x 3+x-1=0 has exactly one real root. Let f(x)=x 3+x-1 ¨ continuous n and differentiable everywhere Since f(-10) is a big negative number and f(10) is a big positive number, the Intermediate Value Th’m says that somewhere on (-10, 10) f(x) = 0. ¨ Therefore there exists at least one root. Calculus, Section 4. 2 5

Using Rolle’s Th’m n n Prove that the equation x 3+x-1=0 has exactly one

Using Rolle’s Th’m n n Prove that the equation x 3+x-1=0 has exactly one real root. Suppose there are two roots a and b ¨ If there are two roots, then f(a)=f(b)=0. ¨ Rolle’s Th’m says that somewhere there is c where f’(c) = 0, but we see the f’(x)=3 x 2+1 which is ALWAYS POSITIVE. ¨ Therefore our supposition must be false. n Therefore there is exactly one root. Calculus, Section 4. 2 6

Mean Value Th’m (or “someone’s got to be average”) Translation: On the interval (a,

Mean Value Th’m (or “someone’s got to be average”) Translation: On the interval (a, b) there is at least one place where the average slope is the instantaneous slope. n Calculus, Section 4. 2 7

Mean Value Th’m (or “someone’s got to be average”) Calculus, Section 4. 2 8

Mean Value Th’m (or “someone’s got to be average”) Calculus, Section 4. 2 8

Mean Value Th’m (or “someone’s got to be average”) n There must be a

Mean Value Th’m (or “someone’s got to be average”) n There must be a place on (a, b) where f’(x) = 1 Calculus, Section 4. 2 9

Warnings! n Don’t apply Rolle’s Th’m or The Mean Value Th’m unless the conditions

Warnings! n Don’t apply Rolle’s Th’m or The Mean Value Th’m unless the conditions are met ¨ Continuous on [a, b] ¨ Differentiable on (a, b) Calculus, Section 4. 2 10

Assignment n Section 4. 2, 1 -25, odd Calculus, Section 4. 2 11

Assignment n Section 4. 2, 1 -25, odd Calculus, Section 4. 2 11