Section 4.2 Rolle’s Theorem & Mean Value Theorem Calculus Winter, 2010.

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Section 4.2 Rolle’s Theorem & Mean Value Theorem Calculus Winter, 2010

Calculus, Section 4.12 Rolle’s Theorem (or “What goes up must come down”) IF (condition)  f is continuous on [a,b]  f is differentiable on (a,b)  f(a)=f(b) THEN (conclusion)  There exist a number c in (a,b) such that f’(c)=0

Calculus, Section 4.13 Rolle’s Theorem (or “What goes up must come down”) IF  f is continuous on [a,b]  f is differentiable on (a,b)  f(a)=f(b) THEN  There exist a number c in (a,b) such that f’(c)=0 a b c

Calculus, Section 4.14 Rolle’s Theorem (or “What goes up must come down”) Since we know such a c exists, we now can solve from c with confidence. a b c

Calculus, Section 4.15 Using Rolle’s Theorem Prove that the equation x 3 +x-1=0 has exactly one real root. Let f(x)=x 3 +x-1  continuous and differentiable everywhere Since f(-10) is a big negative number and f(10) is a big positive number, the Intermediate Value Theorem says that somewhere on (-10,10) f(x) = 0.  Therefore there exists at least one root.

Calculus, Section 4.16 Using Rolle’s Theorem 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 Theorem says that somewhere there is c where f’(c) = 0, but we see the f’(x)=3x 2 +1 which is ALWAYS POSITIVE.  Therefore our supposition must be false. Therefore there is exactly one root.

Calculus, Section 4.17 Mean Value Theorem (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.

Calculus, Section 4.18 Mean Value Theorem (or “someone’s got to be average”)

Calculus, Section 4.19 Mean Value Theorem (or “someone’s got to be average”) There must be a place on (a,b) where f’(x) = -1

Calculus, Section Warnings! Don’t apply Rolle’s Theorem or The Mean Value Theorem unless the conditions are met  Continuous on [a,b]  Differentiable on (a,b)

Calculus, Section Assignment Section 4.2, # 1, 4, 6, 9, 11, 15, 17, 19, 21, 26, 29, 31, 40, 43