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Chapter 4.2Mean Value Theorem

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Rolle's TheoremLet f be a function. If:1. f is continuous on the closed interval [a, b];2. f is differentiable on the open interval (a, b);3. f(a) = f(b)Then ∃ a number c in (a, b) ∋ f '(c) = 0

x

y

a b

1 2

f is differentiable on the open interval (a, b)­to show this, derivative has to exist at every point on (a, b)

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Mean Value TheoremLet f be a function. If:1. f is continuous on [a, b];2. f is differentiable on (a, b);Then ∃ a number c in (a, b) ∋

f '(c) = f(b) ­ f(a) b ­ a

Tangent and Secant line are parallel!

connects instantaneous rate of change with average rate of change

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ex: For the function f(x) = x 2 on the interval [0, 2], show the MVT exists and then find a solution to satisfy the MVT.

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ex: For the function f(x) = x 3 ­ x on the interval [0, 2], show the MVT exists and then find a solution to satisfy the MVT.

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ASSIGNMENTp. 202 #1­3

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Recall: Dig deep for precalc knowledge =)

What does f(x) = |x| look like?

What does f(x) = |x ­ 1| look like?

What does f(x) = |x ­ 1| + 3 look like?

**Absolute value graphs aren't differentiable at their "corner"

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ex: Explain why the MVT fails on the interval [­1, 1]:a. f(x) = |x| + 1

b. f(x) = x3 + 3 for x < 1x2 + 1 for x ≥ 1

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Let f(x) = 1 ­ x 2 on [­1, 1]. Show that MVT exists and if so, find the point where the tangent line is parallel to the secant line AB.

√

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ex: Let f(x) = x on the interval [­1, 1]. Does the MVT apply and if so, find the point where the tangent line is parallel to the secant AB.

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ex: For the equation f(x) = 3x ­ x 2 on the interval [­1, 3], write the equation of the tangent and secant line.

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ASSIGNMENTp. 202 #9­13

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Increasing/Decreasing Functions­With MVT, we can identify exactly where graphs rise and fall ­Functions with positive derivatives are

increasing (f ' > 0) ­Functions with negative derivatives are

decreasing (f ' < 0)

ex: The function f(x) = x 2 is:a. Decreasing on (­ ∞, 0] since f '(x) = 2x < 0 on (­ ∞, 0)b. Increasing on [0, ∞) since f '(x) = 2x > 0 on (0, ∞)

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ex: Find all extreme values of f(x) = x 3 ­ 4x. Then determine where the function is increasing and decreasing.

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ASSIGNMENTp. 202 #15­16, 21, 22, 25