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Applications of Derivatives: Section 4.2

3 Pages

Mathematics and Statistics
Course Code
MATH 1013
Anthony Anthony

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4.2: The Mean Value Theorem:  Many results rely on this one fact. Rolle's Theorem:  Where a function f satisfies these 4 rules: 1. F is continuous at the closed interval [a,b]. 2. F is differentiable on the open interval (a,b). 3. f(a) = f(b) there is a number c, so that f'(c) = 0. (tangent is horizontal) There are 3 cases to this function for these 3 facts! 1. f(x) = k, f'(x) = 0 The function is a Refer to (a) constant. 2. f(x) > f(a) in Extreme Value The function is Refer to (b) (a,b) Theorem: There is a max never below f(a) and (c - both in [a,b] 2 & 3) Fermat Theorem: f'(c) = 0 3. f(x) < f(a) in Extreme Value The function is Refer to (b) (a,b) Theorem: There is a min never above f(a) and (c - both at [a,b] 2 & 3) Fermat Theroem: f'(c) = 0  We use Rolle's Theorem in proving the Mean Value Theorem. Ex. Apply Rolle's Theorem to the position function: s = f(t) of a moving object.  If the object is in the same place at 2 different times, then f(a) = f(b)  There is an instance between a and b where velocity is 0. f'(c) = 0. o Ex. Think of a ball thrown upwards. Ex. Prove that the equation x + x - 1 = 0 has only one real root. 3 f(x) = x + x - 1 Let a = 0, and b = 1, to create interval [a,b] - We want the smallest possible difference from 0. f(0) = -1 which is < 0 f(1) = 1 which is > 0 Therefore, there is a number c between a and b such that f'(c) = 0. There is a root.  But! Since f'(x) = 3x + 1 >= 1 for all x, since (x >= 0), f'(x) can never be 0.  Therefore, this equation cannot have 2 real roots. The Mea
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