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# MATH 1207, Week 1 Notes MATH 1207

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This 2 page Class Notes was uploaded by Yasmeen Roumie on Sunday September 4, 2016. The Class Notes belongs to MATH 1207 at Fordham University taught by Quinn Culver in Fall 2016. Since its upload, it has received 31 views. For similar materials see Calculus II in Math at Fordham University.

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Date Created: 09/04/16

Calculus I Review (09/01) A. Differentiation dy ● f a function → f’ (or dx ), the derivative, is a new function that represents the rate of change of f with respect to x. It’s also the slope of the tangent line. 1. Definition f(x+h) − f(x) f(x) = h → 0 h 2. Recall f(x) > 0 → f is increasing f(x) < 0 → f is decreasing f(x) = 0 → local max or min 3. Rules ● d (x ) = nx n − dx ● dx (cf(x)) = c dxf(x) d d d ● dx (f(x) ± g(x)) = dx f(x) ± dx g(x) d ● Product Rule: (fdx) g(x)) = f (x) g(x) + g(x) f(x) ● ( ) = gf′2− f ′ g g ● Trigonometric ○ dx sin x = cos x d ○ dx cos x = − sin x d 2 ○ dx tan x = sec x ● Chain Rule: f(g(x)) = f (g(x)) ∙ ′(x) B. Antiderivatives ● F(x) = ∫(x) dx is a function such that F ′(x) = f(x) ● There are antidifferentiation rules like the differentiation ones ○ Ex: ∫ − 5x + 7 dx = x − x + 7x + c 4 3 1. Integration b ● ∫ f(x) dx means the “signed” or “net” area. a 2. Remark ● ∫ f(x) dx is a function (actually many) b ● ∫ f(x) dx is a number a 3. Riemann Sums: approximate (signed) area with rectangles then take limit as number of rectangles goes to infinity b n ● ∫ f(x) dx = lim ∑ f(a + iΔx) Δx where Δx = bn− a n → ∞ n = 1 4. The Fundamental Theorem of Calculus b ● f(x) dx = F(b) − F(a) where F(x) is an antiderivative of f(x) a ● Ex: 1 3 1 4 2 1 1 23 ∫ (x − 2x − 5) dx = 4 − x − 5x | 0= 4 1 − 5 − (0 − 0 − 0) = − 4 0

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