chad_sheet - B A x y relative maximum f '(x)=0 zero f (x)=0...

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Unformatted text preview: B A x y relative maximum f '(x)=0 zero f (x)=0 relative minimum f '(x)=0 absolute maximum inflection point f ''(x)=0 a b c d e k b f '(x) d e +- + c f ''(x) e +- Area = ∫ f (x) dx a b Domain: ( - ∞ , e ]- ∞ < x ≤ e Range: ( - ∞ , k ]- ∞ < y ≤ k Sample Function f(x) sin ² x + cos ² x = 1 1 + tan ² x = sec ² x 1 + cot ² x = csc ² x cos (a+b) = cos a cos b - sin a sin b sin (a+b) = sin a cos b + cos a sin b cos (a - b) = cos a cos b + sin a sin b sin (a - b) = sin a cos b - cos a sin b transcendental integrals Double Angle Identities sin 2x = 2 sin x cos x cos 2x = cos ² x - sin ² x cos ² x = 1+ cos 2x 2 sin ² x = 1- cos 2x 2 Odd/Even Identities sin (- x) = - sin x cos (- x) = cos x tan (- x) = - tan x cot (- x) = - cot x sec (- x) = sec x csc (- x) = - csc x C B A a c b x sin x =a/c= opposite/hypotenuse cos x = b/c = adjacent/hypotenuse sec x = c/b = hypotenuse/adjacent csc x = c/a = hypotenuse/opposite tan x = a/b = sin x/cos x = opposite/adjacent cot x = b/a = cos x/sin x = adjacent/opposite trig in a nutshell logs in a nutshell personal notes ln (xy) = ln x + ln y ln (x/y) = lnx - lny ln x = n ln x ln e = e = x ln 1 = 0 ln e = 1 ln x lim n → ∞ n = e (1+ ) 1 n if: a = x log x = b b a e 10 log x = log x log x = ln x 3 step test for continuity: 1. f(c) exists 2. lim exists x->c 3. lim = f(c) x->c derivatives integration by parts velocity & motion volumes & areas disc & shell methods partial fractions trig substitutions transcendental derivatives Product Rule f ' (u v) = udv + vdu Chain Rule (f o g)' = f ' g ' Quotient Rule f ' ( ) = v du - u dv v ² u v (Lo D Hi minus Hi D Lo over Lo Lo) f (x + h) - f (x) h f ' (x) = lim h -> 0 definition of the derivative: Power Rule f ' ( x ) = c x Addition Rule f ' (u + v) = f ' u + f ' v c c -1 Mean Value Theorem f (b) - f (a) b - a = f ' (c) l’Hôpital’s Rule When lim f (x) g(x) x → a lim f ' (x) g ' (x) x → a = ∞ ∞ lim f (x) g(x) x → a = OR trig derivatives Standard Trig (d/dx)(csc u) = - csc u cot u (d/dx)(sec u) = sec u tan u (d/dx)(cot u) = - csc ² u (d/dx)(tan u) = sec ² u (d/dx)(cos u) = - sin u (d/dx)(sin u) = cos u d dx 1 du u dx ln u = d dx du dx e = e u u d dx du dx a = a ln a u u 1 1 f (x) = a x f (x) = log x a (Use h(b 1 +b 2 )/2 for trapezoids of different height) ∫ f (x) dx = F(a) - F(b) where F is the antiderivative of f a b integrals Power Rule: ∫ x dx = x +C a+1 x ≠ - 1 a a+1 Average Value f (x) dx Avg. (f (x)) = a b 1 ....
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This note was uploaded on 04/16/2008 for the course PHYS 40 taught by Professor Ellison during the Winter '08 term at UC Riverside.

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chad_sheet - B A x y relative maximum f '(x)=0 zero f (x)=0...

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