R1_MATH_2006

R1_MATH_2006 - R 1 1 J am B M onal d es cD Br i gham Y oung...

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R. 1 1 James B. McDonal d Br i gham Young Uni ver si t y 8/ 01 R. 1 Mat hemat i cal Pr el i mi nar i es A. Summat i on Not at i on. 1. Si ngl e sums n m = i a i i s a conveni ent shor t hand not at i on f or a m + a m+1 + . . . + a n Pr oper t i es of si ngl e summat i ons ( a) a k = ka i n 1 = i i n 1 = i ( b) n i=1 k = k + k + . .. + k = nk n t er ms ( c) b + a = ) b + a ( i n 1 = i i n 1 = i i i n 1 = i 2. Doubl e Summat i ons a + ... + a + a a nj m 1 j= j 2 m 1 j= j 1 m 1 j= ij m 1 j= n 1 = i = = a 11 + a 12 + . . . + a 1m + a 21 + a 22 + . . . + a 2m . . . . . . + a n1 + a n2 + . . . + a nm j-1 n n n n n 2 2 j j i i j i i j i=1 i=1 i=1 j=2 i=1 i=1 i< j ( ) ( + 2 = + 2 a a a a a a a a ) = ∑ ∑ a a + a = j i j i 2 i n 1 = i
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R. 1 2 B. Di f f er ent i at i on 1. Def i ni t i on sl ope of t angent l i ne = f ' ( x 0 ) f ( x) x 0 The der i vat i ve of f ( x) at t he poi nt x 0 i s def i ned by 0 0 h 0 f( + h) - f( ) x x limit h and i s denot ed by f ' ( x O ) or df ( x) / dx. The der i vat i ve i s a f unct i on of x. I f t he der i vat i ve i s def i ned, i t i s equal t o t he sl ope of a t angent l i ne t o t he cur ve at t he poi nt i n quest i on. 2. Some usef ul r ul es nx = dx ) x d( 1 n- n e = dx ) e d( x x x g ) ( g(x) de = g (x) e dx x 1 = dx ln(x) d d ln(g(x)) g (x) = dx g(x) [ ] d f(x) + g(x) = f (x) + g (x) dx { } d f(x)g(x) = f(x)g (x) + g(x)f (x) dx { } 2 d f(x) /g(x) g(x)f (x) - f(x)g (x) = dx (g(x)) df(g(x)) = f (g(x))g (x) dx
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R. 1 3
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R. 1
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This note was uploaded on 02/29/2012 for the course ECON 388 taught by Professor Mcdonald,j during the Winter '08 term at BYU.

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R1_MATH_2006 - R 1 1 J am B M onal d es cD Br i gham Y oung...

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