131BHandout1 - Handout 1 Calculus Review STA 131B For more...

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Handout 1: Calculus Review STA 131B For more details, please see a calculus text. 1. Series a) 1 + a + · · · + a n = 1 - a n +1 1 - a if a 6 = 1 . b) 1 + a + a 2 + · · · = i =0 a i = lim n →∞ 1 - a n +1 1 - a = 1 1 - a if | a | < 1 . c) i =1 ia i - 1 = i =1 d da a i = d da a 1 - a = 1 (1 - a ) 2 , i =2 i ( i - 1) a i - 2 = i =2 d 2 d 2 a a i = d 2 d 2 a i =2 a i = d 2 d 2 a ( a 2 1 - a ) = 2 (1 - a ) 3 for | a | < 1. d) i =0 a i i ! = e a = exp( a ) . e) n k =0 ( n k ) a k b n - k = ( a + b ) n (Binomial Theorem). 2. Limits a) lim n →∞ (1 + a n ) n = e a . b) lim x →∞ e - x x a for all a > 0 . 3. Differentiation of an inverse function d dx f - 1 ( x ) = 1 f 0 ( f - 1 ( x )) . 4. Integration by parts Z b a u ( x ) dv ( x ) = - Z b a v ( x ) du ( x ) + lim x b u ( x ) v ( x ) - lim x a u ( x ) v ( x ) , or for differentiable functions f and g , Z b a f ( x ) g 0 ( x ) dx = - Z b a f 0 ( x ) g ( x ) dx + f ( x ) g ( x ) | b a . 5. Gamma function Definition: Γ( α ) = R 0 e - x x α - 1 dx for α > 0. a) For α > 1 , Γ( α ) = ( α - 1)Γ( α - 1) . b) For α > 0, let [ α ] = greatest integer not exceeding α. Then by (a), Γ( α ) = ( α - 1)( α - 2) · · · ( α - [ α ])Γ( α - [ α ]) .
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  • Fall '08
  • ztan
  • Trigraph, Inverse function, e−x dx, e−x xn dx, limx→∞ e−x xa

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