formulas - Formulae for Calculus 151 p 1 Differentiation...

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Formulae for Calculus 151 p. 1 Differentiation Definition: f 0 ( a ) = lim h 0 f ( a + h ) - f ( a ) h General formulas: Product : ( uv ) 0 = u 0 v + uv 0 Quotient : ( u/v ) 0 = [ u 0 v - uv 0 ] /v 2 Chain rule : [ f ( u )] 0 = f 0 ( u ) · u 0 Constant multiple : ( cu ) 0 = cu 0 Special functions: Exponential, logarithmic : d dx e x = e x ; d dx ln( x ) = 1 /x d dx a x = ln a · a x Trigonometric : d dx sin( x ) = cos( x ); d dx cos( x ) = - sin( x ) d dx sec( x ) = sec( x ) tan( x ); d dx csc( x ) = - csc( x ) cot( x ) d dx tan( x ) = sec 2 ( x ); d dx cot( x ) = - csc 2 ( x ) d dx sin - 1 ( x ) = 1 1 - x 2 ; d dx cos - 1 ( x ) = - 1 1 - x 2 d dx tan - 1 ( x ) = 1 1 + x 2 ; d dx cot - 1 ( x ) = - 1 1 + x 2 d dx sec - 1 ( x ) = 1 x x 2 - 1 d dx csc - 1 ( x ) = - 1 x x 2 - 1 Mean Value Theorem: f ( x ) cont. on [ a, b ], diff. on ( a, b ): one can solve f 0 ( x ) = f ( b ) - f ( a ) b - a , with a < x < b . Differentials and Newton’s method dy = y 0 dx ; y y 0 + dy f ( a + Δ x ) f ( a ) + f 0 ( a x Newton’s method : x new = x - f ( x ) /f 0 ( x ) (iterate) Intermediate Value Theorem: If f ( x ) is continuous on [ a, b ], f ( a ) < N < f ( b ), then the equation f ( x ) = N is solvable, with a < x < b . Graphing Symmetry: Even : f ( - x ) = f ( x ) (symmetric) Odd : f ( - x ) = - f ( x ) (skew symmetric) Asymptotes: Horizontal : lim x →±∞ f ( x ) = a
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