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# Math - Mech 221 Math Formulae 2010 Trigonometry sin = o/h...

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Mech 221 Math Formulae 2010 Trigonometry θ h o a sin θ = o/h cos θ = a/h tan θ = o/a tan x = sin x cos x sin 2 x + cos 2 x = 1 sin( - x ) = - sin x cos( - x ) = cos x sin( x + y ) = sin x cos y + cos x sin y cos( x + y ) = cos x cos y - sin x sin y e ix = cos x + i sin x Quadratic Equation ax 2 + bx + c = 0 has roots x = - b ± b 2 - 4 ac 2 a Newton’s Method If an initial guess x 0 is close enough to a root of a function g , then the iteration formula x n +1 = x n - g ( x n ) g ( x n ) gives increasingly good estimates x n of the root. 1

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Elementary Derivatives d dx x r = rx r - 1 ( r = 0) d dx sin x = cos x d dx cos x = - sin x d dx tan x = sec 2 x = 1 / cos 2 x d dx e x = e x d dx ln | x | = 1 /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 ) Taylor Polynomials and Series Taylor polynomial approximation: f ( x ) P n ( x ) = f ( a ) + f ( a )( x - a ) + f ( a ) 2 ( x - a ) 2 + · · · + f ( n ) ( a ) n ! ( x - a ) n Residual formula f ( x ) - P n ( x ) = f ( n +1) ( ξ ) ( n + 1)! ( x - a ) n +1 where ξ is a point between a and x (that is not known). Basic Taylor (McLaurin) series: sin x = x - x 3 6 + x 5 5! - x 7 7! + · · · cos x = 1 - x 2 2 + x 4 4! - x 6 6! + · · · e x = 1 + x + x 2 2 + x 3 6 + x 4 4! + · · · 1 1 - x = 1 + x + x 2 + x 3 + · · · for | x | < 1 Linear Interpolation If f ( a ) and f ( b ) are known and c is in [ a, b ] then f ( c ) b - c b - a f ( a ) + c - a b - a f ( b ) 2
Numerical Integration Approximations to I = b a f ( x ) dx starting from a division of [ a, b ] into N sub-intervals of equal length h = ( b - a ) /N : Trapezoidal Rule: I T N = h 2 f ( a ) + hf ( a + h ) +

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