4.4 Formula Sheet (Final) - Formula sheet for Final Exam in Math 106 on Derviative formulas(xp = pxp-1 General properties of derivatives(cf = cf(1\/g = g

# 4.4 Formula Sheet (Final) - Formula sheet for Final Exam in...

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Formula sheet for Final Exam in Math 106 on Monday 14, 2007 Derviative formulas ( x p ) = px p - 1 ( e x ) = e x (ln x ) = 1 x General properties of derivatives ( cf ) = cf ( f + g ) = f + g ( fg ) = f g + f g (1 /g ) = - g g 2 ( f/g ) = f g - f g g 2 ( f ( g )) = f ( g ) g Equation to the tangent line to the graph at x 0 : y = f ( x 0 ) + f ( x 0 )( x - x 0 ) Exponentials. lim n →∞ ( 1 + 1 n ) n = e = 2 . 71828. Properties: x p x q = x p + q e x e y = e x + y 1 /x p = x - p 1 /e x = e - x ( x p ) q = x pq ( e x ) y = e xy h ( t ) = Ae rt satisfies h ( t ) = rh ( t ), h (0) = A . If r > 0, t = (ln(2)) /r is the doubling time . If r < 0, t = (ln(1 / 2)) /r is the half-life . Natural Logarithm. By definition e ln x = x . This implies ln( e x ) = x . Properties: ln( uv ) = ln( u ) + ln( v ) ln(1 /u ) = - ln( u ) ln( u/v ) = ln( u ) - ln( v ) ln( u p ) = p ln( u ) Maxima and minima, increasing, decreasing, convex, concave f is increasing where f ( x ) > 0, decreasing where f ( x ) < 0. If f ( c ) = 0 then c is a critical point. (This differs from the book, but is the commonly accepted definition). This is a necessary condition for the point c to be a local maximum [minimum], i.e., there are a < c < b so that f ( c ) is the largest [smallest] value of f on [ a, b ].
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