Formula sheet for Prelim 2 Thursday March 29 in Uris Hall Aud. f is increasing where f′(x) > 0, f is decreasing where f′(x) < 0 If f′(c) = 0 then c is a critical point. This is a necessary condition for a point to be a local maximum [or mimimum], i.e., so that there are a < c < b so that f(c) is the largest [or smallest] value of f on [a,b]. FIRST DERIVATIVE TEST. If f′(x) > 0 for a < x < c and f′(x) < 0 for c < x < b, x is a local maximum. If f′(x) < 0 for a < x < c and f′(x) > 0 for c < x < b, x is a local minimum. SECOND DERIVATIVE TEST. If f′(c)=0 and f′′(c)<0 then f is concave at c

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Unformatted text preview:c is a local maximum. If f ′ (c)=0 and f ′′ (c)>0 then f is convex at c c is a local minimum. Inflection points have f ′′ (c)=0. 2 2-4 0 has solutions 2 b b ac ax bx c a ± − + + = Antiderivatives (Indefinite integrals) 1 1 for 1 ln | | 1 p kx p kx x e x dx p x dx x e dx p k + − = ≠ − = = + ∫ ∫ ∫ Substitution. Suppose g has antiderivative G. If we let u = f(x), du = f ′ (x) dx ( ( )) '( ) ( ) ( ) ( ( )) g f x f x dx g u du G u G f x = = = ∫ ∫ If there are formulas for derivatives that you want to have with you, write them on the back of this sheet.