3.6 Formula Sheet - Formula sheet for Prelim 3 in Math 106...

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Formula sheet for Prelim 3 in Math 106 on Tuesday April 24, 2007 Antiderivatives(indefinite integrals) xpdx=xp+1p+ 1forp=-1x-1dx= ln|x|erx dx= erx r Substitution.SupposeG=g. Lettingu=f(x),du=f(x)dx g(f(x))f(x)dx=g(u)du=G(u) =G(f(x)) Integration by parts f(x)g(x)dx=f(x)g(x)-f(x)g(x)dx Definite integrals.The area under the curvefbetweenaandbis given by b a f(x)dx=F(b)-F(a) =F(x)|b a whereFis ANY antiderivative. If we rotate the part of the curvefover the interval [a, b] around thex-axis, the result is a solid of revolutionwith volumeV=b a πf(x)2dx. The improper integral 0 f(x)dx= limb→∞ f(x)dx To find the limit it is useful to know that ifp, q, r >0 then asx→ ∞ e-rx,x-q,xqe-rx,(lnx)-p,(lnx)px-q 0 A probability density hasf(x)0 andf(x)dx= 1. The meanμ
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Unformatted text preview:μ = EX = xf ( x ) dx . The second moment E ( X 2 ) = x 2 f ( x ) dx . The variance σ 2 = E ( X 2 )-μ 2 . Partial derivatives. Given f ( x, y ), ∂f ∂x is computed by holding y constant and differentiat-ing the resulting function of x . Higher order partial derivatives such as ∂ 2 f ∂x∂y are computed by first taking the partial dervia-tive with respect to y , then taking the partial derivative of the result with respect of x . Useful derivative formulas. d dx g ( f ) = g ( f ) f d dx fg = f g + fg d dx f g = f g-fg g 2