Exam 2 review sheet

# Exam 2 review sheet - Center of Mass 2-D=M/m M=mixi m=mi...

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much salt in tank y(t) after t mins? –find y(0) –y’=(rate coming in – rate going out kg/min) DE: y’=0.75-0.005y -0.005(y-150) k(y-T) y(t)=T+(y 0 -T)e kt 150+(20-150)e -0.005t oo) [(1/n)-(1/n+1)] (cannot split up b/c individual series are divergent. Telescoping = lim(n to oo) 1-[1/(n+1)] = 1 Ex. Integral test: x) = 1/x ; by definition of derivative at x=1: 1+h) – ln(1)]/[h] ; lim ln(1+h) 1/h ; let n= 1/h ; lim(n->oo) ln[1+(1/n)] n = 1 ; e ln[1+(1/n)]n = e 1 ; [1+(1/n)] n = e r series: T x =f(a)/0! + [f’(a)(x-a)]/1! + … + [f x’ (a)(x-a) x ]/x! + x + [x 2 /2!] + … + [x n /n!] = x – [x 3 /3!] + [x 5 /5!] - … = 1 - [x 2 /2!] + [x 4 /4!] - [x 6 /6!] + … (1 to oo) [1/x p ] p>1 conv. p 1 div. /x p ] p<1 conv. p 1 div. n=1 to oo) [1/n p ] converges when p>1 and diverges when p 1 Arclength L = sqrt.[1 + [f’(x)] 2 ] dx Surface Area S = 2 f(x) sqrt.[1 + [f’(x)] ∫ π 2 ] dx Quotient Rule [uv’ – u’v]/u 2 ; u is bottom func. Cos 2 x = 1 + cos(2x) = 1- sin 2 x Cos(2x) = 1 – 2sin 2 x = cos 2 x – sin 2 x Sin(2x) = 2sinxcosx sin 2 x = 0.5x-0.25sin(2x) cos 2 x = 0.5x + 0.25 sin(2x) c Functions: e x + e -x ]/2 coshx = sinhx +c e x – e -x ]/2 sinhx = coshx +c sinhx/coshx tanhx = ln(coshx) +c [d/dx] tanhx = sech 2 x /coshx sechx = tan -1 (sinhx) +c 1/sinhx
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• Fall '07
• Diniz-Behn
• dx Arclengthsinhx sqrt., neg. infin, w/ convergent series, e-0.005t oo

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