137p_Assignment3.pdf - MATH 137 Winter 2020 u2013...

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MATH 137 Winter 2020 – Assignment 3Due: 5 pm Friday, March 13.1. For each of the following, determinedydx.a)y=x2e-1/x3b)y=p4 + sin2x2c)y=e3xd)y= ln (sinx)e)y=(sin2x)xf)x2ey=y2+ 2g)y=f(3x)3h)y=fg(x)x2i)y=3ssinf(x)cosg(x)wheref(x) andg(x) are differentiable functions.2.Find the radius of the circlex2+ (y-R)2=R2which (as shown infigure to the right) smoothly fits the curvey=f(x) at the origin,wheref(0) = 0 andf0(0) = 0.Hint:Makethefirst-andsecond-derivativesofbothcurvesidentical.3.a) Approximate the exponential function exp[x]exby the polynomiala0+a1x+a2x2, where the threecoefficients are determined from the conditions,dndxnexp [x]x=0=dndxn(a0+a1x+a2x2)x=0forn= 0,1,2.b) How would you generalize this procedure to higher degree polynomials,NXn=0anxn, (N >2)?c) In lectures, we used the monotone convergence theorem to prove that the sequencean+1=an+1n!,a1= 1,converges. From part b), what is the limit of this sequence?4. Newton’s method is an algorithm for numerically estimating the root of a functionf(x) by successive intersec-tions of the tangent line and thex-axis. An implicit sequence is formed by solving for the

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