137_A3.pdf - MATH 137 Winter 2018 u2013 Assignment 3 Due 5...

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MATH 137 Winter 2018 – Assignment 3Due: 5 pm Friday, March 9.1. For each of the following, determinedydx.a)y=x2e-1/xb)y=p2 + cos2xc)y=exd)y= ln (tanx)e)y= (tanx)xf)xey=y-1g)y=f(x)2h)y=fg(x2)x!i)y=scosf(x)sing(x)wheref(x) andg(x) are differentiable functions.2. Ify= (arcsin(x))2and 0< x <1, then prove that(1-x2)dydx2= 4y,and deduce that(1-x2)d2ydx2-xdydx-2 = 0.Hint:To deduce the second expression, try differentiating the first expression with respect tox.3. Consider the functionf(x) =xx,x0.(a) The pointx= 0 is undefined in the original function definition; determine the value off(0) that ensuresf(x) =xxis continuous atx= 0.(b) Find the pointx?where the function achieves its minimum.Hint:a=elnawhena >0.4. 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 thex-intercept,xn+1=xn-f(xn)f0(xn).The hope is thatxnx?asn→ ∞, wherex?

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