Homework 1.pdf - Math 128A Homework 1 Due June 21 1 Determine the limits of the following sequences as n(b an = 3n2 n2 n2 2n 2n 1 n3(c an = sin(n4 6n n

Homework 1.pdf - Math 128A Homework 1 Due June 21 1...

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Math 128A: Homework 1 Due: June 21 1. Determine the limits of the following sequences asn→ ∞.(a)an=3n2+n-2n2-2n.(b)an=(1 +3n)2n.(c)an=sin(n4+6n)n+1.(d)an= lnn2+3n3+3n.2. Show that the following equations have at least one solution in the given intervals.(a) 2xsin(2x) + (x-2)2= 0 in [2, π].(b)e-xsec2(πx) = 1 in [-1,1].3. Definef:RRbyf(x) =x2sin(1x)x6= 00x= 0(a) Show thatfis continuous atx= 0.(b) Isfdifferentiable as well atx= 0?4. Prove that(a) ifx >1, then ln(x)< x-1.(b) the graph off(x) =x5+ 5x+ccrosses thex-axis exactly once, regardless of thevalue of the constantc.5. Find the Taylor polynomial of degree 5 about the pointx= 0 for the following func-tions:(a)f(x) =ex2.(b)f(x) = ln(1 +x).(c)f(x) = cos2(x).6. Find the third Taylor polynomialP3(x) for the functionf(x) =x+ 1 aboutx0= 0.Approximate0.7 and1.1, find upper bounds for the errors and compare with actualerrors. 1
7. Letf(x) = (x-1) ln(x) andx= 1.(a) Find the third Taylor polynomialP3forfaboutx.(b) UseP3to approximatef(0.5). Find an upper bound for the error in the approx-imation and compare it to the actual error.(c) ApproximateR1.5.5f(x)dxusingP3in place off.(d) Find an upper bound for the error in (c).8.BONUS:Suppose a ring is heated so that the temperature varies continuously overit. Show that there exists a pair of diametrically opposite points (also calledantipodalpoints) that have the same temperature. 2

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