Consider the following function. nx)-x2/3, a-1, n.3, 0.7sx 1.3 (a) Approximate f

Question

Consider the following function. nx)-x2/3, a-1, n.3, 0.7sx 1.3 (a) Approximate fby a Taylor polynomial with degree n at the number a 81 (b) Use Taylor's Inequality to estimate the accuracy of the approximation f(x) Tn(x) when x lies in the given interval. (Round your answer to eight decimal places.) R2(x) 0.00012651X (c) Check your result in part (b) by graphing Rn(x)l 0.0004 0.9 1.0 1.1 21.3 0.0003 -0.0001 0.0002 -0.0002 -0.0003 -0.0004 0.0001 0.8 0.9 1.0 1.1 1.2 13 0.0004 0.8 0.9 1.2 13 -0.0001 -0.0002 -0.0003 -0.0004 0.0003 0.0002 0.0001 0.8 0.9 1.0 . .2 13

Explanation / Answer

b>
Taylor's Inequality tells us :

|Rn| <= M/(n+1)! *|x-a|^(n+1)

now we have n = 3 and a = 1

=> The inequality becomes :

|R3(x)| <= (M/4!) *|x-1|^(4)

or |R3(x)| = |f(x) - T3(x)| <= (M/4!) *|x-1|^(4)

M is the upper limit for f ''''(x) on the given interval for x, that is x E [0.7 , 1.3]

Now we are given that f(x) = x^(2/3)

=> f '(x) = (2/3)*x^(-1/3)

f ''(x) = (-2/9)*x^(-4/3)

f '''(x) = (-2/9)(-4/3)*x^(-7/3) = (8/27)*x^(-7/3)

and f ''''(x) = (8/27)(-7/3)*x^(-10/3) = (-56/81)*x^(-10/3)

Lets find the value of the fourth derivative that is f ''''(x) at the end-points of the given interval

for x, that is x E [0.7 , 1.3]

=> f ''''(x) = (-56/81)*x^(-10/3)

=> f ''''(0.7) = (-56/81)*(0.7)^(-10/3) = - 2.27009   

=> f ''''(1.3) = (-56/81)*(1.3)^(-10/3) = - 0.28833

So we could see that as we move from left to right on the interval x E [0.7 , 1.3] the value of

f ''''(x) = (-56/81)*x^(-10/3) is increasing. So its maximum value is maximum value is attained at

x = 1.3

=> As M is the upper limit of f ''''(x) on the givne interal

=> M = f ''''(1.3) = - 0.28833

=> The Taylors inequality becomes :

|R3(x)| <= (M/4!) *|x-1|^(4)

=> |R3(x)| <= (- 0.28833/24) *|x-1|^(4)

As |x-1| <= 1.3 on the interval x E [0.7 , 1.3]

=> |R3(x)| <= (-0.28833/24) *(1.3)^(4)

or |R3(x)| <= - 0.03431247

Hence, the approximation is accurate to within - 0.03431247
   

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