Question

13 points SCalcET8 11 11,015 Consider the following function. f(x)-x57, a 1, n-3, 0.7Sxs 1.3 (a) Approximate fby a Taylor polynomial with degree n at the number a. T3(x) (b) Use Taylo's Inequality to estimate the accuracy of the a pproximation Rx)· (x) when x lies in the given interval. (Round your answer to eight de IR2(x)I s (c) Check your result in part (b) by graphing IR,(). 0.00015 1.3 0 0.9 1.0 1.1 -0.0000S 0.00010 -0.0001o 0.00005 -0,00015 8 0.910 111213

Answer 1

f(x) = 1.6/7
, $a=1$

Then

48 6 f(a) , f'(a) f"(a) 343 49

Its Taylor polynomial is $f(a)+(x-a)f'(a)/1!+(x-a)^2f''(a)/2!+(x-a)^3f'''(a)/3!$

Which comes out to be

(2-1) 343

The remainder term is

AM n+1

Where

(c) M

Note for the given range, this maximum value is 0.92

So and so

0.92

$|R_3(x)|\leq \frac{0.92}{4!}|x-1|^{4}$ over

0.7<r < 1.3

And so

0.92 R3(z)| <-一10.314 0.0003 105 4!

$\blacksquare$

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