Question

Question Suppose you are estimating V7 using the second Taylor polynomial of the function Væ about x = 4. Use Taylor's theorem to bound the error. Round your answer to four decimal places.

Answer 1

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$f'(x)=\frac{\mathrm{d} }{\mathrm{d} x}(\sqrt{x})=\frac{\mathrm{d} }{\mathrm{d} x}(x^{1/2})=\frac{1}{2}x^{\frac{1}{2}-1}=\frac{1}{2\sqrt{x}}$

f'(4) = 214 4

$f''(x)=\frac{\mathrm{d} }{\mathrm{d} x}(\frac{1}{2\sqrt{x}} )=\frac{\mathrm{d} }{\mathrm{d} x}(\frac{1}{2}x^{-1/2})=\frac{1}{2}*\frac{1}{2}x^{-\frac{1}{2}-1}=\frac{1}{4x^{3/2}}$

$f''(4)=\frac{1}{4(4)^{3/2}}=\frac{1}{32}$

Second order Taylor polynomial

$T_2(x)=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2$

$T_2(x)=f(4)+f'(4)(x-4)+\frac{f''(4)}{2!}(x-4)^2$

$T_2(x)=2+\frac{1}{4}(x-4)+\frac{1}{64}(x-4)^2$

at x=7

$\sqrt{7}\approx T_2(7)=2+\frac{1}{4}(7-4)+\frac{1}{64}(7-4)^2\boldsymbol{=2.8906}$