Question

Aer wi rié error and percent relative error. Add terms until the absolute value of the error estimate falls below an error criterion conforming to two significant figures. 3. The following infinite series can be used to approximate ex: e =1+x+ (1.3) 2 3! n! (a) Show that this Maclaurin series expansion is a special case of Taylor expansion with x 0 and h=x (b) Use Taylor series to estimate f(x)=e* at x,=1 for x = 0.20. Employ zero-, first-, second- and third- order approximations and compute the percent relative absolute error for each case.

Answer 1

3) a)

f(1) =e", a = 0

Then

f(1) = f'(x) = ... f(n) (2) =

Which means

f(a) = f'(a) = ... f(n)(a) = e = 1

Thus the Taylor series about $a=0$ is

(ir - a)" f(n)(a) n! + +

That is,

(ix - a) f(n)(a) n! + T + NEO

b) The Taylor series is

for (x – a)"f(n)(a) = -0.2 - 2 -0.2 (2 – 0.2) + (2 n! e-02--0.26.-0.2) - (x -0.2)e-0.2 - + ..

0th order approximate is $e^{-0.2}$

First order approximate is $e^{-0.2}-e^{-0.2}(1-0.2)\approx 0.163$

Second order approximate is

e-0.2(1 -0.2) e-0.2 - e-0.2 (1 -0.2) + 0.426

And third order approximate is $e^{-0.2}-e^{-0.2}(1-0.2)+\frac{e^{-0.2}(1-0.2)^2}{2!}-\frac{e^{-0.2}(1-0.2)^3}{3!}\approx 0.355$

The actual value is

e- 10.368

$\blacksquare$

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