How do you estimate delta f using Linear Approximation and use a calculator to compute both the error and the percentage error given #f(x)=cos(x)# a=pi/4 and delta x=0.06?

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Answer 1 · 2017-04-07T18:28:06

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The linear approximation is:

#deltaf_("lin") = f(x + delta x) - f(x)#

#= (color(red)(f(x) + deltax cdot f'(x)))) - f(x) #

#= deltax cdot f'(x)#

#f(x) = cos x implies f'(x) = -sin x#

So the estimate is:

#deltaf_("lin") = - 0.06 sin (pi/4) approx -0.04color(blue)(243)#, from my calculator.

In actual fact:

#deltaf_("actual") = cos (pi/4 + 0.06) - cos (pi/4) = -0.04color(red)(367)#, again from my calculator.

To finish this off, as I'm sure you know:

The error is: #deltaf_("actual") - deltaf_("lin")#
The %-age error is: #(deltaf_("actual") - deltaf_("lin"))/(deltaf_("actual"))#