GATE Mechanical Engineering (ME) 2019 Shift 2 Solved Paper

Question : 54 of 65

Marks: +1, -0

The derivative of

f(x)=\cos (x)

can be estimated using the approximation

f^{'}(x)=\;{f(x+h)-f(x-h)}/{2 h}

. The percentage error is calculated as

(\;{\;\text " Exact value-Approximate value "\;}/{\;\text " Exact value "\;}) × 100

. The percentage error in the derivative of

f(x)

x=π \/ 6

radian, choosing

{h}=0.1

radian, is

Solution:

f(x)=\cos (x)

⇒ {f}{'}( {x})=-\sin ( {x})=-\sin (π / 6)=-0.5

(exact value)
Approximate value

=f^{'}(x)=\;{f(x+h)-f(x-h)}/{2 h}

f{'}(x)=\;{\cos (x+h)-\cos x-h}/{2 h}

\cos (x+h)=\cos x \cosh -\sin x \sinh

\cos (x-h)=\cos x \cosh +\sin x \sin h

So,

f{'}(x)=\;{\cos x \cosh -\sin x \sin h-[\cos x \cosh +\sin x \sinh ]}/{2 h}

=\;{-2 \sin x \sin h}/{2 h}=\;{-\sin x \sinh }/{h}

Or,

f{'}(x)=\;{-\sin ({x}/{6}) \sin (0.1)}/{0.1}=\;{-0.5 × 0.0999}/{0.1}=-0.4992

So, percentage error

=\;{-0.5-(-0.4992)}/{-0.5} × 100=0.166

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