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Nitella [24]
3 years ago
9

If the local linear approximation of f(x) = 2cos x + e^2x at x = 2 is used to find the approximation for f(2.1), then the % erro

r of this approximation is?
Mathematics
1 answer:
yKpoI14uk [10]3 years ago
5 0

<u>Answer-</u>

The % error of this approximation is 1.64%

<u>Solution-</u>

Here,

\Rightarrow f(x) = 2\cos x + e^{2x}

\Rightarrow f'(x) = -2\sin x + 2e^{2x}

And,

\Rightarrow f(2) = 2\cos 2 + e^{4}

\Rightarrow f'(2) = -2\sin 2 + 2e^{4}

Taking (2, f(2)) as a point and slope as, f'(2), the function would be,

\Rightarrow y-y_1=m(x-x_1)

\Rightarrow y-(2\cos 2 + e^{4})=(-2\sin 2 + 2e^{4})(x-2)

\Rightarrow y=(-2\sin 2 + 2e^{4})(x-2)+(2\cos 2 + e^{4})

The value of f(2.1) will be

\Rightarrow y=(-2\sin 2 + 2e^{4})(2.1-2)+(2\cos 2 + e^{4})

\Rightarrow y=(-2\sin 2 + 2e^{4})(0.1)+(2\cos 2 + e^{4})

\Rightarrow y=64.5946

According to given function, f(2.1) will be,

\Rightarrow f(2.1) = 2\cos 2.1 + e^{2(2.1)}

\Rightarrow f(2.1) = 65.6766

\therefore \%\ error=\dfrac{65.6766-64.5946}{65.6766}=0.0164=1.64\%

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