Mathematics: Use linear approximation, i.e. the tangent line, to approximate √16.4 as follows: Let f(x)=√x. Find the equation of the tangent line

trueStep-by-step explanation:...

Use linear approximation, i.e. the tangent line,

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30.01.2022

$L(x)=\frac{1}{8}x+2\\ L(16.4)=4.05$

Step-by-step explanation:

The equation for a tangent line of f(x) in the point (a,f(a)) can be calculated as:

L(x) = f(a) + f'(a)(x-a)

Where L(x) is also call a linear approximation and f'(a) is the value of the derivative of f(x) in (a,f(a)).

So, the derivative of f(x) is:

$f(x)=\sqrt{x} \\f'(x)=\frac{1}{2\sqrt{x} }$

Then, to find the linear approximation we are going to use the point (16, f(16)). So a is 16 and f(a) and f'(a) are calculated as:

$f(16)=\sqrt{16}=4\\f'(16)=\frac{1}{2\sqrt{16} }=\frac{1}{8}$

Then, replacing the values, we get that the equation of the tangent line in (16,4) is:

$L(x)=4+\frac{1}{8}(x-16)\\L(x) = \frac{1}{8}x+2$

Finally, the approximation for $\sqrt{16.4}$ is:

$L(16.4)=\frac{1}{8}(16.4)+2\\ L(16.4)=4.05$

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