Use linear approximation to approximate 25.3‾‾‾‾√ as follows. Let f(x)=x√. The equation of the tangent line to f(x) at x=25 can be written in the form y=mx+b. Compute m and b.

Approximation f(25.3)=5.03 (real value=5.0299)

The approximation can be written as f(x)=0.1x+2.5

Step-by-step explanation:

We have to approximate with a linear function.

To approximate a function, we can use the Taylor series.

The point a should be a point where the value of f(a) is known or easy to calculate.

In this case, the appropiate value for a is a=25.

Then we calculate the Taylor series with a number of terms needed to make a linear estimation.

The value of f'(a) needs the first derivate:

Then

We evaluate for x=25.3

If we rearrange the approximation to be in the form mx+b we have:

Then, m=0.1 and b=2.5.

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