Question

Find the linearization L(x) of the function at a.
f(x) = x^1/2, a = 25

Answer 1

What you want is a line that approximates the curve at some point (and, by extension, in some small interval).

f'(x) = 1/(2√x)
f'(25) = 1/10

So, you want the line through (25,5) with slope = 1/10

(y-5) = 1/10 (x-25)
y = x/10 + 5/2

If you restrict your interval of x-values, you can show that the line is as close as you want to the curve in that interval.