Use linear approximation, i.e. the tangent line, to approximate √(36.4) as follows: Let f ( x ) = √ x . Find the equation of the tangent line to f( x ) at x = 36.

Question

L(x)=__________

Using this, we find our approximation for √(36.4) is

_________________

NOTE: For this part, give your answer to at least 9 significant figures or use an expression to give the exact answer.

Answer 1

With f(x) = x^1/2, the slope f'(x) = (1/2) x^-1/2. The linear approximation is then f(x) ~ f(a) + (x-a)/(2√a). Using 36 as our point a, L(x) = 6 + (x - a)/12.

For x = 36.4, L(36.4) = 6 + 0.4/12 = 6 + 1/30 = 181/30. This reproduces √36.4 to the third decimal place.