Let f (x) = 4x^{2} + 3.

(a) Find the tangent line to f at x = 3.(b) Use the tangent line from part (a) to approximate f at 3.001.

(a) To find a tangent line we need a point and a slope. The point we want is

(3,f (3)) = (3,4(3)^{2} + 3) = (3,39).

The slope is

f '(3) = 8(3) = 24.

Putting this together (link to point-slope form), we get the line

y = 24x - 33.

(b) We put x = 3.001 into the tangent line and see what we get:

24(3.001) - 33 = 72.024-33 = 29.024.