Answer

(a) Let's find the tangent line. First we find the value of f when x = 1:

f (1) = 1³ + 2(1) + 1 = 4

so our point is (1,4). Now we find the slope. Since f'(x) = 3x² + 2, the slope at the point (1,4) is

f ' (1) = 3(1)² + 2 = 5.

Putting this together, the tangent line is

y = 5x - 1.

For a sanity check, we can put this into our calculator and make sure it looks like a tangent line, which it does.

(b) We want to estimate f (1 + .2), so we put x = 1.2 into the equation for the tangent line:

5(1.2) - 1 = 6 - 1 = 5