For the function,
(a) Find the tangent line to the function at the specified value of x.
(b) Use the tangent line from (a) to estimate the value of the function at (x + Δ x).
f (x) = x3 + 2x + 1, x = 1, Δ x = .2
(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