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) = 4x + ex, x = 0, Δ x = 0.05
(a) We know f (0) = 4(0) + e0 = 1, so the point we need for the tangent line is (0,1). The derivative of f is 4 + ex, so the slope of the tangent line is
f '(0) = 4 + e0 = 5.
Putting this together, the equation for the tangent line is
y = 5x + 1
(b) Plug 0 + 0.05 = 0.05 into the tangent line equation:
5(0.05) + 1 = 1.25