Theory question 109
Differentiation(a) Find, from first principles, the derivative of f(x) = (x + 3)2
(b) Evaluate ∫12 (x + 1)(x2 − 2x + 2)/x2 dx
Show the model answer
Model answer
(a) f(x + h) − f(x) = (x + h + 3)2 − (x + 3)2 = 2h(x + 3) + h2
[f(x + h) − f(x)]/h = 2(x + 3) + h
As h → 0: f'(x) = 2(x + 3) = 2x + 6
(b) (x + 1)(x2 − 2x + 2) = x3 − x2 + 2, so the integrand is x − 1 + 2x−2
∫ = [x2/2 − x − 2/x] from 1 to 2 = (2 − 2 − 1) − (½ − 1 − 2) = −1 + 2.5 = 3/2
Explanation
For (b) multiply out the top first, then divide each term by x².