Let f(x) = x 3 + 2x + 1

Question

Let f(x) = x^3 + 2x + 1

(a) Use the limit definition of the derivative to find f '(x). Hint: Use the Binomial Theorem to expand (x + h)^3

(b) Using f '(x), find the formula for a linear approximation to f(x) at a = 2 (here we use a and not x, remember the formula for linear approximations).

Answer 1

(a) f'(x) = lim (f(x + h) - f(x))/h = lim [(x + h)³ + 2(x + h) + 1 - x³ - 2x - 1]/h = lim (x³ + 3x²h + 3xh² + h³ +2h - x³)

h→0 h→0 h→0

= lim (3x² + 3xh + h² + 2) = 3x² + 2

h→0

(b) Linear approximation at x = aL(x) = f(a) + f'(a)(x - a);

a = 2; f(2) = 2³ + 2·2 + 1 = 13; f'(2) = 3·2² + 2 = 14.

L(x) = 13 + 14(x - 2) = 14x - 15