The derivative fx of a function fx may be approximated as fx

The derivative, f(x) of a function f\'(x) may be approximated as: f\'(x) = f(x + h) - f(x)/h If f(x) = 0.5e^0.5x and h = 0, 3 find: a) The approximate value of f\'(2) b) The true value of f\'(2) c) The true error for part (a)

Solution

(a) f\'(2) = [f(2+h) - f(2)]/h = [0.5e^(0.5×(2+h)) - 0.5e^(0.5×2)]/h , where h=0.3

= (1.5791 - 1.3591)/0.3 = 0.7333

(b) f\'(x) = 0.25e^(0.5x) . So, f\'(2) = 0.25×e^(0.5×2)=0.6796

(c) True error = (a) - (b) = 0.7333 - 0.6796 = 0.0537