Part (a) 1 1 ax ax ax 2 g(x) = e + f(x) g’(x) = e (ln e) (a) + f’(x) g”(x) = ae (ln e) (a) + f”(x) ax 1 1 g’(0) = e (a) + f’(0) g”(0) = ae (a) + f”(0) g’(0) = a + (-4) g”(0) = (a) (a) + 3 g’(0) = a – 4 g”(0) = a + 3 2

Therefore, h(x) also goes through the point (0,2). Part (b) m = -4 y-2 = -4 (x-0) y = -4x + 2 or u = cos (kx) v = f(x) h(x) = cos (kx) * f(x) h(0) = cos (0) * f(0) 1 2 u’ = -k sin (kx) v’ = f’(x) h’(x) = -k sin (kx) * f(x) + cos (kx) * f’(x) Product Rule required!!! Therefore, h(x) also goes through the point (0,2). h’(0) = -k sin (0) * f(0) + cos (0) * f’(0) 2 * 1 + -4 * 2 h’(0) = -4