1. 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

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