Answer:
rc is 1.5 , 1.05 , 1.005 , 1.005, 1.0005 and 1.00005 for h=1, 0.1, 0.01 , 0.001 and 0.0001 respectively
Step-by-step explanation:
for
f(x) = x²/2 ; x=a=1
the average rate of change of f(x) over the time interval [a, a + h] is
rc= [f(a+h) - f(a) ] / [(a+h)-a] = [(a+h)²/2 - a²/2] /h = 1/h [ (a²/2 +a*h + h²/2) - a²/2]
= a + h/2
then
rc= a + h/2
for x=a=1 and h=1
rc= 1 + 1/2 = 1.5
for a=1 and h=1
rc= 1 + 0.1/2 = 1.05
for a=1 and h=0.01
rc= 1 + 0.01/2 = 1.005
for a=1 and h=0.001
rc= 1 + 0.001/2 = 1.0005
for a=1 and h=0.0001
rc= 1 + 0.0001/2 = 1.00005
when h goes smaller , the average rate of change gets closer to the instantaneous rate of change of f(x) in x=a=1 (the derivative of f in a=1) , that is
f'(x) = x
then
f'(a) = a
