Calculate the average rate of change of the given function f over the intervals [a, a + h] where h = 1, 0.1, 0.01, 0.001, and 0.0001. (Technology is recommended for the cases h = 0.01, 0.001, and 0.0001.) HINT [See Example 4.] (Round your answers to seven decimal places.)

f (x) = x^2/2; a = 1

a. h = 1

b. h = 0.1

c. h = 0.01

d. h = 0.001

e. h = 0.000

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