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

why can't you?

step-by-step explanation:

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step-by-step explanation: vsfdvsfvd