f(x) = 3 − 3x2; x = 2, x = 2 + h (a) Determine the net change between the given values of the variable.(b) Determine the average rate of change between the given values of the variable.

Answer 1

For f(x) = 3x², we'd like to find f(2 + h) minus f(2). Follow the algebraic rules you already know, while substituting 2 + h for x.

f(2 + h) = 3 - 3(2 + h)(2 + h) = -9 - 12h - 3h²

f(2) = -9

So f(2 + h) minus f(2) = -18 - 12h - 3h², which answers (a).

For (b), the average rate of change is the total change in the function values (output values) divided by the change in the input values:

(f(2 + h) - f(2)) / (2 + h) - 2 = (-18 - 12h - 3h²)/h

Answer 2

f(x) = 3-3x**2; x1=2, x2=2+h

(a) net change of f(x):

f1= f(x=2) = 3-3*4 = -9

f2= f(x=2+h) = 3-3(2+h)**2 = 3-3(h**2+4h+4) = 3-3h**2-12h-12 = -3h**2-12h-9

net change = f2 - f1 = (-3h**2-12h-9)-(-9) = -3h**2-12h = -3h(h+4)

(b) avg rate of change = slope = (f2 - f1)/(x2-x1) = -3h(h+4)/(2+h-2) = -3(h+4)