Step-by-step explanation:
Sure,
To find the difference quotient, we use the formula:
(f(x+h) - f(x)) / h
Given f(x) = -4x^2, let's first find f(x+h):
f(x+h) = -4(x+h)^2
Expanding this:
f(x+h) = -4(x^2 + 2xh + h^2)
f(x+h) = -4x^2 - 8xh - 4h^2
Now, subtract f(x) = -4x^2 from f(x+h):
f(x+h) - f(x) = (-4x^2 - 8xh - 4h^2) - (-4x^2)
f(x+h) - f(x) = -8xh - 4h^2
Now, divide by h to get the difference quotient:
(f(x+h) - f(x)) / h = (-8xh - 4h^2) / h
(f(x+h) - f(x)) / h = -8x - 4h
Now, to find f'(x), we take the limit as h approaches 0:
lim (h -> 0) (f(x+h) - f(x)) / h = lim (h -> 0) (-8x - 4h)
f'(x) = -8x
Now, to find f'(-2), f'(1), and f'(0), simply plug in the respective values:
f'(-2) = -8(-2) = 16
f'(1) = -8(1) = -8
f'(0) = -8(0) = 0
