Compute F'(-4) where F(x) = x^3 + X^2 (using the definition of the derivative)
F'(-4)
= lim(h->0) [((-4+h)^3 + (-4+h)^2) - ((-4)^3+(-4)^2)]/h
= lim(h->0) [(h^3-11h^2+40h-48)-(-48)]/h
= lim(h->0) (h^3-11h^2+40h)/h
= lim(h->0) (h^2-11h+40)
= 40
To compute F'(-4) using the definition of the derivative, we need to calculate the limit as h approaches 0 of the difference quotient:
F'(-4) = lim(h->0) [(F(-4 + h) - F(-4))/h]
First, let's compute F(-4 + h):
F(-4 + h) = (-4 + h)^3 + (-4 + h)^2
Now, let's calculate F(-4):
F(-4) = (-4)^3 + (-4)^2
Next, let's subtract F(-4) from F(-4 + h):
F(-4 + h) - F(-4) = [(-4 + h)^3 + (-4 + h)^2] - [(-4)^3 + (-4)^2]
= (h^3 + 3h^2*(-4) + 3h*(-4)^2 + (-4)^3) + (h^2 + 2h*(-4) + (-4)^2) - (64 + 16)
= h^3 - 12h^2 - 32h - 48
Now, we substitute the difference quotient into the limit:
F'(-4) = lim(h->0) [(F(-4 + h) - F(-4))/h]
= lim(h->0) [(h^3 - 12h^2 - 32h - 48)/h]
Finally, we evaluate the limit. To do this, we factor out h from the numerator:
F'(-4) = lim(h->0) [h^2 - 12h - 32 - 48/h]
As h approaches 0, the last term, 48/h, goes to infinity. So we can ignore it. This leaves us with:
F'(-4) = lim(h->0) [h^2 - 12h - 32]
Plugging in h=0, we get:
F'(-4) = (0)^2 - 12(0) - 32
F'(-4) = -32
Therefore, the value of F'(-4) using the definition of the derivative is -32.