evaluate the different quotient for the function given simplify your answer. f(x)=-x^3
there is no quotient.
-x^3 is a polynomial
ah -- maybe you meant "difference" quotient.
well, geez -- you have the formula, right?
(f(x+h)-f(x))/h
now plug in -x^3 for f(x)
To evaluate the difference quotient for the function f(x) = -x^3, we first need to understand what the difference quotient is.
The difference quotient measures the average rate of change of a function over a small interval. It is written as:
[f(x + h) - f(x)] / h
In this case, we substitute f(x) with -x^3.
So, the difference quotient for f(x) = -x^3 is:
[(-x - h)^3 - (-x)^3] / h
To simplify this expression, we need to expand the cube of a binomial (-x - h)^3. Using the binomial formula, we get:
(-x - h)^3 = (-x - h)(-x - h)(-x - h) = (-x - h)(x^2 + 2xh + h^2) = -x^3 - 3x^2h - 3xh^2 - h^3
Now we substitute this expression back into the difference quotient:
[(-x^3 - 3x^2h - 3xh^2 - h^3) - (-x^3)] / h
Simplifying further, we get:
[-x^3 - 3x^2h - 3xh^2 - h^3 + x^3] / h
= (-3x^2h - 3xh^2 - h^3) / h
= -3x^2 - 3xh - h^2
Therefore, the simplified form of the difference quotient for f(x) = -x^3 is:
-3x^2 - 3xh - h^2