Evaluate the difference quotient for the given function. Simplify your answer.f(x) = −x3, f(a + h) − f(a)h
Question
Evaluate the difference quotient for the given function. Simplify your answer.
Given the function:
Evaluate:
rac{f(a + h) - f(a)}{h}
Solution
Sure, let's evaluate the difference quotient for the function f(x) = -x^3.
The difference quotient formula is given by:
(f(a + h) - f(a)) / h
We can substitute the function f(x) = -x^3 into the difference quotient formula:
f(a + h) = -(a + h)^3 f(a) = -a^3
Substitute these into the difference quotient formula:
= [-(a + h)^3 - (-a^3)] / h = [-a^3 - 3a^2h - 3ah^2 - h^3 + a^3] / h = [-3a^2h - 3ah^2 - h^3] / h
Now, we can simplify by dividing each term in the numerator by h:
= -3a^2 - 3ah - h^2
So, the difference quotient for the function f(x) = -x^3 is -3a^2 - 3ah - h^2.
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