Evaluate the difference quotient for the given function. Simplify your answer.

*f*(*x*) = −*x*3,    

*f*(*a* + *h*) − *f*(*a*)*/h*

To evaluate and simplify the difference quotient \(\frac{f(a + h) - f(a)}{h}\) for the function \(f(x) = -x^3\), follow these steps:

### Step 1: Find \(f(a + h)\)
Substitute \(x = a + h\) into the function \(f(x) = -x^3\):

\[
f(a + h) = -(a + h)^3
\]

Expanding \((a + h)^3\):

\[
(a + h)^3 = a^3 + 3a^2h + 3ah^2 + h^3
\]

Now, apply the negative sign:

\[
f(a + h) = -(a^3 + 3a^2h + 3ah^2 + h^3)
\]
\[
f(a + h) = -a^3 - 3a^2h - 3ah^2 - h^3
\]

### Step 2: Find \(f(a)\)
Substitute \(x = a\) into the function \(f(x) = -x^3\):

\[
f(a) = -a^3
\]

### Step 3: Calculate the Difference Quotient
Now, use the difference quotient formula:

\[
\frac{f(a + h) - f(a)}{h} = \frac{[-a^3 - 3a^2h - 3ah^2 - h^3] - [-a^3]}{h}
\]

Simplify the expression:

\[
\frac{-a^3 - 3a^2h - 3ah^2 - h^3 + a^3}{h}
\]

The \(a^3\) terms cancel out:

\[
\frac{-3a^2h - 3ah^2 - h^3}{h}
\]

### Step 4: Factor and Simplify
Factor \(h\) out of the numerator:

\[
\frac{h(-3a^2 - 3ah - h^3)}{h}
\]

Cancel the \(h\) (assuming \(h \neq 0\)):

\[
-3a^2 - 3ah - h^2
\]

### Final Answer:
The simplified difference quotient \(\frac{f(a+h) - f(a)}{h}\) is:

\[
-3a^2 - 3ah - h^2
\]

Would you like further details or have any questions?

### Related Questions:
1. How would the difference quotient change if the function were \(f(x) = x^3\) instead of \(-x^3\)?
2. Can you find the derivative of \(f(x) = -x^3\) using the difference quotient approach?
3. What is the role of the \(h\) term in the difference quotient as \(h\) approaches zero?
4. How does this difference quotient relate to the slope of the tangent line at \(x = a\)?
5. What would the difference quotient look like for a quadratic function, such as \(f(x) = x^2\)?

### Tip:
Simplifying expressions before dividing by \(h\) is key in difference quotients. Always check for terms that cancel out to avoid unnecessary complexity.

Learn how to evaluate and simplify the difference quotient for the function f(x) = -x^3. This problem involves polynomial functions and requires understanding of the difference quotient concept in calculus. Suitable for advanced high school students in Grades 11-12.

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