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.