Questions: Compute the difference quotient (f(x+h)-f(x))/h for the function f(x)=-3 x^2-2 x-2. Simplify your answer as much as possible.

Transcript text: Compute the difference quotient $\frac{f(x+h)-f(x)}{h}$ for the function $f(x)=-3 x^{2}-2 x-2$. Simplify your answer as much as possible. \[ \frac{f(x+h)-f(x)}{h}=\square \]

Solution

Solution Steps

To compute the difference quotient $\frac{f(x+h)-f(x)}{h}$ for the function $f(x) = -3x^2 - 2x - 2$, follow these steps:

Substitute $x + h$ into the function to get $f(x + h)$.
Compute $f(x + h) - f(x)$.
Divide the result by $h$.
Simplify the expression.

Step 1: Define the Function

Given the function $ f(x) = -3x^2 - 2x - 2 $.

Step 2: Compute $ f(x + h) $

Substitute $ x + h $ into the function: \[ f(x + h) = -3(x + h)^2 - 2(x + h) - 2 \] Expanding this, we get: \[ f(x + h) = -3(x^2 + 2xh + h^2) - 2x - 2h - 2 = -3x^2 - 6xh - 3h^2 - 2x - 2h - 2 \]

Step 3: Compute $ f(x + h) - f(x) $

Subtract $ f(x) $ from $ f(x + h) $: \[ f(x + h) - f(x) = (-3x^2 - 6xh - 3h^2 - 2x - 2h - 2) - (-3x^2 - 2x - 2) \] Simplifying, we get: \[ f(x + h) - f(x) = -6xh - 3h^2 - 2h \]

Step 4: Divide by $ h $

Divide the result by $ h $: \[ \frac{f(x + h) - f(x)}{h} = \frac{-6xh - 3h^2 - 2h}{h} \] Simplifying, we get: \[ \frac{f(x + h) - f(x)}{h} = -6x - 3h - 2 \]

Final Answer

\[ \boxed{-6x - 3h - 2} \]

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