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.

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. \[ \frac{f(x+h)-f(x)}{h}=\square \]
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Solution

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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:

  1. Substitute \(x + h\) into the function to get \(f(x + h)\).
  2. Compute \(f(x + h) - f(x)\).
  3. Divide the result by \(h\).
  4. 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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