To find the Average Rate of Change (ARC) of $f(x)$ as $\Delta x \to 2$, given that $f(x) = 3x^2 - 2x + 11$, we can use the formula:

$\text{ARC} = \frac{f(x + \Delta x) - f(x)}{\Delta x}$

Since we want the ARC specifically as $\Delta x \to 2$, let’s proceed by substituting $\Delta x = 2$ into the formula.

Step 1: Calculate $f(x + 2)$

1. Substitute $x + 2$ into $f(x) = 3x^2 - 2x + 11$: $f(x + 2) = 3(x + 2)^2 - 2(x + 2) + 11$

2. Expand $(x + 2)^2$: $(x + 2)^2 = x^2 + 4x + 4$

3. Substitute this back into $f(x + 2)$: $f(x + 2) = 3(x^2 + 4x + 4) - 2(x + 2) + 11$

4. Distribute and simplify: $f(x + 2) = 3x^2 + 12x + 12 - 2x - 4 + 11$ $f(x + 2) = 3x^2 + 10x + 19$

Step 2: Substitute $f(x)$ and $f(x + 2)$ into the ARC formula

Now we calculate the difference and divide by $\Delta x = 2$: $\text{ARC} = \frac{f(x + 2) - f(x)}{2} = \frac{(3x^2 + 10x + 19) - (3x^2 - 2x + 11)}{2}$

Step 3: Simplify

1. Simplify the expression in the numerator: $f(x + 2) - f(x) = 3x^2 + 10x + 19 - 3x^2 + 2x - 11$ $= 12x + 8$

2. Divide by $\Delta x = 2$: $\text{ARC} = \frac{12x + 8}{2} = 6x + 4$

Answer

The Average Rate of Change (ARC) of $f(x) = 3x^2 - 2x + 11$ as $\Delta x \to 2$ is:

$\boxed{6x + 4}$

Would you like further clarification, or do you have any questions?

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Here are some related questions you might find helpful:

1. How would we find the instantaneous rate of change of $f(x)$ instead?
2. How would changing $\Delta x$ to a different value affect the ARC?
3. Can we apply the same method if $f(x)$ was a cubic function?
4. What is the relationship between ARC and the derivative of $f(x)$?
5. How would the ARC change if $f(x)$ were linear?

Tip: For quadratic functions, the ARC over any interval approaches the slope of the tangent as the interval narrows.