To solve the problem, we are tasked with finding the average rate of change of $f(x)$ using the formula:

$\frac{f(x+h) - f(x)}{h}$

where $f(x) = 4x^2 - 3x + 2$. Let’s compute step by step.

Step 1: Find $f(x+h)$

Substitute $x+h$ into $f(x)$: $f(x+h) = 4(x+h)^2 - 3(x+h) + 2$

Expand $(x+h)^2$: $(x+h)^2 = x^2 + 2xh + h^2$ $f(x+h) = 4(x^2 + 2xh + h^2) - 3(x + h) + 2$ $f(x+h) = 4x^2 + 8xh + 4h^2 - 3x - 3h + 2$

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

Using $f(x) = 4x^2 - 3x + 2$, subtract $f(x)$ from $f(x+h)$: $f(x+h) - f(x) = (4x^2 + 8xh + 4h^2 - 3x - 3h + 2) - (4x^2 - 3x + 2)$

Simplify: $f(x+h) - f(x) = 8xh + 4h^2 - 3h$

Step 3: Divide by $h$

The average rate of change is: $\frac{f(x+h) - f(x)}{h} = \frac{8xh + 4h^2 - 3h}{h}$

Factor $h$ out from the numerator: $\frac{f(x+h) - f(x)}{h} = \frac{h(8x + 4h - 3)}{h}$

Cancel $h$ (for $h \neq 0$): $\frac{f(x+h) - f(x)}{h} = 8x + 4h - 3$

Final Answer:

$\frac{f(x+h) - f(x)}{h} = 8x + 4h - 3$

Would you like me to clarify any steps, or provide further examples or explanations?

Related Questions:

1. How is the average rate of change formula connected to the derivative?
2. What happens as $h \to 0$ in this formula?
3. How do you expand and simplify $f(x+h)$ for polynomial functions in general?
4. Can this formula be applied to find the slope of the tangent line at a specific point?
5. How do you interpret the average rate of change geometrically?

Tip:

Always expand $(x+h)^n$ carefully using the binomial theorem or direct multiplication to avoid errors in calculations.