We are given the function $f(x) = 2x^2 + C$, where $C$ is a constant, and we are tasked with finding the limit of the difference quotient:

$\frac{f(2+h) - f(2)}{h}$

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

Using the given function $f(x) = 2x^2 + C$, we calculate $f(2+h)$:

$f(2+h) = 2(2+h)^2 + C = 2(4 + 4h + h^2) + C = 8 + 8h + 2h^2 + C$

Step 2: Calculate $f(2)$

Next, we calculate $f(2)$:

$f(2) = 2(2^2) + C = 8 + C$

Step 3: Difference of $f(2+h)$ and $f(2)$

Now, subtract $f(2)$ from $f(2+h)$:

$f(2+h) - f(2) = (8 + 8h + 2h^2 + C) - (8 + C) = 8h + 2h^2$

Step 4: Expression $\frac{f(2+h) - f(2)}{h}$

Now, divide by $h$:

$\frac{f(2+h) - f(2)}{h} = \frac{8h + 2h^2}{h} = 8 + 2h$

Step 5: Express in the form $Ah + B$

The expression $8 + 2h$ is already in the form $Ah + B$, where $A = 2$ and $B = 8$.

Thus:

• (a) $A = 2$
• (b) $B = 8$

Step 6: Find $f'(2)$

The derivative $f'(x)$ of the function $f(x) = 2x^2 + C$ is:

$f'(x) = 4x$

Therefore, $f'(2)$ is:

$f'(2) = 4(2) = 8$

Thus, the answers are:

• (a) $A = 2$
• (b) $B = 8$
• (c) $f'(2) = 8$

---

Would you like more details on any of the steps, or do you have any questions?

Here are 5 related questions to extend your understanding:

1. How do you find the derivative of any quadratic function?
2. What is the difference quotient and why is it useful in calculus?
3. How does the constant $C$ affect the derivative of the function?
4. Can you find $f'(x)$ for other values of $x$ in this problem?
5. How would the difference quotient change if the function was cubic?

Tip: The difference quotient is a key concept in calculus that leads to the derivative, which measures the rate of change of a function at a point.