We are given the function $f(x) = 3x^2$ and are asked to find:

(A) The slope of the secant line through the points $(2, f(2))$ and $(5, f(5))$ on the graph of $y = f(x)$.

(B) The slope of the secant line through the points $(2, f(2))$ and $(2 + h, f(2 + h))$, where $h \neq 0$, and we need to simplify the answer.

(C) The slope of the graph at $(2, f(2))$, which is the derivative of $f(x)$ evaluated at $x = 2$.

Solution:

(A) The slope of the secant line through $(2, f(2))$ and $(5, f(5))$:

The slope of a secant line between two points $(x_1, f(x_1))$ and $(x_2, f(x_2))$ is given by the formula:

$m = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$

Here, $x_1 = 2$ and $x_2 = 5$. Let's calculate $f(2)$ and $f(5)$:

$f(2) = 3(2)^2 = 3 \times 4 = 12$ $f(5) = 3(5)^2 = 3 \times 25 = 75$

Now, apply the slope formula:

$m = \frac{f(5) - f(2)}{5 - 2} = \frac{75 - 12}{3} = \frac{63}{3} = 21$

So, the slope of the secant line is 21.

(B) The slope of the secant line through $(2, f(2))$ and $(2 + h, f(2 + h))$, where $h \neq 0$:

The slope of the secant line in this case is:

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

We already know that $f(2) = 12$. Now, we need to calculate $f(2 + h)$:

$f(2 + h) = 3(2 + h)^2 = 3(4 + 4h + h^2) = 12 + 12h + 3h^2$

Now, substitute this into the slope formula:

$m = \frac{(12 + 12h + 3h^2) - 12}{h} = \frac{12h + 3h^2}{h} = 12 + 3h$

So, the slope of the secant line is $12 + 3h$.

(C) The slope of the graph at $(2, f(2))$:

The slope of the graph at a point is given by the derivative of the function. Let's compute the derivative of $f(x)$:

$f(x) = 3x^2$ $f'(x) = \frac{d}{dx}(3x^2) = 6x$

Now, evaluate the derivative at $x = 2$:

$f'(2) = 6(2) = 12$

So, the slope of the graph at $(2, f(2))$ is 12.

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Would you like any further details or have any questions?

Here are 5 related questions to expand on this:

1. What is the geometric interpretation of the slope of a secant line?
2. How does the secant slope change as $h \to 0$ in part (B)?
3. How would you find the equation of the tangent line at $x = 2$?
4. What is the second derivative of $f(x) = 3x^2$, and what does it represent?
5. Can you generalize the slope of the secant line for $f(x) = ax^n$?

Tip: The derivative gives you the slope of the tangent line, which shows the rate of change of the function at any point.