https://file.aiquickdraw.com/mathbot/file/1726838926727u6jqdc85.jpg

Let's analyze the image to determine which statements are equivalent to "Find the derivative of \( f(x) = x^2 \) at \( x = 3 \)."

### The derivative of a function at a point \( x = a \) is defined as:
\[
f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}
\]
For \( f(x) = x^2 \), this becomes:
\[
f'(3) = \lim_{h \to 0} \frac{(3+h)^2 - 3^2}{h}
\]
Thus, calculating this limit gives us the derivative of \( f(x) = x^2 \) at \( x = 3 \).

### Now, let's go through the choices:

1. **Compute the limit \( \lim_{h \to 0} \frac{(3+h)^2 - 3^2}{h} \):**
   - This is the definition of the derivative at \( x = 3 \), so **this is equivalent**.

2. **Find the difference quotient of \( f(x) = x^2 \) at \( x = 3 \):**
   - The difference quotient is used to compute the derivative, so **this is equivalent**.

3. **Find the slope of the tangent line to the graph of \( f(x) = x^2 \) at the point \( (3,9) \):**
   - The derivative at a point gives the slope of the tangent line at that point, so **this is equivalent**.

4. **Find the average velocity of \( f(x) = x^2 \) on the interval \( [0, 3] \):**
   - The average velocity refers to the slope of the secant line between two points, not the derivative at a specific point, so **this is not equivalent**.

5. **Find \( f'(3) \) if \( f(x) = x^2 \):**
   - This directly asks for the derivative at \( x = 3 \), so **this is equivalent**.

6. **Compute the limit \( \lim_{x \to 3} \frac{(3+x)^2 - 3^2}{x-3} \):**
   - This is another form of computing the derivative using the alternative definition of the derivative at a point, so **this is equivalent**.

### Correct answers:
- The equivalent statements are:
  - Compute the limit \( \lim_{h \to 0} \frac{(3+h)^2 - 3^2}{h} \).
  - Find the difference quotient of \( f(x) = x^2 \) at \( x = 3 \).
  - Find the slope of the tangent line to the graph of \( f(x) = x^2 \) at the point \( (3,9) \).
  - Find \( f'(3) \) if \( f(x) = x^2 \).
  - Compute the limit \( \lim_{x \to 3} \frac{(3+x)^2 - 3^2}{x-3} \).

Would you like more details or clarifications? Here are five related questions you might consider:

1. What is the geometric interpretation of a derivative?
2. How does the difference quotient relate to secant lines?
3. Can we always use limits to find the derivative of any function?
4. What does the derivative represent in terms of rates of change?
5. How is the average velocity different from instantaneous velocity?

**Tip:** The limit definition of the derivative can be interpreted as the slope of the tangent line at a particular point on the graph of a function.

Which of the following are equivalent to asking 'Find the derivative of f(x) = x^2 at x = 3?'

Discover how to find the derivative of f(x) = x^2 at x = 3 using limit definitions and tangent lines. This advanced calculus problem explores key mathematical concepts including limits, the difference quotient, and the geometric interpretation of derivatives.

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