https://file.aiquickdraw.com/mathbot/file/1726978630549k0z4jr30.jpg

The problem asks to identify statements that are equivalent to "Find the derivative of \( f(x) = x^2 \) at \( x = 3 \)." This can be interpreted as calculating \( f'(3) \), which is the slope of the tangent line to \( f(x) = x^2 \) at \( x = 3 \). 

Let's go through the given options:

1. **Find the average velocity of \( f(x) = x^2 \) on the interval \([0, 3]\):**
   - This calculates the average rate of change of the function over an interval, which is different from finding the derivative at a single point. This is **not equivalent** to finding \( f'(3) \).

2. **Find \( f'(3) \) if \( f(x) = x^2 \):**
   - This is exactly the same as what is asked in the problem: finding the derivative at \( x = 3 \). This is **equivalent**.

3. **Compute the limit \( \lim\limits_{h \to 0} \frac{(3+h)^2 - 3^2}{h} \):**
   - This is the definition of the derivative at \( x = 3 \), using the difference quotient formula. This is **equivalent**.

4. **Find the difference quotient of \( f(x) = x^2 \) at \( x = 3 \):**
   - The difference quotient is used to approximate the derivative. If taken in the limit as \( h \to 0 \), this would give the derivative. This is **equivalent**.

5. **Compute the limit \( \lim\limits_{x \to 3} \frac{x^2 - 3^2}{x - 3} \):**
   - This is another form of the derivative definition at \( x = 3 \) (simplified difference quotient). This is **equivalent**.

6. **Find the slope of the tangent line to the graph of \( f(x) = x^2 \) at the point \( (3,9) \):**
   - The slope of the tangent line is the same as the derivative at \( x = 3 \). This is **equivalent**.

### Summary:
The following options are equivalent to finding the derivative at \( x = 3 \):
- \( \boxed{2, 3, 4, 5, 6} \)

Would you like further clarification or a detailed explanation of any specific part?

Here are 5 related questions to expand on this topic:
1. How do you compute the average rate of change of a function?
2. What is the difference between a difference quotient and a derivative?
3. Why does the limit definition of a derivative give the slope of the tangent line?
4. How would you compute the derivative of \( f(x) = x^2 \) at an arbitrary point \( x = a \)?
5. Can the derivative at a point be negative, and what does that imply about the graph?

**Tip:** Understanding the limit definition of a derivative is crucial for grasping many fundamental concepts in calculus!

Which of the following are equivalent to asking "Find the derivative of f(x) = x^2 at x = 3"? (Select all that apply.)

Understand which statements are equivalent to finding the derivative of f(x) = x^2 at x = 3. Explore concepts like limits, difference quotients, and the definition of the derivative to identify the correct options.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation