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Let's go through each of these problems step-by-step.

### Problem 2
The question asks for the slope of the tangent line to the graph of \( f(x) = x^2 \) at the point \( A(2, 4) \).

1. First, find the derivative of \( f(x) = x^2 \):  
   \[ f'(x) = 2x \]

2. Substitute \( x = 2 \) into \( f'(x) \) to find the slope at that point:  
   \[ f'(2) = 2 \times 2 = 4 \]

**Answer for Problem 2: A) 4**

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### Problem 3
We need to match each function with its derivative.

Given functions:
1. \( y = x^4 \)
2. \( y = \sin x \)
3. \( y = a^x \)

List of derivatives:
- \( A. \, y = -\cos x \)
- \( B. \, y = \frac{a^x}{\ln a} \)
- \( C. \, y = \frac{x^5}{5} \)
- \( D. \, y = \cos x \)
- \( E. \, y = a^x \ln a \)
- \( F. \, y = 4x^3 \)

Matching:
1. For \( y = x^4 \), the derivative is \( y' = 4x^3 \), which corresponds to **F**.
2. For \( y = \sin x \), the derivative is \( y' = \cos x \), which corresponds to **D**.
3. For \( y = a^x \), the derivative is \( y' = a^x \ln a \), which corresponds to **E**.

**Answer for Problem 3: D) I-F, II-D, III-E**

---

### Problem 4
The question asks to compute the following limit:
\[
\lim_{x \to 3} \frac{x^2 - 5x + 6}{x - 3}
\]

1. Factor the numerator:
   \[ x^2 - 5x + 6 = (x - 3)(x - 2) \]

2. Substitute back into the limit:
   \[
   \lim_{x \to 3} \frac{(x - 3)(x - 2)}{x - 3}
   \]

3. Simplify by canceling \( x - 3 \):
   \[
   = \lim_{x \to 3} (x - 2)
   \]

4. Now substitute \( x = 3 \):
   \[ = 3 - 2 = 1 \]

**Answer for Problem 4: C) 1**

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### Summary of Answers:
- **Problem 2:** A) 4
- **Problem 3:** D) I-F, II-D, III-E
- **Problem 4:** C) 1

Would you like further details on any of these steps, or do you have additional questions?

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### Related Questions:
1. What is the derivative of \( f(x) = x^3 \) at any point \( x \)?
2. How do you find the tangent line to a curve at a specific point?
3. What is the chain rule in differentiation?
4. How can you simplify limits involving indeterminate forms?
5. What is the derivative of an exponential function with base \( a \)?

### Tip:
When calculating a limit, always check if the function can be factored or simplified first to avoid indeterminate forms like \( \frac{0}{0} \).

Реши и дай правильные ответы

Find solutions for calculus problems involving derivatives and limits. Learn how to calculate the slope of a tangent, match functions to derivatives, and solve limits with factoring. Suitable for high school calculus students.

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