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The image you provided contains a mathematical problem written in Japanese. Here is a translation and breakdown of the problem:

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**Problem 3**: 

Let \( f(x) \) be a function defined as:
\[
f(x) = 
\begin{cases} 
\int_0^x (t^2 + 1) dt & \text{for } x \leq 2 \\
0 & \text{for } x > 2
\end{cases}
\]
Here, the function \( f(x) \) is continuous on the entire real line.

1. **(1)** Calculate \( f(2) \).
2. **(2)** Calculate the derivative \( f'(x) \) for \( x \leq 2 \).
3. **(3)** Given that \( M(y) = \int_0^y f(t) dt \), calculate \( M(2) \) and express \( M'(y) \) for \( y \leq 2 \).
4. **(4)** Based on the results of **(1)**, **(2)**, and **(3)**, calculate \( \lim_{y \to 2} M'(y) \).
5. **(5)** Prove whether \( M(y) \) is a continuous function on the interval \( 0 \leq y \leq 2 \).

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Would you like a detailed solution to any part of this problem or further assistance with the steps? Here are some related questions you might be interested in:

1. How do you evaluate the integral \( \int_0^x (t^2 + 1) dt \)?
2. What does it mean for a function to be continuous?
3. How do you find the derivative of an integral function?
4. What is the significance of calculating \( M(2) \) in the problem?
5. How can you prove the continuity of a function using limits?
6. How do you apply the Fundamental Theorem of Calculus in this context?
7. Why is it necessary to consider the derivative of \( f(x) \) separately for \( x \leq 2 \)?
8. What strategies can you use to evaluate \( \lim_{y \to 2} M'(y) \)?

**Tip:** For solving integrals, always remember to check if the integrand can be simplified or if substitution can make the calculation easier.

Explore an advanced calculus problem involving integral functions, continuity, and derivatives. This problem covers the Fundamental Theorem of Calculus and the Mean Value Theorem for Integrals, with step-by-step solutions and explanations.

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