Advanced Calculus Problem: Integral Function and Continuity

Math Problem Statement

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

Problem 3:

Let $f(x)$ be a function defined as:

\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$$. --- Would you like a detailed solution to any part of this problem or further assistance with the steps? Here are some related questions you might be interested in: 1. How do you evaluate the integral $$\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.

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