This is a handwritten set of mathematical problems, covering topics like definite integrals, limits, derivatives, and function evaluation. Here’s a breakdown of what each section is asking:

1. Integrals:

   • a. $\int_0^2 250 \, u \, du$
   • b. $\int_0^2 150 \, du$
   • c. $\int_1^2 16 \, u \, du$

2. Derivatives (Turunan):

   • Find the derivative of each function.
   • a. $f(u) = (4u^2 - 12u)(2u+2)$
   • b. $f(u) = (6u^7)(8u)$
   • c. $f(u) = 2u - 1$

3. Limits:

   • a. $\lim_{u \to \infty} \left(\sqrt{9u^2 + 5u + 5} - \sqrt{9u^2 - 7u - 4}\right)$
   • b. $\lim_{u \to \infty} \left(3u + 1 - \sqrt{9u^2 + 9u + 10}\right)$
   • c. $\lim_{u \to 1} \frac{u^2 + 32u + 2}{u + 3}$

4. Integration:

   • a. $\int_1^t u^2 - 8u + 9 \, du$
   • b. $\int_5^t 5u^2 \, du$

5. Derivatives of Complex Functions:

   • a. $f(u) = 5000 u^{20} - 7u$
   • b. $f(u) = 100u^{10} + 5u^7 - 7u - 8$
   • c. $f(u) = \frac{8}{u^{10}} (10u)$

6. More Limits:

   • a. $\lim_{u \to 3} \frac{u^2 + 20u - 12}{u^2}$
   • b. $\lim_{u \to 1} \frac{5u^2 - 3u + 2}{9u - 1}$
   • c. $\lim_{u \to \infty} \frac{2u^2 - u - 6}{3u^2 - 5u - 2}$

7. Function Composition:

   • $F(u) = \frac{u - 1}{u}$, find $F'(u)$.

Do you want to go through solving any of these problems in particular? Let me know if you'd like explanations or solutions for any section.

Related Questions:

1. What is the process for evaluating definite integrals?
2. How do you apply the limit laws when dealing with infinity?
3. What are the rules for differentiating products and composite functions?
4. How can we simplify expressions involving square roots before taking limits?
5. What strategies are effective in solving rational functions as limits approach infinity?

Tip: When calculating derivatives, always look for opportunities to simplify the function first using product or quotient rules.