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The image you uploaded contains a set of mathematical exercises from a test or practice sheet. Here’s a breakdown of the problems and how we can solve them:

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### **Problem 1: Definite Integrals**
1. \[
a. \int_0^2 250 \, u \, du
\]
   **Solution**: Apply the basic integration rule \(\int u \, du = \frac{u^2}{2}\).
   
2. \[
b. \int_0^5 150 \, du
\]
   **Solution**: The integral of a constant \( c \) with respect to \( u \) is \( c \cdot u \).

3. \[
c. \int_5^6 16 \, u^3 \, du
\]
   **Solution**: Use the power rule for integrals \(\int u^n \, du = \frac{u^{n+1}}{n+1}\).

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### **Problem 2: Derivatives**
This section involves taking derivatives (Turunan) of functions:
1. \[
a. \frac{d}{du} \left( 4u^2 - 12u \right)(2u + 2)
\]
   **Solution**: Apply the product rule \(\frac{d}{du}(f(u)g(u)) = f'(u)g(u) + f(u)g'(u)\).
   
2. \[
b. \frac{d}{du} \left( 6u^7 \right)(8u)
\]
   **Solution**: Use the power rule for differentiation and simplify the product.
   
3. \[
c. \frac{d}{du} \left( 2u - 1 \right) \div \left( 5u + 2 \right)
\]
   **Solution**: Apply the quotient rule \(\frac{d}{du}\left( \frac{f(u)}{g(u)} \right) = \frac{f'(u)g(u) - f(u)g'(u)}{(g(u))^2}\).

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### **Problem 3: Limits**
1. \[
a. \lim_{u \to \infty} \left( \sqrt{9u^2 + 5u + 5} - \sqrt{9u^2 - 7u - 4} \right)
\]
   **Solution**: For large values of \( u \), factor out the highest powers of \( u \) inside the square roots.

2. \[
b. \lim_{u \to \infty} \left( 3u + 1 - \sqrt{9u^2 + 4u + 10} \right)
\]
   **Solution**: Analyze the behavior as \( u \to \infty \), focusing on the dominant terms of the expressions.

3. \[
c. \lim_{u \to \infty} \frac{u^2 + 3u + 2}{u + 3}
\]
   **Solution**: Divide both the numerator and denominator by \( u \), then evaluate the limit as \( u \to \infty \).

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### **Problem 4: Solve**
\[
u - 1 = 5u + 2
\]
   **Solution**: Solve the equation for \( u \) by isolating \( u \).

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### **Problem 5: Integral**
\[
\int_1^5 \left( u^2 - 8u + 9 \right) du
\]
   **Solution**: Apply the power rule for integration to each term inside the integral, then evaluate the definite integral from 1 to 5.

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Would you like me to provide detailed solutions for any of these problems?

Here are 5 related questions for further practice:
1. How do you evaluate definite integrals with variable limits?
2. What is the product rule in differentiation?
3. How do you simplify expressions with square roots when taking limits as \( u \to \infty \)?
4. Can you derive the quotient rule for derivatives from basic principles?
5. How can you solve a system of linear equations?

**Tip**: When solving limits involving square roots, try multiplying by the conjugate to simplify the expressions.

User submitted a mathematics test sheet containing various problems related to calculus, derivatives, and limits.

Explore detailed solutions to calculus exercises involving integrals, derivatives, and limits. These problems are designed for college-level students and cover essential concepts and theorems, such as the Fundamental Theorem of Calculus and L'Hopital's Rule.

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