https://r2.erweima.ai/mathbot/image/1726869996720kcqpdc2j.jpg

Let's go over the questions in the image.

1. **First problem:**
   The task is to find the area under the function \( f(x) = \frac{1}{3}x - 1 \) within the limits \( x = 3 \) and \( x = 9 \). The options for answers are:
   - \( 6u^2 \)
   - \( 5u^2 \)
   - \( 7u^2 \)
   - \( 8u^2 \)

   **Steps to solve:**
   To find the area under the curve, you need to compute the definite integral of the function between the given limits.
   \[
   A = \int_{3}^{9} \left( \frac{1}{3}x - 1 \right) dx
   \]
   
   **Integral Calculation:**
   - The integral of \( \frac{1}{3}x \) is \( \frac{1}{6}x^2 \).
   - The integral of \( -1 \) is \( -x \).

   Applying limits:
   \[
   A = \left[ \frac{1}{6}x^2 - x \right]_{3}^{9}
   \]
   Substituting \( x = 9 \) and \( x = 3 \) and subtracting the results will give the area.

2. **Second problem:**
   The task is to determine the derivative of the function \( y = \frac{1}{2}x^8 \). The options are:
   - \( \frac{1}{2}x^7 \)
   - \( \frac{1}{2}x \)
   - \( 8x^7 \)
   - \( 4x^7 \)

   **Derivative Calculation:**
   Using the power rule \( \frac{d}{dx} \left( x^n \right) = nx^{n-1} \), we find the derivative of \( y = \frac{1}{2}x^8 \):
   \[
   \frac{dy}{dx} = \frac{1}{2} \cdot 8x^{7} = 4x^7
   \]
   Thus, the correct derivative is \( 4x^7 \).

Would you like further details for the integration steps or more explanation? Here are 5 related questions you can explore:

1. How do you solve definite integrals step by step?
2. What is the geometric interpretation of the area under a curve?
3. How does the power rule for differentiation apply to polynomials?
4. What are the fundamental theorems of calculus, and how do they relate to integration?
5. Can definite integrals result in negative values?

**Tip:** For definite integrals, always substitute the limits into the antiderivative function carefully to avoid sign mistakes.

Find the area under the function f(x) = (1/3)x - 1 with limits x = 3 and x = 9, and determine the derivative of the function y = (1/2)x^8.

Learn how to find the area under the curve f(x) = (1/3)x - 1 between x = 3 and x = 9, and how to calculate the derivative of y = (1/2)x^8 using the power rule. This problem covers key concepts such as definite integrals and differentiation, suitable for students in Grades 10-12.

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