https://file.aiquickdraw.com/mathbot/file/17342327910569x4nokhk.jpg

The problem presents a piecewise-defined function \( f(x) \), its graph, and integrals/derivatives over specific intervals. Let's analyze the given information step by step:

---

### **Function Definition**
The function \( f(x) \) is defined as:
\[
f(x) =
\begin{cases}
2x^2 & \text{if } -2 \leq x < 2, \\
(x - 2)^3 & \text{if } 2 \leq x < 4, \\
-\frac{2}{3}(x - 4) + 4 & \text{if } 4 \leq x \leq 10.
\end{cases}
\]

### **Graph Observations**
The graph provided visually represents \( f(x) \), where:
1. Solid points show function values that are taken at the boundaries.
2. Hollow points indicate missing values at boundary transitions.

---

### **Integration Results**
1. **Integral over \([-2, 2]\):**
   \[
   \int_{-2}^2 f(x) \, dx = \frac{32}{3}.
   \]
   Here, \( f(x) = 2x^2 \), so the integral evaluates to the given result.

2. **Integral over \([2, 4]\):**
   \[
   \int_{2}^4 f(x) \, dx = 5.
   \]
   In this case, \( f(x) = (x - 2)^3 \), and the integral results in \( 5 \).

---

### **Derivative Results**
1. **For the interval \((2, 4)\):**
   The derivative is given as:
   \[
   f'(x) = 3(x^2 - 4x + 4).
   \]

2. **For the interval \((4, 10)\):**
   The derivative is:
   \[
   f'(x) = \frac{2}{3}.
   \]

This derivative corresponds to the linear segment \( -\frac{2}{3}(x - 4) + 4 \), which has a constant slope \( \frac{2}{3} \).

---

### **Key Insights**
- The function is continuous except for boundary behavior indicated by hollow points.
- The integrals are correctly evaluated for each piecewise-defined part.
- The derivative expressions align with the behavior of the function across the specified intervals.

Let me know if you need specific calculations or further clarifications!

---

### **Related Questions**
1. How would you calculate the derivative of \( f(x) \) at specific points within each interval?
2. How can you verify the given integral values step-by-step for each segment?
3. What is the behavior of \( f(x) \) as \( x \to 4 \) from the left and right?
4. How would you check the continuity of \( f(x) \) at the boundaries \( x = 2 \) and \( x = 4 \)?
5. How can the function's graph help verify its derivative in each interval?

**Tip:** For piecewise-defined functions, always analyze continuity and differentiability at the boundaries between pieces.

Define a piecewise function f(x) over the interval [-2, 10], analyze its graph, integral, and derivative behavior across specified intervals. Specifically:
- f(x) is given as 2x^2 for [-2, 2), (x - 2)^3 for [2, 4), and -2/3(x - 4) + 4 for [4, 10].
- Evaluate ∫[−2, 2] f(x) dx = 32/3 and ∫[2, 4] f(x) dx = 5.
- Analyze the derivative f'(x) = 3(x^2 - 4x + 4) in (2, 4) and f'(x) = 2/3 in (4, 10).

Analyze a piecewise function f(x) over the interval [-2, 10], including integrals ∫[−2, 2] f(x) dx = 32/3 and ∫[2, 4] f(x) dx = 5. Explore derivatives in specific intervals: f'(x) = 3(x^2 - 4x + 4) for (2, 4) and f'(x) = 2/3 for (4, 10). Learn key concepts in calculus, such as continuity, differentiability, and definite integration.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation