Math Problem Statement

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).

Solution

The problem presents a piecewise-defined function f(x)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)f(x) is defined as:

\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.

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Math Problem Analysis

Mathematical Concepts

Piecewise functions
Definite integration
Derivatives
Continuity and differentiability

Formulas

∫ f(x) dx for definite integrals
Derivative rules for polynomials: d/dx [x^n] = n*x^(n-1)
Slope of a linear function: y = mx + c → m = derivative

Theorems

Fundamental Theorem of Calculus
Piecewise function continuity and differentiability

Suitable Grade Level

Grades 11-12

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