Math Problem Statement

A delivery service charges based on the following piecewise function:

C(x)={10,if 0≤x≤510+2(x−5),if x>5C(x)={10,10+2(x−5),​if 0≤x≤5if x>5​

Graph the piecewise function.

Calculate and interpret the average rate of change from x=3x=3 to x=7x=7

Solution

Let's break this down and approach each part of the problem.

1. Graphing the Piecewise Function

The piecewise function is given as:

\begin{cases} 10 & \text{if } 0 \leq x \leq 5 \\ 10 + 2(x - 5) & \text{if } x > 5 \end{cases}$$ - **For $$0 \leq x \leq 5$$**: The function is constant, meaning $$C(x) = 10$$. - **For $$x > 5$$**: The function follows the equation $$C(x) = 10 + 2(x - 5)$$, which simplifies to $$C(x) = 2x$$. To graph this: - For $$0 \leq x \leq 5$$, plot a horizontal line at $$C(x) = 10$$. - For $$x > 5$$, the graph will start at $$C(5) = 10$$, and then increase according to the equation $$C(x) = 2x$$ for values greater than 5. ### 2. **Calculating the Average Rate of Change from $$x = 3$$ to $$x = 7$$** The average rate of change of a function from $$x = a$$ to $$x = b$$ is given by: $$\text{Average Rate of Change} = \frac{C(b) - C(a)}{b - a}$$ Here, $$a = 3$$ and $$b = 7$$, so we need to find $$C(3)$$ and $$C(7)$$. - **For $$x = 3$$**: Since $$3 \leq 5$$, we use the first piece of the function $$C(x) = 10$$. Thus, $$C(3) = 10$$. - **For $$x = 7$$**: Since $$7 > 5$$, we use the second piece of the function $$C(x) = 2x$$. So, $$C(7) = 2(7) = 14$$. Now, apply the formula for the average rate of change: $$\text{Average Rate of Change} = \frac{C(7) - C(3)}{7 - 3} = \frac{14 - 10}{7 - 3} = \frac{4}{4} = 1$$ ### 3. **Interpretation** The average rate of change from $$x = 3$$ to $$x = 7$$ is **1**. This means that, on average, the cost increases by 1 unit for every 1 unit increase in $$x$$ between $$x = 3$$ and $$x = 7$$. --- ### Would you like to see the graph or need any further explanation? Here are some follow-up questions you can explore: 1. How would the graph change if the piecewise function was defined differently (e.g., using a different slope for $$x > 5$$)? 2. How do we calculate the instantaneous rate of change for this function at $$x = 6$$? 3. How does the average rate of change compare for other intervals, such as $$x = 0$$ to $$x = 5$$? 4. How would you interpret the behavior of the delivery service's cost for $$x > 5$$? 5. What would happen to the average rate of change if the function were continuous at $$x = 5$$? **Tip:** When working with piecewise functions, always check which piece of the function applies at the given $$x$$-values to ensure correct calculations.

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Math Problem Analysis

Mathematical Concepts

Piecewise Functions
Average Rate of Change
Linear Functions

Formulas

C(x) = { 10 if 0 ≤ x ≤ 5; 10 + 2(x - 5) if x > 5 }
Average Rate of Change = (C(b) - C(a)) / (b - a)

Theorems

Piecewise Function Definition
Rate of Change Formula

Suitable Grade Level

Grades 8-10