Verified Answer :
- Identify the function \(h(x) = -x^2 + 3x + 10\) and the interval \([0, 4]\).
- Calculate the values at the endpoints: \(h(0) = 10\) and \(h(4) = 6\).
- Apply the average rate of change formula \(\frac{h(x_2) - h(x_1)}{x_2 - x_1} = \frac{6 - 10}{4 - 0}\).
- Obtain the result: \(\boxed{-1}\).
Explanation
- Definition and Problem Analysis
The function is given by \(h(x) = -x^2 + 3x + 10\). We are analyzing its behavior over the interval \([x_1, x_2] = [0, 4]\). The average rate of change of a function over an interval is defined as the slope of the secant line connecting the points on the curve at the endpoints of the interval. Mathematically, this is expressed as:
\[\text{Average Rate of Change} = \frac{h(x_2) - h(x_1)}{x_2 - x_1}\]
where \(x_1 = 0\) and \(x_2 = 4\).
- Evaluating Function Values at Endpoints
First, evaluate the function \(h(x)\) at the lower bound \(x_1 = 0\):
\[h(0) = -(0)^2 + 3(0) + 10 = 0 + 0 + 10 = 10\]
Next, evaluate the function at the upper bound \(x_2 = 4\):
\[h(4) = -(4)^2 + 3(4) + 10 = -16 + 12 + 10 = 6\]
So, the points on the curve are \((0, 10)\) and \((4, 6)\).
- Calculating the Average Rate of Change
Now, substitute these values into the average rate of change formula:
\[\text{Average Rate of Change} = \frac{h(4) - h(0)}{4 - 0} = \frac{6 - 10}{4 - 0} = \frac{-4}{4} = -1\]
The average rate of change of the function over the given interval is \(-1\).
- Final Answer
The calculated average rate of change is \(-1\). This means that on average, for every unit increase in \(x\) in the interval \([0, 4]\), the value of the function \(h(x)\) decreases by 1 unit.
The final answer is \(\boxed{-1}\).
Examples
Understanding the average rate of change is essential in many practical scenarios, such as tracking the speed of a car or the growth of a population. For example, if you know the position of a car at two different times, calculating the average rate of change of its position gives you its average velocity over that time period. This simple calculation allows us to make predictions about future trends based on past data, which is a fundamental tool for engineers, economists, and data analysts.