Questions: The height (in feet) of a soccer ball that was kicked is modeled by the function f(x)=-x^2+9x, where x is time (in seconds) that the ball was in the air. Estimate the average rate of change over the interval [0.7,4.3]. (1 point) feet per second

Solution Steps

To estimate the average rate of change of the function \( f(x) = -x^2 + 9x \) over the interval \([0.7, 4.3]\), we will use the formula for the average rate of change, which is \(\frac{f(b) - f(a)}{b - a}\), where \(a\) and \(b\) are the endpoints of the interval. In this case, \(a = 0.7\) and \(b = 4.3\).

The height of the soccer ball is modeled by the function \( f(x) = -x^2 + 9x \). We are tasked with estimating the average rate of change of this function over the interval \([0.7, 4.3]\).

First, we calculate the function values at the endpoints of the interval:

• \( f(0.7) = -(0.7)^2 + 9 \times 0.7 \)
• \( f(4.3) = -(4.3)^2 + 9 \times 4.3 \)

The average rate of change of the function over the interval \([a, b]\) is given by: \[ \frac{f(b) - f(a)}{b - a} \] Substituting the values: \[ \frac{f(4.3) - f(0.7)}{4.3 - 0.7} = \frac{4.0}{3.6} \]

Simplifying the expression, we find: \[ \frac{4.0}{3.6} = 1.1111 \]

Final Answer