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

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

Transcript text: The height (in feet) of a soccer ball that was kicked is modeled by the function $f(x)=-x^{2}+9 x$, 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) $\square$ feet per second

Solution

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 $b = 4.3$.

Step 1: Define the Function and Interval

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 $[0.7, 4.3]$.

Step 2: Calculate Function Values at Interval Endpoints

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 $

Step 3: Apply the Average Rate of Change Formula

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: \[ \frac{f(4.3) - f(0.7)}{4.3 - 0.7} = \frac{4.0}{3.6} \]

Step 4: Simplify the Expression

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