Use the formula f(b)−f(a)b−a to calculate the average rate of change over the interval [−2,0] given the function table for f(x)=x2+6x+8 . Express your answer as an integer.

Question

x f(x)
−3 −1
−2 0
−1 3
0 8
(1 point)
The average rate of change is
2.Tyreke is an engineer and needs to design a rocket for an experiment. He has calculated that the height of the rocket, in feet, with respect to time, in seconds, can be modeled by the function h(t)=−16t2+160t
. Find the average rate of change of the rocket over the interval [4,5]
. (1 point)
The average rate of change is
feet per second.

Answer 1

To find the average rate of change of the function \( f(x) = x^2 + 6x + 8 \) over the interval \([-2, 0]\), we can use the formula:

\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]

Here, \( a = -2 \) and \( b = 0 \).

First, we need to calculate \( f(-2) \) and \( f(0) \):

1. For \( f(-2) \): \[ f(-2) = (-2)^2 + 6(-2) + 8 = 4 - 12 + 8 = 0 \]

2. For \( f(0) \): \[ f(0) = (0)^2 + 6(0) + 8 = 0 + 0 + 8 = 8 \]

Now, we can plug these values into our formula:

\[ \text{Average Rate of Change} = \frac{f(0) - f(-2)}{0 - (-2)} = \frac{8 - 0}{0 + 2} = \frac{8}{2} = 4 \]

So the average rate of change of \( f(x) \) over the interval \([-2, 0]\) is 4.

Now, for Tyreke's rocket height function \( h(t) = -16t^2 + 160t \), we need to find the average rate of change over the interval \([4, 5]\):

Using the same formula:

1. Calculate \( h(4) \): \[ h(4) = -16(4)^2 + 160(4) = -16(16) + 640 = -256 + 640 = 384 \]

2. Calculate \( h(5) \): \[ h(5) = -16(5)^2 + 160(5) = -16(25) + 800 = -400 + 800 = 400 \]

Now apply the formula for average rate of change:

\[ \text{Average Rate of Change} = \frac{h(5) - h(4)}{5 - 4} = \frac{400 - 384}{1} = 16 \]

So the average rate of change of the rocket over the interval \([4, 5]\) is 16 feet per second.

Final Answers:

1. The average rate of change of \( f(x) \) over \([-2, 0]\) is 4.
2. The average rate of change of the rocket over \([4, 5]\) is 16 feet per second.