Answer: 536
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Explanation:
The average rate of change is the same as the slope of the line through the endpoints mentioned.
The interval [6, 8] means that we're talking about [tex]6 \le x \le 8[/tex]
- When x = 6, we have y = f(x) = 1178
- When x = 8, we have y = f(x) = 2742
So we're tasked to find the slope of the line through (6,1178) and (8,2742)
We'll use the aptly named slope formula.
[tex](x_1,y_1) = (6,1178) \text{ and } (x_2,y_2) = (8,2742)\\\\m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}\\\\m = \frac{2742 - 1178}{8 - 6}\\\\m = \frac{1564}{2}\\\\m = 782\\\\[/tex]
The slope through those two endpoints is 782, which is the average rate of change on the interval [6, 8].
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For the interval [5,7] we have [tex]5 \le x \le 7[/tex]
According to the function table,
- When x = 5, y = 690
- When x = 7, y = 1852
Like before, use the slope formula to find the slope through (5,690) and (7,1182)
[tex](x_1,y_1) = (5,690) \text{ and } (x_2,y_2) = (7,1182)\\\\m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}\\\\m = \frac{1182 - 690}{7 - 5}\\\\m = \frac{492}{2}\\\\m = 246\\\\[/tex]
This is the average rate of change on the interval [5,7].
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The last step is to subtract the slopes:
782 - 246 = 536 which is the final answer
