Average Rate of Change

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What is the average rate of change?

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The average rate of change of a function over an interval [a, b] is defined as (f(b) - f(a)) / (b - a). It represents the change in the function's value divided by the change in the input value.

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How do you calculate the average rate of change for f(x) = 2x^2 + 12x + 16 on the interval [-3, -2]?

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1. Calculate f(-3) = 2(-3)^2 + 12(-3) + 16 = 2(9) - 36 + 16 = -2. 2. Calculate f(-2) = 2(-2)^2 + 12(-2) + 16 = 2(4) - 24 + 16 = -4. 3. Average rate of change = (f(-2) - f(-3)) / (-2 - (-3)) = (-4 - (-2)) / (1) = -2 / 1 = 2.

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What is the significance of the average rate of change in real-world applications?

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The average rate of change can represent speed, growth rate, or any change over time, making it useful in fields like physics, economics, and biology.

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What is the average rate of change of a linear function?

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For a linear function, the average rate of change is constant and equal to the slope of the line.

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How do you find the interval with the largest average rate of change?

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To find the interval with the largest average rate of change, calculate the average rate of change for each interval and compare the results.

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If f(x) = x^2, what is the average rate of change from x = 1 to x = 3?

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Average rate of change = (f(3) - f(1)) / (3 - 1) = (9 - 1) / 2 = 8 / 2 = 4.

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What is the average rate of change of a function over an interval [a, b] if f(a) = 5 and f(b) = 15?

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Average rate of change = (f(b) - f(a)) / (b - a) = (15 - 5) / (b - a) = 10 / (b - a).

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What does a positive average rate of change indicate?

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A positive average rate of change indicates that the function is increasing over the interval.

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What does a negative average rate of change indicate?

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A negative average rate of change indicates that the function is decreasing over the interval.

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What is the average rate of change of f(x) = 3x + 2 from x = 0 to x = 4?

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Average rate of change = (f(4) - f(0)) / (4 - 0) = (14 - 2) / 4 = 12 / 4 = 3.