Average rate of change over an interval

Front

What is the average rate of change?

Back

The average rate of change of a function over an interval is the change in the function's value divided by the change in the input value over that interval.

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How do you calculate the average rate of change from x=a to x=b?

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The average rate of change from x=a to x=b is calculated using the formula: \( \frac{f(b) - f(a)}{b - a} \)

Front

Find the average rate of change for the function f(x) = 2x + 3 from x=1 to x=4.

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Average rate of change = \( \frac{f(4) - f(1)}{4 - 1} = \frac{(2(4) + 3) - (2(1) + 3)}{3} = \frac{8 - 2}{3} = 2 \)

Front

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 in quantity over time, making it useful in fields like physics, economics, and biology.

Front

If a function is increasing, what can be said about its average rate of change?

Back

If a function is increasing over an interval, the average rate of change will be positive.

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If a function is decreasing, what can be said about its average rate of change?

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If a function is decreasing over an interval, the average rate of change will be negative.

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What does it mean if the average rate of change is zero?

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An average rate of change of zero indicates that the function's value does not change over the interval, meaning it is constant.

Front

Calculate the average rate of change for the function h(t) = -16t^2 + 48t + 60 from t=1 to t=3.

Back

Average rate of change = \( \frac{h(3) - h(1)}{3 - 1} = \frac{(h(3) = 60) - (h(1) = 92)}{2} = \frac{-32}{2} = -16 \) feet per second.

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What is the average rate of change from x=2 to x=4 for the function f(x) = 2x^2?

Back

Average rate of change = \( \frac{f(4) - f(2)}{4 - 2} = \frac{(2(4)^2) - (2(2)^2)}{2} = \frac{32 - 8}{2} = 12 \)

Front

Find the average rate of change for the interval [2, 6] for the function f(x) = x^2.

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Average rate of change = \( \frac{f(6) - f(2)}{6 - 2} = \frac{(6^2) - (2^2)}{4} = \frac{36 - 4}{4} = 8 \)