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What is the average rate of change of a function?

Back

The average rate of change of a function over an interval [a, b] is given by the formula: \( \frac{f(b) - f(a)}{b - a} \). 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 the function f(x) = 2x^2 + 12x + 16 on the interval [-3, -2]?

Back

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 = \( \frac{f(-2) - f(-3)}{-2 - (-3)} = \frac{-4 - (-2)}{1} = -2 \).

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What is the average rate of change of a function on an interval where the function is undefined?

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If the function is undefined on the interval, the average rate of change cannot be calculated as it requires valid function values at both endpoints.

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What is the average rate of change of the function f(x) = x^2 from x = 0 to x = 1?

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1. Calculate f(0) = 0^2 = 0. 2. Calculate f(1) = 1^2 = 1. 3. Average rate of change = \( \frac{f(1) - f(0)}{1 - 0} = \frac{1 - 0}{1} = 1 \).

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What does a negative average rate of change indicate about a function on an interval?

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

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What is the average rate of change of the function f(x) = -x^2 + 4 on the interval [1, 3]?

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1. Calculate f(1) = -1^2 + 4 = 3. 2. Calculate f(3) = -3^2 + 4 = -5. 3. Average rate of change = \( \frac{f(3) - f(1)}{3 - 1} = \frac{-5 - 3}{2} = -4 \).

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

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How do you find the average rate of change from a graph?

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To find the average rate of change from a graph, identify two points on the curve, (x1, y1) and (x2, y2), and use the formula: \( \frac{y2 - y1}{x2 - x1} \).

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What is the average rate of change of the function f(x) = 3x - 2 from x = 2 to x = 5?

Back

1. Calculate f(2) = 3(2) - 2 = 4. 2. Calculate f(5) = 3(5) - 2 = 13. 3. Average rate of change = \( \frac{f(5) - f(2)}{5 - 2} = \frac{13 - 4}{3} = 3 \).

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

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The average rate of change of a constant function is always 0, as the function's value does not change regardless of the input.