Average Rate of Change Over an Interval

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What is the average rate of change of a function over an interval?

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 \( f(x) = \frac{1}{x-1} \) over the interval [-4, -3]?

Back

To find the average rate of change, calculate \( f(-3) \) and \( f(-4) \): \( f(-3) = -\frac{1}{2} \) and \( f(-4) = -\frac{1}{3} \). Then use the formula: \( \frac{f(-3) - f(-4)}{-3 - (-4)} = \frac{-\frac{1}{2} + \frac{1}{3}}{1} = -\frac{1}{20} \).

Front

What is the average rate of change of the function \( f(x) = -x^2 - x - 1 \) over the interval [-2, -1]?

Back

Calculate \( f(-1) = -1 \) and \( f(-2) = -3 \). Then use the formula: \( \frac{f(-1) - f(-2)}{-1 - (-2)} = \frac{-1 + 3}{1} = 2 \).

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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 the function is constant on that interval.

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How do you find the average rate of change for the function \( f(x) = -2x^2 + 2x - 2 \) over the interval [-1, 1]?

Back

Calculate \( f(1) = -1 \) and \( f(-1) = -1 \). Then use the formula: \( \frac{f(1) - f(-1)}{1 - (-1)} = \frac{-1 - (-1)}{2} = 0 \).

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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.

Front

What is the average rate of change from [-7, 7] for 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, which can be calculated using the formula: \( \frac{f(7) - f(-7)}{7 - (-7)} \).

Front

Define the term 'interval' in the context of average rate of change.

Back

An interval is a range of values, typically represented as [a, b], where 'a' is the starting point and 'b' is the endpoint for which the average rate of change is calculated.

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What is the average rate of change for a constant function over any interval?

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

Front

How can you determine if a function is increasing or decreasing over an interval using average rate of change?

Back

If the average rate of change is positive, the function is increasing; if it is negative, the function is decreasing over that interval.