AP Precalculus 1.1 Change in Tandem Review

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What are decreasing intervals on a graph?

Back

Decreasing intervals are ranges on a graph where the function's value decreases as the input increases. For example, if a function is decreasing on the interval (-∞, -2) & (2, ∞), it means that as x moves from -∞ to -2 and from 2 to ∞, the function's output decreases.

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What is the average rate of change?

Back

The average rate of change of a function over an interval [a, b] is calculated as (f(b) - f(a)) / (b - a). It represents the slope of the secant line connecting the points (a, f(a)) and (b, f(b)).

Front

What is a point of inflection?

Back

A point of inflection is a point on the graph of a function where the curvature changes direction. This means the function changes from concave up to concave down or vice versa.

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How do you determine where f(x) < 0?

Back

To find where f(x) < 0, identify the intervals on the x-axis where the function's graph lies below the x-axis. For example, if f(x) < 0 on the interval (1, 3), the function is negative between x = 1 and x = 3.

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What does it mean for a function to be positive and increasing?

Back

A function is positive when its values are greater than zero (f(x) > 0) and increasing when its output values rise as the input values increase. For example, if f(x) is positive and increasing on the intervals (-5, -3) & (2, ∞), it means the function is above the x-axis and rising in those ranges.

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What is the significance of intervals in calculus?

Back

Intervals in calculus help identify where functions are increasing, decreasing, positive, or negative. They are crucial for understanding the behavior of functions and for solving optimization problems.

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What is the first derivative test?

Back

The first derivative test is a method used to determine whether a function is increasing or decreasing on an interval. If f'(x) > 0, the function is increasing; if f'(x) < 0, it is decreasing.

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What is the second derivative test?

Back

The second derivative test is used to determine the concavity of a function. If f''(x) > 0, the function is concave up; if f''(x) < 0, it is concave down.

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How do you find critical points?

Back

Critical points occur where the first derivative f'(x) is zero or undefined. These points are important for finding local maxima and minima.

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What is a local maximum?

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A local maximum is a point on the graph of a function where the function value is higher than the values of the function at nearby points.