Increasing, Decreasing, Positive, and Negative Intervals

Front

What is an increasing interval in a function?

Back

An increasing interval is a range of x-values where the function's output (y-values) rises as x increases. For example, if f(x) is increasing on the interval (1, 3), then f(1) < f(2) < f(3).

Front

What is a decreasing interval in a function?

Back

A decreasing interval is a range of x-values where the function's output (y-values) falls as x increases. For example, if f(x) is decreasing on the interval (3, 5), then f(3) > f(4) > f(5).

Front

Define positive interval in the context of a function.

Back

A positive interval is a range of x-values where the function's output (y-values) is greater than zero. For example, if f(x) is positive on the interval (0, 2), then f(x) > 0 for all x in (0, 2).

Front

Define negative interval in the context of a function.

Back

A negative interval is a range of x-values where the function's output (y-values) is less than zero. For example, if f(x) is negative on the interval (-2, 0), then f(x) < 0 for all x in (-2, 0).

Front

How can you determine increasing and decreasing intervals from a graph?

Back

To determine increasing and decreasing intervals from a graph, observe the slope of the curve. If the curve rises as you move from left to right, it is increasing; if it falls, it is decreasing.

Front

What is the first derivative test?

Back

The first derivative test is a method used to determine the local maxima and minima of a function by analyzing the sign of the first derivative. If f'(x) changes from positive to negative at a point, that point is a local maximum.

Front

What does it mean if a function is constant on an interval?

Back

If a function is constant on an interval, it means that the output (y-values) does not change as the input (x-values) varies. For example, f(x) = 5 is constant for all x.

Front

How do you find the intervals of increase and decrease using the first derivative?

Back

To find intervals of increase and decrease, calculate the first derivative of the function. Set the derivative equal to zero to find critical points, then test intervals around these points to determine where the function is increasing or decreasing.

Front

What is a critical point?

Back

A critical point is a value of x in the domain of a function where the derivative is either zero or undefined. Critical points are potential locations for local maxima, minima, or points of inflection.

Front

Explain the concept of local maximum and minimum.

Back

A local maximum is a point where the function value is higher than all nearby points, while a local minimum is a point where the function value is lower than all nearby points.