1.2 Increasing/Decreasing Intervals

Front

What is an increasing function?

Back

A function is increasing on an interval if, for any two points x1 and x2 in that interval, where x1 < x2, the function value at x1 is less than the function value at x2 (f(x1) < f(x2)).

Front

What is a decreasing function?

Back

A function is decreasing on an interval if, for any two points x1 and x2 in that interval, where x1 < x2, the function value at x1 is greater than the function value at x2 (f(x1) > f(x2)).

Front

How do you determine the intervals where a function is increasing or decreasing?

Back

To determine the intervals where a function is increasing or decreasing, find the derivative of the function, set it to zero to find critical points, and analyze the sign of the derivative in the intervals defined by these critical points.

Front

What does it mean for a function to have critical points?

Back

Critical points are values of x where the derivative of the function is zero or undefined. These points are potential locations for local maxima, minima, or points of inflection.

Front

What is the significance of the first derivative test?

Back

The first derivative test helps determine whether a critical point is a local maximum, local minimum, or neither by analyzing the sign of the derivative before and after the critical point.

Front

What is the interval notation for increasing functions?

Back

Interval notation for increasing functions indicates the ranges of x-values where the function is increasing, typically expressed as (a, b) where 'a' and 'b' are the endpoints of the interval.

Front

What is the interval notation for decreasing functions?

Back

Interval notation for decreasing functions indicates the ranges of x-values where the function is decreasing, typically expressed as (a, b) where 'a' and 'b' are the endpoints of the interval.

Front

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

Back

On a graph, increasing intervals are where the graph rises as you move from left to right, while decreasing intervals are where the graph falls as you move from left to right.

Front

What is the relationship between the first derivative and the behavior of a function?

Back

The first derivative indicates the slope of the function. If the first derivative is positive, the function is increasing; if it is negative, the function is decreasing.

Front

What does the notation (-∞, -1) represent?

Back

The notation (-∞, -1) represents an interval that includes all real numbers less than -1, extending indefinitely to the left.