Intervals of increase are sections of the domain where the function's value rises as the independent variable increases.
How do you identify intervals of increase on a graph?
On a graph, intervals of increase are identified by looking for sections where the graph moves upwards from left to right.
Why are intervals of increase important in mathematics?
They help in understanding the behavior of the function, such as trends and solutions to certain problems, including maximizing or minimizing values.
What mathematical notation is used to express intervals of increase?
Intervals are usually expressed using interval notation, such as (a, b), where the function increases between the points 'a' and 'b'.
Can an interval of increase include endpoints?
Typically, intervals of increase do not include endpoints in strict math terms, but endpoints can be included if given function properties allow it.
What role does the derivative of a function play in finding intervals of increase?
If the derivative of a function, f'(x), is positive within an interval, then the function is increasing on that interval.
What is the relationship between intervals of increase and critical points?
Intervals of increase are bounded by critical points, where the derivative changes sign, indicating a change from increasing to decreasing or vice versa.
How can you determine intervals of increase using a table of values?
By evaluating the function at various points and seeing where the function values increase as the input values increase.
What is an example of a function with multiple intervals of increase?
The cubic function f(x) = x³ - 3x has multiple intervals, such as increasing on (-∞, -1) and (1, ∞).
How do you test intervals for increase or decrease?
Choose test points between critical points and calculate the derivative. If the derivative is positive, the function is increasing at that interval.
What graphical feature indicates the start or end of an interval of increase?
A local minimum typically indicates the starting point of an interval of increase, while a local maximum indicates the end.
What happens if the graph levels off or has a horizontal tangent within an interval of increase?
An interval of increase doesn't include sections where the tangent is horizontal unless it's part of a broader increasing pattern surrounding a critical point.
Can there be intervals where the function neither increases nor decreases?
Yes, such intervals are called constant intervals where the function maintains the same value.
What kind of function does not have any intervals of increase?
A constant function, where f(x) = c, does not have any intervals of increase as it remains the same throughout its domain.
How would you describe an interval where a function is not increasing?
If a function is not increasing on an interval, it is either decreasing, constant, or undefined at any point within that interval.