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