3.5 Intervals of increase and decrease

Front

What is an interval of increase?

Back

An interval of increase is a range of x-values over which a function's output (y-value) is rising as x increases.

Front

What is an interval of decrease?

Back

An interval of decrease is a range of x-values over which a function's output (y-value) is falling as x increases.

Front

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

Back

To determine where a function is increasing or decreasing, analyze the function's derivative. If the derivative is positive, the function is increasing; if negative, it is decreasing.

Front

What does it mean if a function is increasing on the interval (-∞, ∞)?

Back

It means that the function's output values rise as the input values increase across the entire domain.

Front

What does it mean if a function is decreasing nowhere?

Back

It means that the function does not have any intervals where the output values fall as the input values increase.

Front

What is the significance of critical points in determining intervals of increase and decrease?

Back

Critical points are where the derivative is zero or undefined. They help identify potential intervals of increase or decrease.

Front

If a function has a critical point at x = 3, how do you test intervals around this point?

Back

Choose test points in the intervals (-∞, 3) and (3, ∞) and evaluate the derivative at these points to determine if the function is increasing or decreasing.

Front

What is the interval of increase for the function f(x) = x^2?

Back

The interval of increase for f(x) = x^2 is (0, ∞).

Front

What is the interval of decrease for the function f(x) = -x^2?

Back

The interval of decrease for f(x) = -x^2 is (-∞, ∞).

Front

What does it mean if a function is increasing on the interval (1, ∞)?

Back

It means that for any x-value greater than 1, the function's output values are greater than those for x-values less than or equal to 1.