Monotonicity: Increasing and Decreasing Functions A function is called increasing if its graph lies above its tangent line for all values of its input. I...
A function is called increasing if its graph lies above its tangent line for all values of its input. In other words, the function's output will always be greater than its input for all values.
A function is called decreasing if its graph lies below its tangent line for all values of its input. In other words, the function's output will always be less than its input for all values.
Examples:
Increasing:
f(x) = x^2
f(x) = x + 1
f(x) = 3x - 1
Decreasing:
f(x) = x
f(x) = x^3
f(x) = 2x + 1
Key Properties of Monotonicity:
A function is strictly increasing if its derivative is always positive.
A function is strictly decreasing if its derivative is always negative.
A function can be both increasing and decreasing on the same interval.
Applications of Monotonicity:
Monotonicity is used in various real-world applications, including economics, finance, physics, and biology.
It helps us identify intervals where a function is either increasing or decreasing.
It can be used to solve problems involving rates of change and relative extrema.
Additional Notes:
The derivative of a function tells us the slope of its tangent line at any point.
If the derivative is positive, the function is increasing.
If the derivative is negative, the function is decreasing.
A function's graph can have points where its slope is undefined, but that doesn't mean it's not increasing or decreasing