15.1 Increasing and decreasing functions
3 min read•july 22, 2024
The of a function reveals its behavior, showing where it increases, decreases, or stays constant. By analyzing the sign of , we can determine how changes across different intervals.
Critical points are key values where a function's behavior shifts. By finding where f'(x) equals zero or is undefined, we can identify these points and classify them as local maxima, minima, or neither using derivative tests.
Increasing and Decreasing Functions
First derivative for function behavior
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- Determines whether a function is increasing or decreasing on an interval based on the sign of its first derivative
- Function is increasing on an interval when for all in that interval, meaning as increases, also increases (positive slope)
- Function is decreasing on an interval when for all in that interval, meaning as increases, decreases (negative slope)
- Function is constant on an interval when for all in that interval, meaning the function maintains the same value (zero slope)
Intervals of function change
- Find the first derivative of the function, denoted as , to analyze its behavior
- Solve to find critical points where the function changes from increasing to decreasing or vice versa
- Evaluate the sign of on the intervals between critical points to determine whether the function is increasing or decreasing on each interval
- on an interval indicates the function is increasing on that interval
- on an interval indicates the function is decreasing on that interval
- Utilize a sign chart or number line to visualize and clearly represent the intervals of increase and decrease
Real-world applications of function behavior
- Identify the function that accurately models the real-world situation (position, velocity, population growth)
- Analyze the intervals where the function is increasing or decreasing to understand the behavior of the modeled situation
- Interpret the meaning of increasing and decreasing intervals in the context of the problem
- Height of a projectile over time: increasing intervals represent the projectile rising, while decreasing intervals represent the projectile falling
- Population growth: increasing intervals indicate population expansion, while decreasing intervals indicate population decline
Critical Points of Functions
Critical points of functions
- Critical points are values of where the first derivative equals zero or does not exist (undefined)
- Find critical points by solving for and identifying values of where is undefined
- Classify critical points as , , or neither using the following tests:
- First derivative test:
- Local maximum: changes from positive to negative at the
- Local minimum: changes from negative to positive at the critical point
- Neither: does not change sign at the critical point
- Second derivative test (applicable if exists):
- Local maximum: and at the critical point
- Local minimum: and at the critical point
- Inconclusive: and at the critical point
- First derivative test:
Key Terms to Review (17)
Concavity: Concavity refers to the direction in which a function curves, either concave up or concave down. A function is concave up on an interval if its second derivative is positive, indicating that the slope of the tangent line is increasing, while it is concave down if its second derivative is negative, indicating that the slope is decreasing. Understanding concavity helps identify the behavior of a function, particularly in determining inflection points and analyzing the nature of extrema.
Critical Point: A critical point is a point on the graph of a function where the derivative is either zero or undefined, indicating potential locations for local maxima, minima, or inflection points. These points are essential for understanding the behavior of functions, particularly when analyzing slopes, determining intervals of increase or decrease, and conducting tests for concavity.
Decreasing Function: A decreasing function is a type of function where, as the input values increase, the output values decrease. This behavior can be observed on a graph where the line moves downward from left to right. Decreasing functions are essential for understanding the overall shape and behavior of graphs, particularly in identifying intervals where a function is not increasing and analyzing the overall trend of the function.
F'(x): The notation f'(x) represents the derivative of the function f at the point x, indicating the rate at which the function's value changes as x changes. This concept is crucial for understanding how functions behave, particularly in determining slopes of tangent lines, rates of change, and overall function behavior, which are foundational in various applications such as motion analysis and optimization problems.
F(x): The notation f(x) represents a function, which is a rule that assigns each input from a set of numbers (domain) to exactly one output in another set (codomain). This notation allows us to easily express relationships between variables and understand how changes in the input affect the output. It also serves as a foundation for more complex concepts like derivatives, rates of change, and function behavior.
First Derivative: The first derivative of a function represents the rate at which the function's value changes with respect to changes in its input. It provides essential information about the behavior of the function, such as its slope at any given point, and is fundamental in analyzing how functions increase or decrease, as well as in understanding relationships between variables through implicit differentiation.
Increasing Function: An increasing function is a type of function where, as the input values (or x-values) increase, the output values (or y-values) also increase. This means that for any two points within the domain of the function, if the first point has a smaller x-value than the second, then the function's value at the first point is less than or equal to its value at the second point. Understanding increasing functions is crucial for analyzing their graphs, determining their behavior with derivatives, and identifying intervals where a function rises or falls.
Inflection Point: An inflection point is a point on a curve where the concavity changes, meaning the curve switches from being concave up to concave down, or vice versa. Identifying these points is crucial as they can indicate where the function's growth behavior changes, which connects deeply to understanding slopes, critical points, increasing or decreasing functions, and utilizing second derivatives for further analysis.
Interval of decrease: An interval of decrease refers to a section of a function where the output values are getting smaller as the input values increase. This means that as you move along the x-axis in this interval, the corresponding y-values drop, indicating that the function is losing value. Recognizing these intervals is essential for understanding the overall behavior of functions, particularly when analyzing their increasing and decreasing patterns.
Interval of Increase: An interval of increase is a range of values for which a function's output increases as the input increases. Understanding where a function increases helps identify its behavior and is crucial in analyzing its overall shape and properties. This concept connects directly to determining critical points, the first derivative test, and recognizing local maxima and minima.
Linear Function: A linear function is a type of function that can be represented by a straight line on a graph, described by the equation $$f(x) = mx + b$$, where $$m$$ is the slope and $$b$$ is the y-intercept. This function maintains a constant rate of change, meaning for every unit increase in the input, the output changes by a fixed amount. Linear functions are fundamental in understanding relationships between variables and are key in topics such as slope, graphing different types of functions, and identifying increasing or decreasing behavior.
Local maximum: A local maximum refers to a point on a function where the function's value is higher than that of its immediate neighbors. This means that, in a small interval around this point, the function does not exceed this value, making it a crucial concept for identifying peaks in graphs. Understanding local maxima is essential in various contexts, including analyzing critical points, determining increasing or decreasing behavior of functions, and applying second derivative tests for concavity.
Local minimum: A local minimum is a point in a function where the value of the function is lower than the values of the function at nearby points. This concept is vital in understanding the behavior of functions, as local minima help identify potential points of interest where a function may change from decreasing to increasing. It connects with the idea of critical points, the nature of extrema, how functions increase or decrease, and further analysis through concavity.
Mean Value Theorem: The Mean Value Theorem states that for a function that is continuous on a closed interval and differentiable on the open interval, there exists at least one point where the derivative equals the average rate of change of the function over that interval. This theorem provides a bridge between the behavior of a function and its derivatives, showing how slopes relate to overall changes.
Quadratic function: A quadratic function is a type of polynomial function that can be expressed in the standard form $$f(x) = ax^2 + bx + c$$, where $$a$$, $$b$$, and $$c$$ are constants and $$a$$ is not equal to zero. This function forms a parabola when graphed, which can open either upwards or downwards depending on the value of $$a$$. Understanding the properties of quadratic functions helps in analyzing their behavior, such as finding their vertex, determining their maximum or minimum values, and exploring their increasing and decreasing intervals.
Rolle's Theorem: Rolle's Theorem states that if a function is continuous on a closed interval and differentiable on the open interval between the endpoints, and if the function takes the same value at both endpoints, then there exists at least one point in the open interval where the derivative of the function is zero. This theorem highlights a crucial relationship between differentiability, continuity, and the behavior of functions on intervals.
Sign of the derivative: The sign of the derivative indicates whether a function is increasing or decreasing at a particular point. If the derivative is positive, the function is increasing; if it is negative, the function is decreasing. Understanding the sign of the derivative helps identify critical points, local maxima and minima, and overall behavior of functions.