13.2 Critical points and the First Derivative Test
2 min read•july 22, 2024
Critical points are key to understanding a function's behavior. They occur where the derivative is zero or undefined, indicating potential turning points or discontinuities in the graph.
The helps determine if these points are local maxima, minima, or neither. By examining how the derivative's sign changes around critical points, we can identify peaks, valleys, and saddle points.
Critical Points and the First Derivative Test
Critical points of functions
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- Points where the derivative is zero (stationary points) or undefined (non-differentiable points)
- Stationary points occur when the is horizontal ()
- Non-differentiable points arise from vertical tangents, cusps, or discontinuities ( is undefined)
- Found by setting and solving for , then checking for any -values that make undefined
First derivative test conditions
- Determines the nature of critical points and relative extrema for a continuous function on an open interval containing the critical point
- If changes from positive to negative at , then is a (peak)
- If changes from negative to positive at , then is a (valley)
- If does not change sign at , then is neither a local maximum nor a local minimum ( or )
Nature of critical points
- Apply the first derivative test to determine if a critical point is a local maximum, local minimum, or neither
- Find the critical points of the function by setting and solving for , and identifying any -values that make undefined
- Evaluate the sign of on the left and right sides of each critical point using test points
- If the sign changes from positive to negative, the critical point is a local maximum
- If the sign changes from negative to positive, the critical point is a local minimum
- If the sign does not change, the critical point is neither a local maximum nor a local minimum
Relative extrema using derivatives
- Relative extrema are the local maxima and minima of a function
- Found by applying the first derivative test to each critical point
- If changes from positive to negative at a critical point, it is a local maximum
- If changes from negative to positive at a critical point, it is a local minimum
- The -coordinates of the local maxima and minima are the relative maximum and minimum values of the function, respectively
Key Terms to Review (17)
Continuity: Continuity in mathematics refers to a property of a function where it does not have any breaks, jumps, or holes over its domain. A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. This concept is crucial because it ensures that the behavior of functions can be analyzed smoothly, impacting several important mathematical principles and theorems.
Decreasing Interval: A decreasing interval is a range of x-values over which a function's output values are falling as you move from left to right. This characteristic indicates that the slope of the function is negative during this interval, suggesting that the first derivative of the function is less than zero. Understanding decreasing intervals helps identify critical points where a function changes behavior, which is key in analyzing overall function behavior and applying the First Derivative Test.
Differentiability: Differentiability refers to the property of a function being differentiable at a point or on an interval, which means it has a defined derivative at that point or throughout that interval. This concept is essential in understanding how functions behave, as it indicates smoothness and continuity, allowing for the application of various calculus principles. Differentiability also plays a crucial role in analyzing inverse functions, exponential functions, critical points, limits, and iterative methods for finding roots of equations.
Finding Derivatives: Finding derivatives is the process of calculating the rate at which a function changes at any given point. It is a fundamental concept in calculus that helps determine the behavior of functions, including their increasing or decreasing nature, and identifies critical points where the function's slope is zero or undefined. This is crucial for analyzing the properties of functions, optimizing values, and understanding their graphical representations.
First Derivative Test: The first derivative test is a method used to determine the local extrema of a function by analyzing its first derivative. By finding critical points, where the first derivative equals zero or is undefined, and then testing the sign of the derivative on intervals around these points, one can identify whether each critical point is a local maximum, local minimum, or neither. This approach connects to understanding absolute and relative extrema, determining concavity, analyzing inflection points, and applying optimization in various contexts.
Function graph: A function graph is a visual representation of a mathematical function, illustrating the relationship between input values (usually represented on the x-axis) and output values (represented on the y-axis). It shows how each input is associated with exactly one output, allowing for analysis of the function's behavior, including its critical points, increasing and decreasing intervals, and overall shape. The function graph is essential for understanding concepts like local maxima and minima as well as determining the behavior of numerical methods.
Horizontal tangent: A horizontal tangent is a straight line that touches a curve at a specific point, where the slope of the curve at that point is zero. This indicates that the function has a critical point, which can be a local maximum, local minimum, or a saddle point. Identifying horizontal tangents is crucial because they help determine where a function changes direction and can indicate potential points of interest for further analysis.
Increasing Interval: An increasing interval is a range of values for which a function's output rises as its input increases. In terms of calculus, this concept is vital for understanding the behavior of functions, particularly in identifying where a function is gaining value and how this relates to critical points and derivative analysis.
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.
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.
Polynomial function: A polynomial function is a mathematical expression consisting of variables raised to whole number powers and coefficients, combined using addition, subtraction, and multiplication. These functions can take various forms, such as linear, quadratic, cubic, or higher degree polynomials, and they play a crucial role in calculus for understanding shapes of graphs and behaviors of functions. Their properties are foundational for concepts such as differentiation, critical points, and integral applications.
Saddle Point: A saddle point is a critical point on a surface where the slope is zero, but it is not a local extremum. It can be characterized as a point that is a minimum along one direction and a maximum along another. This unique nature of saddle points makes them important in understanding the behavior of functions, especially when analyzing critical points and using second derivative tests.
Setting the derivative to zero: Setting the derivative to zero is a fundamental technique in calculus used to find critical points of a function, which are points where the function's slope is either zero or undefined. This action helps identify where a function may change its behavior, such as transitioning from increasing to decreasing or vice versa. Critical points are essential for determining local maxima, local minima, and points of inflection, which are crucial for understanding the overall shape and characteristics of the graph of the function.
Tangent Line: A tangent line is a straight line that touches a curve at a single point without crossing it, representing the instantaneous rate of change of the curve at that point. This concept connects deeply with the ideas of slope, derivatives, and the behavior of functions, providing crucial insight into how functions change locally.
Trigonometric function: A trigonometric function is a mathematical function related to the angles and sides of triangles, primarily used in geometry and calculus. These functions, such as sine, cosine, and tangent, help in analyzing periodic phenomena and can be applied to various fields including physics and engineering. Understanding these functions is crucial for studying concepts such as critical points, derivative tests, and initial value problems.