BackFirst Derivatives and Graphs: Extrema, Critical Points, and Applications
Study Guide - Smart Notes
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First Derivatives and Graphs
Understanding Derivatives and Graphs
The first derivative of a function provides valuable information about the shape and behavior of its graph. By analyzing the slope of the graph, we can determine where the function increases, decreases, and where it attains maximum or minimum values (extrema).
Slope of the Graph: The derivative at a point gives the slope of the tangent line to the graph at that point.
Extrema: Maximum and minimum values of a function occur where the slope is zero or undefined.
Increasing/Decreasing Intervals: The sign of indicates whether the function is increasing () or decreasing ().
Example: For , . The function increases for and decreases for .
Critical Numbers and Partition Numbers
A critical number of a function is a value in the domain of $f$ where or does not exist. These points are candidates for local extrema.
Critical Numbers: Values in the domain where or is undefined.
Partition Numbers: Points where is discontinuous or does not exist, but not all partition numbers are critical numbers.
Local Extrema: Occur only at critical numbers, but not every critical number is an extremum.
Example: For , the domain is . , which is undefined at (not in domain), so no critical numbers from undefined derivative. Setting gives no solution, so is always increasing.
Determining Increasing and Decreasing Intervals
To find where a function is increasing or decreasing, analyze the sign of its derivative.
Increasing: on an interval.
Decreasing: on an interval.
Horizontal Tangent Lines: Occur where .
Example: For , . Setting gives , which corresponds to a horizontal tangent line.
Local Maximum and Minimum
A function has a local maximum at if is greater than nearby values, and a local minimum if is less than nearby values. The sign change of around helps identify these points.
Local Maximum: changes from positive to negative at .
Local Minimum: changes from negative to positive at .
Neither: If does not change sign, is neither a max nor a min.
Example: For , . At , changes from negative to positive, indicating a local minimum.
Application: Marginal Revenue and Extrema in Business
In business calculus, the concept of marginal revenue is closely related to derivatives. The marginal revenue function describes how revenue changes with each additional unit sold.
Total Revenue Graph: Shows revenue as a function of quantity sold.
Marginal Revenue: The derivative of the revenue function, .
Critical Points: Where or changes sign, indicating maximum revenue.
Example: If the revenue function has a maximum at , then . Selling 400 desks yields maximum revenue of .
Discriminant and Real Zeros
The discriminant of a quadratic function is . It determines the nature of the roots (zeros) of the function.
: Two distinct real zeros.
: One repeated real zero.
: No real zeros.
Example: For , , so there is one repeated real zero.
Summary Table: Critical Points and Extrema
Type of Point | Condition | Result |
|---|---|---|
Critical Number | or undefined, in domain | Possible local max/min or neither |
Local Maximum | changes from positive to negative at | is a local maximum |
Local Minimum | changes from negative to positive at | is a local minimum |
Neither | does not change sign at | No extremum at |
Additional info: Academic context was added to clarify the definitions, examples, and business applications, as the original notes were fragmented and partially incomplete.
