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# 3: Using Derivatives

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• 3.1: Using Derivatives to Identify Extreme Values
The critical numbers of a continuous function f are the values of p for which f′(p)=0 or f′(p) does not exist. These values are important because they identify horizontal tangent lines or corner points on the graph, which are the only possible locations at which a local maximum or local minimum can occur.
• 3.2: Using Derivatives to Describe Families of Functions
Given a family of functions that depends on one or more parameters, by investigating how critical numbers and locations where the second derivative is zero depend on the values of these parameters, we can often accurately describe the shape of the function in terms of the parameters. In particular, just as we can created first and second derivative sign charts for a single function, we often can do so for entire families of functions.
• 3.3: Global Optimization
To find relative extreme values of a function, we normally use a first derivative sign chart and classify all of the function’s critical numbers. If instead we are interested in absolute extreme values, we first decide whether we are considering the entire domain of the function or a particular interval. If we are working to find absolute extremes on a restricted interval, then we first identify all critical numbers of the function that lie in the interval
• 3.4: Applied Optimization
While there is no single algorithm that works in every situation where optimization is used, in most of the problems we consider, the following steps are helpful: draw a picture and introduce variables; identify the quantity to be optimized and find relationships among the variables; determine a function of a single variable that models the quantity to be optimized; decide the domain on which to consider the function being optimized; use calculus to identify the absolute maximum and/or minimum.
• 3.5: Related Rates
When two or more related quantities are changing as implicit functions of time, their rates of change can be related by implicitly differentiating the equation that relates the quantities themselves.
• 3.E: Using Derivatives (Exercises)
These are homework exercises to accompany Chapter 3 of Boelkins et al. "Active Calculus" Textmap.

## Contributors

Matt Boelkins (Grand Valley State University), David Austin (Grand Valley State University), Steve Schlicker (Grand Valley State University)