The many useful applications of the derivative are the focus of this unit. Beginning with an interpretation of the derivative as the instantaneous rate of change of a function, we show how the derivative can be used to calculate the instantaneous velocity and acceleration of an object if an expression for the position is known. Next is a discussion of a method for solving related rate problems, in which the rate of change of one quantity is deduced from the rate of change of another.
The derivative is then presented as a powerful tool to analyze the behavior of functions, including their direction, concavity, and points where they reach extreme values. After showing how the first derivative can be used as an indicator of direction and how the second derivative can be used as an indicator of concavity, we attempt to find important features of the graph that do not depend on the derivative, including intercepts and horizontal and vertical asymptotes. The ultimate goal of these sections is to learn how to sketch a graph of a function.
Finally, constrained optimization is presented as an important practical application of the concepts behind curve-sketching.
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