Modeling with Calculus

The derivative of a differentiable function describes the original function's instantaneous rate of change at any particular input. When a differentiable function is used as a model to describe an application, the first and second derivatives of that function can provide valuable information about the situation being modeled. This chapter focuses on how derivatives can be used to learn more about applications. Optimization itself means the process of optimizing a situation or creating the best outcome possible. One powerful tool in mathematical optimization is recognizing that a continuous function has relative high or low points when the derivative of that function changes sign. The chapter provides a review of calculus related to optimization as well as investigating situations where multiple aspects need to be considered to create the best mathematical outcome. It shows how to view an integral as the accumulation of a particular quantity and how to connect the accumulation concept to applications.