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Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 37

Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 37. A major complaint of professors teaching calculus is that students don't have the appropriate background to work through the calculus course successfully. This text is targeted directly at this underprepared audience. This is a single-variable (2-semester) calculus text that incorporates a conceptual re-introduction to key precalculus ideas throughout the exposition as appropriate. This is the ideal resource for those schools dealing with poorly prepared students or for schools introducing a slower paced, integrated precalculus/calculus course | C H A Optimization EXAMPLE 10.1 10.1 ANALYSIS OF EXTREMA Introduction Via Examples Many problems we deal with every day involve a search for some optimal arrangement. What route must be taken to travel the distance between two cities in the shortest amount of time When should a farmer harvest his crop in order to maximize his profit What dimensions will minimize the amount of material required to construct a can of a given volume Example 6.1 was such an optimization problem a gardener had a fixed amount of fencing and wanted to find out how to maximize the area of her garden. These and many other questions require the optimization of some quantity. If the quantity we aim to optimize can be expressed as a function of one variable on a particular domain then knowing about the graph of the function can help us locate the point at which the function achieves its maximum or minimum depending on the problem value. Knowing the rate of change of the function can be very useful in finding this point regardless of whether or not we actually produce a graph. In this chapter we will look at some basic types of optimization problems to get an idea of how to set them up and use both calculus-based and non-calculus-based methods to solve them. Throughout the course as we study new kinds of functions and their derivatives we will return to the topic of optimization. The Beta Shuttle flies passengers between Boston and New York City. Currently it charges 150 for each one-way ticket. At this price an average of 190 people buy tickets on the Shuttle so the company receives 150 per person - 190 people 28 500 for each flight. The owners of the Beta Shuttle hire a consulting firm to help them figure out how the number of people buying tickets would change if the price were changed. The consultants conclude that for each one dollar increase in the price of the ticket two fewer people would 341 342 CHAPTER 10 Optimization be willing to fly the Beta Shuttle similarly for each one dollar .