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Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 74. 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 | PART VIII Integration An Introduction Net Change in Amount and Area Introducing the Definite Integral 22.1 FINDING NET CHANGE IN AMOUNT PHYSICAL AND GRAPHICAL INTERPLAY Introduction The derivative allows us to answer two related problems. How do we calculate the instantaneous rate of change of a quantity How do we calculate the slope of the line tangent to a curve at a point The physical and graphical questions are intertwined the slope of the graph of an amount function can be interpreted as the instantaneous rate of change of the function. Given an amount function we can derive a rate function we now shift our viewpoint and investigate the problem of how to recover an amount function when given a rate function.1 If we know the rate of change of a quantity how can we flnd the net change in the quantity over a certain time period Suppose for example that we know an object s velocity over a specified time interval. Then we ought to be able to use that information to 1 We did this when we looked at projectile motion in Section 20.7. 711 712 CHAPTER 22 Net Change in Amount and Area Introducing the Definite Integral determine the object s net change in position during that time. To figure out how to do this we begin by looking at some simple examples where the rate of change is constant and then apply what we learn to cases where the rate of change is not constant. Let s clarify what is meant by net change. If you take two steps forward and one step back your net change in position is one step forward. If over the course of a day the stock market falls 100 points and then gains 40 points the net change for the day is -60 points. We now prepare to tackle the following two questions. Given a rate function how do we calculate the net change in amount How do we calculate the area under the graph of a function In this chapter we aim to convince you that the physical and graphical questions are closely related. We ll approach the questions using the strategy that served us .
