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Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 75. 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 | 22.1 Finding Net Change in Amount Physical and Graphical Interplay 721 Let s return for a moment to the problem of calculating the cheetah s net change in position and approach it with a different mindset. Let s i the cheetah s position at time i. Then n i 5 i . In other words f 2i 5 s i is a function whose derivative is 2i 5. Because 5 i is linear we might suspect that s i is quadratic. If 5 i i2 5i then s i 2i 5. In fact if two functions have the same derivative then the functions must differ by an additive constant if x g x then x g x C for some constant C. Therefore 5 i i2 5i C for some constant C. Then the change in position from i 1 to i 4 is given by 5 4 - 5 1 42 5 4 C - 12 5 1 C 16 20 C - 6 - C 30. In Chapter 24 we will explore the relationship between the two mindsets presented in Example 22.5b and arrive at a wonderful theorem that unifies them. For the time being there is a lot to be learned from the first mindset we will stick with it for a while. EXAMPLE 22.6 A gazelle s velocity is given by the graph below. n i is increasing on 0 3 and decreasing on 3 6 . How can we find the net change in the gazelle s position over the interval 0 5 SOLUTION Because v is increasing on the interval 0 3 we can find lower bounds for the gazelle s net change in position by using left-hand sums inscribed rectangles and upper bounds by using right-hand sums circumscribed rectangles . Figure 22.14 722 CHAPTER 22 Net Change in Amount and Area Introducing the Definite Integral Because v is decreasing on the interval 3 5 we can find lower bounds for the net change in position by using right-hand sums inscribed rectangles and upper bounds by using left-hand sums circumscribed rectangles . See Figure 22.14. We can obtain lower and upper bounds for the gazelle s net change in position on 0 5 by treating the intervals 0 3 and 3 5 independently. For instance L L on 0 3 Rm on 3 5 gives a lower bound while U R on 0 3 Lm on 3 5 gives an upper bound. If we compute the limit as n and m .
