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Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 32

Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 32. 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 | 8.3 Derivatives of Sums Products Quotients and Power Functions 291 Suppose limx a x L1 and limx a g x L2 where L1 and L2 are flnite. Then 1 lim x g x Li L2 x- a The limit of a sum difference is the sum difference of the limits. 2 lim x x L1 L2 x a The limit of a product is the product of the limits in particular g x may be constant limx ak x kL1 . We can use principles 1 and 2 given above to prove two properties of derivatives the Constant Multiple Rule and the Sum Rule stated in Section 7.4 and worked with in the Exploratory Problem for Chapter 7. Properties of Derivatives k x k x or k x k x where k is any constant. dx dx Multiplying by a constant k multiplies its derivative by k. Sum Rule r x g x x g x or g x x g x ax ax ax The derivative of a sum is the sum of the derivatives. These properties are natural from a real-world point of view. Sum Rule . Suppose a juice manufacturer is packaging cranapple juice by pouring cranberry and apple juice simultaneously into juice containers. Let c t be the number of liters of cranberry juice in a container at time t. Let a t be the number of liters of apple juice in a container at time t. Then c t a t is the amount of juice in the container at time t. It follows that da and dc are the rates at which apple and cranberry juice respectively are entering the container. d a t c t is the rate at which juice is entering. d da dc a t c t F -dt dt dt Constant Multiple Rule . An athlete and a Sunday jogger start at the same place and run down a straight trail. Let s t be the position of the Sunday jogger at time t. If the position of the athlete is always k times t then her velocity is k times that of the jogger. 292 CHAPTER 8 Fruits of Our Labor Derivatives and Local Linearity Revisited EXERCISE 8.5 EXERCISE 8.6 EXERCISE 8.7 EXAMPLE 8.4 The Sum and Constant Multiple Rules are very important results and were the topic of the exploratory problems for the previous chapter. Proofs are provided at the end of the section. We will assume in