TAILIEUCHUNG - Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 54

Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 54. 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 | Exploratory Problems for Chapter 15 511 a Let N N t be the number of moles of substance A at time t. Translate the statement above into mathematical language. Note The number of moles of substance B should be expressed in terms of the number of moles of substance A. b N t is a decreasing function. The rate at which N is changing is a function of N the number of moles of substance A. When the rate at which A is being converted to B is highest how many moles are there of substance A PART V Adding Sophistication to Your Differentiation A P T E R 16 Taking the Derivative of Composite Functions THE CHAIN RULE We can construct conglomerate functions in two different ways. One way is to combine the functions outputs by taking for example their sum or product. We can differentiate a sum by summing the derivatives and differentiate a product by applying the Product Rule. Another way to construct a conglomerate is to have functions operate in an assemblyline manner. In this configuration the output of one function becomes the input of the next creating a composite function. In this section we will look at the derivatives of composite functions. The work we do will have plentiful rewards as the results have extensive application. Our goal is to express the derivative of the composite function f g x in terms of f g and their derivatives. Our assumption throughout is that and g are differentiable functions. Many functions we can t yet differentiate can be decomposed and expressed as the composite of simpler functions whose derivatives we know. For example .

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