tailieunhanh - Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 58

Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 58. 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 | Implicit Differentiation in Context Related Rates of Change 551 EXAMPLE In this section we will demonstrate the power and versatility of implicit differentiation and the Chain Rule in context. The example that follows makes the transition from Section to the applied problems of this section. Suppose that x and y are functions of i they vary with time and x2 y2 25. In other words the point x y is moving on the circle of radius 5 centered at the origin. Think of a bug crawling around the circle the coordinates of the bug at time i are given by x y or x t y t . Suppose that jjy -6 units second when x 3 and y 0. What is Ji at this moment SOLUTION Let s think about the question in terms of the bug. We know that when x 3 y 4 because x2 y2 25 and y 0 . When the bug is at the point 3 4 its y-coordinate is decreasing at a rate of 6 units per second the minus sign indicating a decrease . This tells us that the bug is traveling in a clockwise direction. Knowing that the bug stays on the circle we want to flnd the rate of change of its x-coordinate. By thinking about the bug s path you can see that at 3 4 must be positive. Thinking harder can lead you to realize that Jx 6. Let s make the fact that x and y vary with i more explicit by writing x i 2 y i 2 25. What We Know Jjy 6. What We Want Jx when x 3 and y 0. Using principle i we differentiate both sides of this equation with respect to i. x t 2 y t 2 j 25 Jy Jy Jx Jy 2x i 2y i - 0 Jy Jy 552 CHAPTER 17 Implicit Differentiation and its Applications We are interested in at the point where x 3 and y 0. When x 3 y s 25 - 32 V25 - 9 4. dx 3 4 -6 0 di dx 24 3 8 di REMARK At the moment we are interested in we know that x 3 and y 4 but because x isn t always 3 and y isn t always 4 we can t substitute x 3 and y 4 until after differentiating. Similarly if we are interested in 2 we can t evaluate at 2 until after differentiating. If we were to evaluate at 2 before differentiating we would think that 2 is always zero .