tailieunhanh - Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 99

Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 99. 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 | Working with Series and Power Series 961 x Co Cix------0x2----1 x3 4---0x4 4---1 x5-----0x6-----1 x7 2 3 4 5 6 7 v2 .4 x C0 1 - 2 4 - -1 .2 2 cos x 357 2 1 3 5 7 2 1 sin x x C0 cos x C1 sin x EXERCISE Verify that x Co cos x C1 sin x is a solution to the differential equation y y. We have shown that if a solution to y y has a power series representation then that solution must be of the form Co cos x C1 sin x where Co and C1 are constants. In the example just completed we recognized the Maclaurin series for sin x and cos x. It is entirely possible that we can solve for all the coefficients of a power series and simply have the solution expressed as and defined by the power series expansion. There are well-known functions defined by power series that arise in physics astronomy and other applied sciences. An example of such functions are the Bessel functions named after the astronomer Bessel who came up with them in the early 1800s while working with Kepler s laws of planetary motion. The Bessel function T0 x is defined by œ 2 7 x 5Z -1 222 . 0 As is often the case in mathematics while Bessel functions arose in a particular astronomical problem they are now used in a wide array of situations. One such example is in studying the vibrations of a drumhead. A graph of the partial sum T0 x 0 -1 x 222fr is given in Figure . k 0 k 2 2- Figure 962 CHAPTER 30 Series Transition to Convergence Tests Because this chapter began with Taylor polynomials it was natural to move on to Taylor series directly without the traditional lead-in of convergence tests for inflnite series. Taylor s Theorem enables us to deal with some convergence issues quite efficiently. Not only are we able to show that the series for ex sin x and cos x converge but we can determine that each converges to its generating function. Our previous work with geometric series allows us to conclude that the series for 11x converges to its generating function on -1 1 . When we flnd a Taylor series by