10 - Contents CHAPTER 9 9.1 9.2 9.3 9.4 Polar Coordinates...

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Contents CHAPTER 9 9.1 9.2 9.3 9.4 CHAPTER 10 10.1 10.2 10.3 10.4 10.5 CHAPTER 11 11.1 11.2 11.3 11.4 11.5 CHAPTER 12 12.1 12.2 12.3 12.4 CHAPTER 13 13.1 13.2 13.3 13.4 13.5 13.6 13.7 Polar Coordinates and Complex Numbers Polar Coordinates 348 Polar Equations and Graphs 351 Slope, Length, and Area for Polar Curves 356 Complex Numbers 360 Infinite Series The Geometric Series Convergence Tests: Positive Series Convergence Tests: All Series The Taylor Series for ex, sin x, and cos x Power Series Vectors and Matrices Vectors and Dot Products Planes and Projections Cross Products and Determinants Matrices and Linear Equations Linear Algebra in Three Dimensions Motion along a Curve The Position Vector 446 Plane Motion: Projectiles and Cycloids 453 Tangent Vector and Normal Vector 459 Polar Coordinates and Planetary Motion 464 Partial Derivatives Surfaces and Level Curves 472 Partial Derivatives 475 Tangent Planes and Linear Approximations 480 Directional Derivatives and Gradients 490 The Chain Rule 497 Maxima, Minima, and Saddle Points 504 Constraints and Lagrange Multipliers 514
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CHAPTER Infinite Series Infinite series can be a pleasure (sometimes). They throw a beautiful light on sin x and cos x. They give famous numbers like n and e. Usually they produce totally unknown functions-which might be good. But on the painful side is the fact that an infinite series has infinitely many terms. It is not easy to know the sum of those terms. More than that, it is not certain that there is a sum. We need tests, to decide if the series converges. We also need ideas, to discover what the series converges to. Here are examples of convergence, divergence, and oscillation: The first series converges. Its next term is 118, after that is 1116-and every step brings us halfway to 2. The second series (the sum of 1's) obviously diverges to infinity. The oscillating example (with 1's and - 1's) also fails to converge. All those and more are special cases of one infinite series which is absolutely the most important of all: = - The geometric series is 1 + x + x2 + x3 + 1 1 -x' This is a series of functions. It is a "power series." When we substitute numbers for x, the series on the left may converge to the sum on the right. We need to know when it doesn't. Choose x = 4 and x = 1 and x = - 1: 1 1 + 1 + 1 + is divergent. Its sum is - - - - 1 -a - 1-1 0 1 --• 1 + (- 1) + + is the oscillating series. Its sum should be - - - - 1 1-(-1 2' The last sum bounces between one and zero, so at least its average is 3. At x = 2 -a- there is no way that 1 + 2 + 4 + 8 + agrees with 1/(1 - 2). This behavior is typical of a power series-to converge in an interval of x's and
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10.1 The Geometric Series to diverge when x is large. The geometric series is safe for x between -1 and 1. Outside that range it diverges. The next example shows a repeating decimal 1.1 1 1. . .: 1 Set x = - The geometric series is 1 + - + 10' 10 The decimal 1.1 11 . . . is also the fraction 1/(1 - &), which is 1019.
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This note was uploaded on 06/13/2009 for the course TAM 455 taught by Professor Petrina during the Fall '08 term at Cornell.

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10 - Contents CHAPTER 9 9.1 9.2 9.3 9.4 Polar Coordinates...

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