12 - 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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-- C H A P T E R 12 Motion Along a Curve I [ 12.1 The Position Vector I - , This chapter is about "vector functions." The vector 2i + 4j + 8k is constant. The vector R(t) = ti + t2j + t3k is moving. It is a function of the parameter t, which often represents time. At each time t, the position vector R(t) locates the moving body: position vector = R(t) = x(t)i + y(t)j + z(t)k. (1) Our example has x = t, y = t2, z = t3. As t varies, these points trace out a curve in space. The parameter t tells when the body passes each point on the curve. The constant vector 2i + 4j + 8k is the position vector R(2) at the instant t = 2. What are the questions to be asked? Every student of calculus knows the first question: Find the deriuatiue. If something moves, the Navy salutes it and we differen- tiate it. At each instant, the body moving along the curve has a speed and a direction. This information is contained in another vector function-the velocity vector v(t) which is the derivative of R(t): Since i, j, k are fixed vectors, their derivatives are zero. In polar coordinates i and j are replaced by moving vectors. Then the velocity v has more terms from the product rule (Section 12.4). Two important cases are uniform motion along a line and around a circle. We study those motions in detail (v = constant on line, v = tangent to circle). This section also finds the speed and distance and acceleration for any motion R(t). Equation (2) is the computing rulefor the velocity dR/dt. It is not the definition of dR/dt, which goes back to basics and does not depend on coordinates: dR AR lim R(t + At) - R(t) - lim - = dt at+o At At+O At We repeat: R is a vector so AR is a vector so dR/dt is a vector. All three vectors are in Figure 12.1 (t is not a vector!). This figure reveals the key fact about the geometry: 446 The velocity v = dR/dt is tangent to the curve.
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12.1 The PosMon Vector The vector AR goes from one point on the curve to a nearby point. Dividing by At changes its length, not its direction. That direction lines up with the tangent to the curve, as the points come closer.
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