MAT 169: Calculus III with Analytic Geometry James V. Lambers December 7, 2012 2 Contents 1 Sequences and Series 7 1.1 Introduction . 7 1.1.1 Sequences and Series . 7 1.1.2 Vectors and the Geometry of Space . 8 1.1.3 Parametric Equations and Polar Coordinates . 9 1.2 Review of Calculus . 10 1.2.1 Limits . 10 1.2.2 Continuity . 11 1.2.3 Derivatives . 12 1.2.4 Riemann Sums and the Definite Integral . 13 1.2.5 Extreme Values . 14 1.2.6 The Mean Value Theorem . 16 1.2.7 The Mean Value Theorem for Integrals . 17 1.3 Taylor's Theorem . 18 1.4 Sequences . 21 1.4.1 What is a Sequence? . 21 1.4.2 Why Do We Need Sequences? . 23 1.4.3 Recognizing Sequences . 23 1.4.4 Limits of Sequences . 24 1.4.5 Relation to Limits of Functions . 26 1.4.6 Testing Convergence of Sequences . 27 1.4.7 Alternating Sequences . 28 1.4.8 Growth Rates of Functions . 29 1.4.9 Geometric Sequences . 30 1.4.10 Recursively Defined Sequences . 30 1.4.11 Bounded and Monotonic Sequences . 31 1.4.12 Summary . 35 1.5 Series . 37 1.5.1 What is a Series? . 37 3 4 CONTENTS 1.5.2 Why Do We Need Series? . 39 1.5.3 Geometric Series . 40 1.5.4 Telescoping Series . 43 1.5.5 Harmonic Series . 44 1.5.6 Basic Convergence Tests . 44 1.5.7 Summary . 45 1.6 The Integral and Comparison Tests . 46 1.6.1 The Integral Test . 46 1.6.2 The Comparison Test . 51 1.7 Other Convergence Tests . 54 1.7.1 The Alternating Series Test . 55 1.7.2 Estimating Error in Alternating Series . 56 1.7.3 Absolute Convergence . 58 1.7.4 The Ratio Test . 59 1.7.5 The Root Test . 60 1.7.6 Summary . 61 1.8 Power Series . 62 1.8.1 What is a Power Series? . 62 1.8.2 Convergence of Power Series . 63 1.8.3 The Radius of Convergence . 64 1.8.4 Representing Functions as Power Series . 64 1.8.5 Differentiation and Integration of Power Series . 66 1.8.6 Summary . 68 1.9 Taylor and Maclaurin Series . 70 1.9.1 Summary . 81 1.10 Review . 83 2 Vectors and the Geometry of Space 87 2.1 Three-Dimensional Coordinate Systems . 87 2.1.1 Points in Three-Dimensional Space . 87 2.1.2 Planes in Three-Dimensional Space . 88 2.1.3 Plotting Points in xyz-space . 89 2.1.4 The Distance Formula . 89 2.1.5 Equations of Surfaces . 90 2.1.6 Summary . 92 2.2 Vectors . 93 2.2.1 Combining Vectors . 94 2.2.2 Components . 95 2.2.3 Summary . 102 2.3 The Dot Product . 103 CONTENTS 5 2.3.1 Properties . 105 2.3.2 Orthogonality . 106 2.3.3 Projections . 106 2.3.4 Summary . 109 2.4 The Cross Product . 110 2.4.1 Parallel Vectors . 113 2.4.2 Properties . 113 2.4.3 Areas . 113 2.4.4 Volumes . 115 2.4.5 Summary . 115 2.5 Equations of Lines and Planes . 116 2.5.1 Equations of Lines . 116 2.5.2 Systems of Linear Equations . 120 2.5.3 Equations of Planes . 126 2.5.4 Intersecting Planes . 128 2.5.5 Distance from a Point to a Plane . 129 2.5.6 Summary . 131 2.6 Review . 132 3 Parametric Curves and Polar Coordinates 143 3.1 Parametric Curves . 143 3.1.1 Summary . 147 3.2 Calculus with Parametric Curves . 148 3.2.1 Arc Length . 148 3.2.2 Arc Length of Parametrically Defined Curves . 151 3.2.3 Tangents of Parametric Curves . 155 3.2.4 Areas Under Parametric Curves . 161 3.2.5 Summary . 164 3.3 Polar Coordinates . 165 3.3.1 Conversion Between Cartesian and Polar Coordinates 166 3.3.2 Polar Equations . 168 3.3.3 Tangents to Polar Curves . 171 3.3.4 Summary . 174 3.4 Areas and Lengths in Polar Coordinates . 174 3.4.1 Area . 174 3.4.2 Arc Length . 179 3.4.3 Summary . 180 3.5 Review . 181 Index 184 6 CONTENTS Chapter 1 Sequences and Series 1.1 Introduction This course is the third course in the calculus sequence, following MAT 167 and MAT 168. Its purpose is to prepare students for more advanced mathematics courses, particularly those pertaining to multivariable calculus (MAT 280) and numerical analysis (MAT 460 and 461). The course will focus on three main areas, which we briefly discuss here. 1.1.1 Sequences and Series Nearly all students have had to use a scientific calculator. Consider the fol- lowing: how does a calculator efficiently evaluate many of its functions, such as sin, cos or exp, when its hardware is only able to perform the four basic arithmetic operations, addition, subtraction, multiplication and division? The answer stems from the fact that it is generally not possible to eval- uate such functions exactly; rather, one has to settle for an approximation. However, this is no problem, because a calculator or computer can only rep- resent real numbers to limited accuracy anyway. To approximate a given function in a manner that is suitable for a calculator, we use infinite series, which is a sum of infinitely many terms. For example, we can write 1 1 1 X 1 exp x = 1 + x + x2 + x3 + ··· = xk: 2 6 k! k=0 The last term in the above equation uses sigma notation to express a sum of infinitely many terms in a concise way, when those terms can be described by a pattern. The above series is called a power series because each of its 7 8 CHAPTER 1. SEQUENCES AND SERIES terms features a power of x. When we study infinite series, we will consider important questions such as • When does an infinite series sum, or converge, to a finite number? We will see that in many cases, a sum of infinitely many terms does not converge at all, but rather continues growing, or diverging. For example, consider the two series 1 1 X 1 X 1 ; : n n1:01 n=1 n=1 Although the individual terms in the series may not differ by very much, especially for larger values of n, the first series does not con- verge, while the second one does. • If we truncate a series after a given number of terms, how well does the sum of the retained terms approximate the sum of the entire series? This is particularly important when using series to evaluate functions such as those implemented in scientific calculators. For example, sup- pose we take the abovementioned series for exp x and compute only the first 20 terms. If x = 0:1, then the result is correct to at least 16 significant digits. However, if x = 10, then we only obtain two correct digits. 1.1.2 Vectors and the Geometry of Space Next, we will become acquainted with three-dimensional space, in order to prepare you for later coursework in multivariable calculus. Our basic tools will be vectors, which can be used to represent either a position or direction in space. For example, if we represent three-dimensional space using Cartesian coordinates x, y and z, then the origin is the point with coordinates x = 0, y = 0 and z = 0, typically denoted by the ordered triple (0; 0; 0). Then, the point in space (1; 0; −2) has coordinates x = 1, y = 0 and z = −2, meaning that it is located 1 unit from the origin along the positive x-axis, 0 units from the origin along the y-axis, and 2 units from the origin along the negative z-axis. This is illustrated in Figure 1.1. We will use vectors to facilitate the description of, and operations on, lines and planes. This is particularly useful in computer graphics. Consider the problem of rendering a two-dimensional image, say for a frame in a film, of a collection of three-dimensional objects. To what point in 2-D space does any given point in 3-D space correspond? This question is answered by computing 1.1. INTRODUCTION 9 Figure 1.1: The origin in three-dimensional space, and the point (1; 0; −2). the projection of points in 3-D space onto a given plane in 2-D space that corresponds to the \screen". We will learn about an operation called the dot product that is used to compute projections. Another operation, called the cross product, is very useful for describing planes. 1.1.3 Parametric Equations and Polar Coordinates Finally, will study curves and surfaces in space. Often, it is necessary to describe the position of an object as a function of time. Therefore, we will study curves that are described by parametric equations, where the parameter is usually time. For example, in combat, the military needs to track the position of enemy projectiles over time. Using radar data, they can then construct parametric equations for the projectile's position in 3-D space, and then differentiate the equations with respect to time in order to estimate its velocity and then its trajectory, so that it can be intercepted. While we will work with Cartesian coordinates x, y and z for most of the course, we will find that it is often useful to represent curves or functions in the xy-plane using polar coordinates r and θ, where r represents distance from the origin, and θ represents the angle that the point makes with the origin and the positive x-axis.