CONTENTS
CHAPTER 1
SYSTEMS OF LINEAR EQUATIONS AND MATRICES
1 1 10 22 27 38 51 56
1.1 Introduction to Systems of Linear Equations 1.2 Gauss-Jordan Elimination 1.3 Homogeneous Systems of Linear Equations 1.4 Matrices and Matrix Operations 1.5 Rules of Matr
Chapter 15: Topics in Vector Calculus
Summary: This chapter is the culmination of the discussion of multivariable
scalar and vector-valued functions and the applications of double and triple integrals. The main focus in this chapter is to extend integrati
Chapter 14: Multiple Integrals
Summary: The previous chapter focused upon taking derivatives and limits of multivariable functions. In this chapter, the integration of multivariable functions is considered. This naturally leads to integrating functions wi
Chapter 13: Partial Derivatives
Summary: While Chapter 12 introduced vector-valued functions, the components of these functions still only depended upon a single variable. In this chapter, functions are considered where there may be multiple independent v
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Chapter 9: Infinite Series
Summary: Three main ideas are discussed in this chapter: sequences of numbers, infinite series (a series being the sum of numbers) and the representation of functions using a Taylor Series or a power series. The underlying i
206
Chapter 10: Parametric And Polar Curves; Conic Sections
Summary: This chapter begins by introducing the idea of representing curves using parameters. These parametric equations of the curves can then be used to graph, find tangent lines, and find arc
Chapter 11: Three Dimensional Space; Vectors
Summary: This chapter introduces three dimensional spaces and the concept of
a vector. Much of the material here is focused upon describing vectors, how they interact with each other and their significance in b
Chapter 12: Vector-Valued Functions
Summary: Having defined vectors and their properties, now vector-valued
functions are considered. These are simply functions that have two or more outputs that are the components of a vector. As functions, the usual ide
Chapter 8: Mathematical Modeling with Differential Equations
Summary: This chapter brings together the two important ideas of differentiation and integration of functions. These ideas are related by looking at equations that involve both a function, y(x)
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Chapter 7: Principles of Integral Evaluation
Summary: The primary focus of this chapter is to introduce and explain more
advanced integration techniques. With all of the techniques that are introduced, however, the focus is usually to try and find a w
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Chapter 6: Applications of the Definite Integral in Geometry, Science and Engineering
Summary: This chapter focuses upon using the methods of evaluating definite integrals and applying them in various problems. The first problem considered is that of
90
Chapter 5: Integration
Summary: The central concept introduced in this chapter is that of integration and how to find the area under a curve. There are two varieties of integration represented by indefinite integrals and definite integrals. One of the
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Chapter 4: The Derivative in Graphing and Applications
Summary: The main purpose of this chapter is to use the derivative as a tool
to assist in the graphing of functions and for solving optimization problems. The most prominent use of the derivative i
Chapter 0: Functions
Summary: This chapter sets out to describe mathematical functions. This idea
is central to the rest of the book as virtually all concepts will be framed in the context of a function. The important concepts that are covered here are th
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Chapter 1: Limits and Continuity
Summary: This chapter explores the behavior of functions as they approach
certain x-values. First, some graphical and numerical methods are used to try to ascertain the behavior of a functions output values near a parti
Chapter 2: The Derivative
Summary: Chapter 2 builds upon the ideas of limits and continuity discussed
in the previous chapter. By using limits, the instantaneous rate at which a function changes with respect to its inputs can be investigated. This leads t
Chapter 3: Topics in Differentiation
Summary: Having investigated the derivatives of common functions in Chapter 2 (i.e., polynomials, rational functions, trigonometric functions, and their combinations), this chapter begins by asking if the slope of a ge