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Unformatted text preview: 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 out- puts that are the components of a vector. As functions, the usual ideas of limits, derivatives, and integration are applied to vector-valued functions. One natural way to describe a vector-valued function is as a parameterization where each com- ponent is a function of a parameter such as time, t . Parameterization and derivatives together lead to the ideas of unit tangent and unit normal vectors and ways to describe the curvature of a given curve that can be defined parametrically. The chapter finishes by discussing motion along a curve and how this relates to Kepler’s Laws of Planetary Motion. OBJECTIVES: After reading and working through this chapter you should be able to do the following: 1. Identify the curve of a parametric equation (§12 . 1). 2. Represent a curve as a vector-valued function (§12 . 1). 3. Evaluate limits, derivatives, and integrals of vector-valued functions (§12 . 2). 4. Take derivatives of dot and cross products (§12 . 2). 5. Find an arc length parameterization of a curve (§12 . 3). 6. Use the chain rule to take the derivative of a vector-valued function (§12 . 3). 7. Find unit normal, unit tangent, and unit binormal vectors of a curve or vector-valued function in 2-space or 3-space (§12 . 4). 8. Calculate the curvature of a parametric curve or vector-valued function at a point (§12 . 5). 9. Relate position, velocity, and acceleration vectors of a particle traveling along a curve (§12 . 6). 10. Find the normal and tangential components of acceleration (§12 . 6). 11. Understand the relationship between acceleration, velocity, and curvature (§12 . 5 , 12 . 6). 12. Understand and apply Kepler’s three laws of planetary motion (§12 . 7). 255 256 12.1 Introduction to Vector-Valued Functions PURPOSE: To describe vector-valued functions which are them- selves vectors and to introduce how vector-valued functions may be defined parametrically. A function that has one input and only one output is called a scalar-valued func- scalar-valued function tion since the output is a scalar value. This section describes how curves can be describe parametrically and how curves may represent functions that have vector values or vector-valued functions . vector-valued function In the last chapter, lines in 3-space were described using a parameterization for each coordinate in terms of some parameter or variable such as t . Curves in 3- curves in 3-space space can also be parameterized in the same way. Each coordinate of a point on the curve is determined by some parameter such as t ....
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