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Unformatted text preview: Chapter 13: Partial Derivatives Summary: While Chapter 12 introduced vector-valued functions, the compo- nents of these functions still only depended upon a single variable. In this chap- ter, functions are considered where there may be multiple independent variables. Functions that have multiple inputs are often called multivariable functions. The usual ideas about these functions are discussed. Limits are described as each input approaches a particular value. Continuity is also discussed in a similar fash- ion. Derivatives are also discussed although they take on new meaning now. Derivatives may be taken with respect to only one input and so they are called partial derivatives. The idea of a directional derivative is a consequence of this since partial derivatives can be thought of as derivatives along a particular line or derivatives in a particular direction. This idea is extended to allow a derivative to be taken in any direction. The chain rule was an important concept for single variable functions and a cor- responding idea is investigated for multivariable functions. With single variable functions, derivatives lead to the ideas of tangent lines, local linearizations and optimization problems. Similar topics are discussed for multivariable functions such as tangent planes, gradients and finding the extreme values of a function of two variables. OBJECTIVES: After reading and working through this chapter you should be able to do the following: 1. Define functions of two and three variables and understand what a level curve is. (§13 . 1). 2. Graph a function of two or three variables (§13 . 1). 3. Evaluate limits along a curve and limits of a function with two or three variables (§13 . 2). 4. Recognize when a function of two or three variables is continuous (§13 . 2). 5. Find partial derivatives of functions of two or three variables (§13 . 3). 6. Determine if a function is differentiable and find the local linearization of a function of two or three variables at a point (§13 . 4). 279 280 7. Use the chain rule to find derivatives of a function of two or three variables when the variables are represented using parameterizations (§13 . 5). 8. Find directional derivatives and the gradient of a function of two or three variables at a point (§13 . 6). 9. Find tangent planes to surfaces and their normal vectors (§13 . 7). 10. Use gradients to find tangent lines to the intersections of surfaces (§13 . 7). 11. Find the absolute maximum and minimum of a function of two variables on a bounded set (§13 . 8) 12. Use the method of Lagrange Multipliers to solve constrained optimization problems for functions of two or three variables (§13 . 9). 13.1 Functions of Two or More Variables PURPOSE: To describe functions of two or more variables and to discuss level curves and level surfaces....
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- Spring '10