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Unformatted text preview: 1 Information The review slides are posted on my webpage www.math.vanderbilt.edu/ mrinalr This is the version of these for printing. Do not print the on-screen slides . It is a waste of ink, paper, and electricity. Exam information 120 minutes No calculators, no notes, no formulas. Chapter 15, 16, 17.1 17.5. However, you still need to know the material from 13 and 14. Two parts I 150 points. II 100 points. Part I will replace the lowest of your grades from exam 1 and exam 2, if part I is higher. Remember points are awarded for showing work and an understanding of the concepts. Office Hours: M R, 4 5 pm. Practice problems = Homework + Previous exams. You can see your exam the first weekday after your final. 2 Limits and continuity Limits and continuity Definition. Let A be a subset of R 2 and let f : A R be a function of two variables. If ( a,b ) A , then we say that the limit as ( x,y ) approaches ( a,b ) exists and is equal to L if and only if for every sequence ( x n ,y n ) A such that ( x n ,y n ) ( a,b ) we have, lim n f ( x n ,y n ) = L. We write this as lim ( x,y ) ( a,b ) f ( x,y ) = L . Definition. In addition, if L = f ( a,b ), then f is said to be continuous at ( a,b ). Examples of continuous functions 1. Polynomials in x,y , e.g., ( x 2 y- 3 xy 2 ). 2. Rational functions R ( x,y ) = P ( x,y ) Q ( x,y ) at points where Q ( x,y ) 6 = 0, e.g., f ( x,y ) = x 2- y 2 x 2 + y 2 is continuous for ( x,y ) 6 = (0 , 0). 3. The product, sum and composite of two continuous functions, e.g., e x 2 + y 2 , sin( x + y ). Checking limits To check that a function does not have a limit at a particular point ( a,b ) we usually need to show that f has different limits along two different paths through ( a,b ). Sometimes a change of variables allows you to compute a limit by turning the problem into a one-variable problem. Examples include: switching to polar coordinates, spherical coordinates. Remember ( x,y ) (0 , 0) is the same as r 0. 3 Partial and directional derivatives Partial derivatives Definition. Let f be a function of two or three variables. We define the partial derivatives of f , denoted f x and f y , by f x ( a,b ) = lim h f ( a + h,b )- f ( a,b ) h f y ( a,b ) = lim h f ( a,b + h )- f ( a,b ) h . You need to know the limit definition for the exam!...
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