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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 onscreen 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 onevariable 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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 Spring '08
 ALDROUBI
 Math, Derivative, Multivariable Calculus, Qj, fxx

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