mat203_finals_complied

mat203_finals_complied - Previous Final Problems Andrei...

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Unformatted text preview: Previous Final Problems Andrei Jorza December 11, 2009 Contents 1 Basics 1 1.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Functions 2 2.1 Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.3 Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3.1 div and curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3.2 Using Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 Integrals 4 3.1 Basic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.1.1 Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.1.2 Iterated Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.2 Change of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.3 Path and Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.4 Surface and Volume Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.4.1 Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.4.2 Green’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.4.3 Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.4.4 Gauss’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1 Basics 1.1 Geometry 1. ( 203Sfnl07-08-1) Let A = (0 , 1 , − 2) ,B = (5 , 2 , − 1) ,C = ( − 2 , − 1 , 3) ,D = (2 , 2 , − 1). Let −−→ AB etc denote the vector from A to B . (a) What is the volume of the parallelopiped spanned by −−→ AB , −→ AC and −−→ AD ? (b) What is the distance from the line through A and B to the line through C and D ? (c) What is the distance from A to the plane though B , C and D ? 1 1.2 Limits 1. ( 203Ffnl07-08-2) Compute the limits (if they exist): (a) lim ( x,y ) → (0 , 0) x 4 − y 2 + x 2 y x 4 − y 2 − xy 2 . (b) lim ( x,y ) → (0 , 0) e − y 2 − e x 2 x 2 + y 2 . 2. ( 203Ffnl06-07-3-b) Determine whether the following limits exist, and if they do, compute them: lim ( x,y ) → (0 , 0) x 2 + xy 2 x 2 + y 3 lim ( x,y ) → (0 , 0) log(1 + x 2 + y 2 ) x 2 + y 2 2 Functions 2.1 Extrema 1. ( 203Sfnl05-06-6)Find the local and global maxima and minima of the function f ( x,y,z ) = xy + yz + zx in the closed unit ball B = { ( x,y,z ) : x 2 + y 2 + z 2 ≤ 1 } . Find both the locations of the extrema and the minimum and maximum value that...
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This note was uploaded on 11/05/2011 for the course MAT 203 at Princeton.

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mat203_finals_complied - Previous Final Problems Andrei...

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