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# 22FNLf07aweb - Math 22 Calculus of Several Variables M...

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Math 22 - Calculus of Several Variables Final Exam - Form A M. Eastman - Fall 2007 1. Given the points , , and , find:  œ Ð &ß \$ß  " Ñ œ Ð %ß &ß  # Ñ œ Ð !ß )ß  " Ñ a. parametric equations for the line through and and is parallel to the line through and . b. the angle . n ‚ c. the equation of the plane (in the form through these three points. +B  ,C  -D œ .Ñ 2. Given the curve defined by , find: r Ð>Ñ œ  > ß #>ß 68 >  # a. (the unit tangent to this curve when ). T Ð#Ñ > œ # b. the length of this curve between the points and ab a b "ß #ß ! "'ß )ß 68 % c. the vector equation for the line tangent to this curve when . >œ% 3. Given , find: 0ÐBß Cß DÑ œ #B C  (BC D  %B D \$# \$ # a. b. 0 ÐBß Cß DÑ f0Ð  "ß "ß # Ñ BD c. the directional derivative of at in the direction of the point . 0 Ð" ß" ß#Ñ Ð# ß& ß%Ñ d. The equation of the tangent plane to the level surface at the point . 0ÐBß Cß DÑ œ ! Ð  "ß "ß # Ñ 4. Find the critical points for . Find the value of at each 0ÐBßCÑœB C )B #C#% 0 %# #

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## This note was uploaded on 10/10/2009 for the course MATH 22 taught by Professor Castro during the Spring '08 term at UCSC.

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22FNLf07aweb - Math 22 Calculus of Several Variables M...

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