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Multivariable Calculus

by: Kavon Feest

Multivariable Calculus MATH 53

Kavon Feest

GPA 3.93


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Class Notes
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This 4 page Class Notes was uploaded by Kavon Feest on Thursday October 22, 2015. The Class Notes belongs to MATH 53 at University of California - Berkeley taught by Staff in Fall. Since its upload, it has received 8 views. For similar materials see /class/226611/math-53-university-of-california-berkeley in Mathematics (M) at University of California - Berkeley.

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Date Created: 10/22/15
Practice Midterm MATH 53 CALCULUS HI FALL 2010 Student ID Name Treat this as if you were taking the actual midterm Sit yourself down for 90 minutes and try to write complete and correct solutions just as you would on the exam Do not use a calculator and do not look in your book No justi cation no points Problem 18 points Find the point in R3 closest to the plane z 2y 32 1 Problem 27 points Determine the tangent plane to the surface zyz 23 3 at the point 1172 Problem 310 points Determine the limit 3 7 3 lim wya00 W 7 2 Hint Use polar coordinates eventually Problem 47 points Let ab be two real numbers with ab 3A 0 Show that the polar equation 7 asin bcos l represents a Circle Find its center and radius Problem 510 points Let P 123Q 010 and R 111 Determine the area of the triangle PQR Also nd the equation of a plane containing all three points Problem 615 points Find the absolute maximum and minimum values of f 4x6y7x27y2 onthe set Dzy 0 470y 5 2 2 Problem 710 points Let Fzyz y2 Z7 7 3 Determine the tangent plane at the point 727 17 73 Use this to approximate F7197 117 73 Problem 810 points Let x tt2 7 1 and y t2 7 1 Find the equations of the tangent lines of the parametric curve passing through the point 00 Problem 98 points Let ag2 26 In which direction does f increase the fastest What is the maximum rate of increase Problem 1015 points Three non collinear points P7 Q7 R in R3 determine a triangle Find the maximum and minimum area of the triangle given that the perimeter is 12 units long Could you have guessed your answer beforehand7 using geometry


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