quiz2summer2009 - year-old grandmother, who can only...

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MAT237Y – Multivariable Calculus Summer 2009 Quiz #2 Tuesday, July 9 from 6:10pm to 7:00pm. Instructors: A. Hammerlindl and J. Uren 1. (a) [3 marks] State the deFnition of the ±r´ echet derivative. (b) [7 marks] Show that if f : R n R m and g : R n R m are ±r´ echet di²erentiable at a R n , then the sum f + g : R n R m , x m→ f ( x ) + g ( x ) is ±r´ echet di²erentiable at a . 2. (a) [6 marks] Suppose you are picnicking in a beautiful valley where the height of the terrain is given by h : R 2 R , ( x, y ) m→ e 1 10 ( x 2 - y ) , where all distances are in kilometers. As you enjoy the picturesque and mathematically precise scenery from the spot (1 , 1) R 2 , a thunderstorm suddenly erupts and the valley starts to ³ood. Luckily, there is a 0 . 8 kilometer tall radio tower at (2 , 3) which acts as a lightning rod. In which direction, given as a unit vector in R 2 , should you walk to head straight uphill from your position? (b) [4 marks] You are not picnicking alone, but with your dear, 98-
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Unformatted text preview: year-old grandmother, who can only negotiate a slope with a grade of twenty-Fve percent or less. Will she be able to start walking uphill with you, or will you have to abandon her to ensure your cucumber sandwiches do not get soggy? 3. [10 marks] Suppose a = (1 , 2 , 3) and h : R 3 R is a C 1 function such that h ( a ) = 3 and h ( a ) = (2 , 4 , 5) . Compute d dt p h ( t 3 , 3 t 2 1 , h ( t, 2 , t + t 2 + t 3 ) ) P when t = 1. 1 4. (a) [5 marks] Consider the set C R 3 which is the solution of the system of equations arctan( x ) + ( y + 1) 5 = ( z + 1) 3 x 4 cos( x ) + z 2 = 1 + ( x 3) y. Show that in a neighbourhood of the origin, y and z can be rep-resented as functions of x . (b) [5 marks] Find the line tangent to C at the origin. Give it as a subset of R 3 . 2...
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This note was uploaded on 10/21/2010 for the course MATHEMATIC MAT237Y1 taught by Professor Romauldstanczak during the Fall '09 term at University of Toronto- Toronto.

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quiz2summer2009 - year-old grandmother, who can only...

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