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quiz2summer2009

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 definition of the Fr´ echet derivative. (b) [7 marks] Show that if f : R n R m and g : R n R m are Fr´ echet differentiable at a R n , then the sum f + g : R n R m , x mapsto→ f ( x ) + g ( x ) is Fr´ echet differentiable 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 ) mapsto→ 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 flood. 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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