Assignment 1 PDF

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MTH2010 Assignment 1 Semester 1 2019 Important instructions and remarks (a) Your completed assignment must be submitted in the designated assignment box on the ground floor of 9 Rainforest Walk by 4 pm on Friday, 29 March (Week 4). (b) Late assignments must be submitted directly to the current Lectu...

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MTH2010 Assignment 1 Semester 1 2019 Important instructions and remarks (a) Your completed assignment must be submitted in the designated assignment box on the ground floor of 9 Rainforest Walk by 4 pm on Friday, 29 March (Week 4). (b) Late assignments must be submitted directly to the current Lecturer. A penalty of 10% per day will be applied to late assignments until the solutions are released after which they will be worth 0. (c) You are expected to put sufficient effort and thought into the presentation of your solutions so that they are clear, neat and concise. They should have a completed cover sheet and be stapled. Poorly presented assignments will be penalized.

Problem 1 Determine if the following limits exist:1 xy 2 + sin(x)x2 , (i) lim x2 + y 2 (x,y)→(0,0) cos x − 1 + x2 /2 . (ii) lim x4 + y 4 (x,y )→(0,0)

[5 marks] [5 marks]

Problem 2 Consider the function 1

f (x, y) = (x − 1) 3 y 2 . ∂f ∂f and ∂y exist at the points (x, y) = (1, 0)? If they do, (i) Do the partial derivatives ∂x then compute them. [4 marks] (ii) Is f (x, y) differentiable at (x, y) = (1, 0)? [4 marks] (iii) Compute the linearization of f (x, y) at the point (x, y) = (1, 0). Does the linearization approximate f (x, y) near the point (1, 0)? Justify your answer. [2 marks]

Problem 3 Consider the function Let   x2 y 2 z(x, y) = e f ye 2y ,

(x, y) ∈ R2 \ { (t, 0) | t ∈ R },

where f (t) is differentiable for all t ∈ R. Show that z satisfies the partial differential equation ∂z ∂z = xyz (x2 − y 2 ) + xy ∂y ∂x for all (x, y) ∈ R2 \ { (t, 0) | t ∈ R }. [10 marks]

1Please

clearly note all the properties of limits and continuous functions that you use when presenting your solutions. For example, if you want to use the fact that polynomials are continuous or properties of the limit such as lim(x,y)→(a,b) f (x, y)g(x, y) = lim(x,y )→(a,b) f (x, y) lim(x,y)→(a,b) g(x, y) whenever the limits lim(x,y)→(a,b) f (x, y) and lim(x,y )→(a,b) g(x, y) exist, then state this clearly in your solutions. 1

2

Problem 4 Find the equation of the tangent plane and the normal vector to the graph of the following functions at the given point: 2 (i) f (x, y) = (x + y)e−x , (x, y) = (0, 1), [5 marks]   x (ii) f (x, y) = sin , (x, y) = (π, 4). [5 marks] y...