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DEPARTMENT OF MATHEMATICS

MATH1052

Assignment 2

Summer 2022

1. A bomb suddenly explodes within earshot of Adriana, Brinda, and Chimalsi. The instant they hear it, all three immediately look at their watches.  The times that each person hears the explosion are as follows:

– Adriana (A): Midday exactly

– Brinda (B): 5 −^17 seconds before midday

– Chimalsi (C) 5 −^5 seconds before midday.

Meanwhile, the locations of all three people are pictured below.

a) Show that the explosion’s coordinates (x,y) must lie on the hyperbola

b) Find a second such hyperbola.

c) Plot these hyperbolas using MATLAB on the same graph. Where do they intersect? (Hint, you may wish to use vpasolve and/or fsolve for this). Include any MATLAB code used in your answer.

d) Where did the explosion occur?

N.B. For this problem, assume the following things:

– Sound travels in a straight line with speed 1 in the distance units used.

– the Earth is approximately flat over the relatively small area in question.

– Adriana, Brinda, and Chimalsi are very diligent and keep their watches accurate, with time displayed in surds.

2. Consider the function

z = f(x,y), where x = r cosθ and y = r sinθ .

a) Use the Chain Rule to find and .

b) Show that

3. Suppose

f(x,y) = e −x2 −y2 .

a) Find the quadratic approximation Q(x,y) for f(x,y) at the point (0,0).

b) Compare the values of Q(x,y) and f(x,y) at (0.9, 0.1).

4. Let

f(x,y) = 2x3 + xy2 + 5x2 + y2 .

a) Find all critical points of the function f(x,y).

b) Classify all critical points of the function f(x,y).

5. a) Find the points on the sphere x2 + y2 + z2  = 4 that are closest to and farthest from

the point (3, 1, −1).

b) Use the method of Lagrange multipliers to show part a).

(Hint: use the square of the distance)