The height of the point on a hill above the point (x,y) in the plane is h(x,y) =
ID: 2848304 • Letter: T
Question
The height of the point on a hill above the point (x,y) in the plane is h(x,y) = -3x^2 + 2xy - y^2 + 2x + y + 500
a) What is the aproximate difference in height between the points (2,3) and (2.01, 3.02)?
b) What is the approximate difference in height between the points (2,3) and (2 + (delta x), 3 + 2 delta(y))?
The height of the point on a hill above the point (x,y) is (-6x^2+3xy-2y^2+15x-42y+5000)/100
a) At the point (2.3) which of the eight compass directions would you go down the hill as quickly as possible? The x-axis points east and the y-axis points north.
b) At which points (x,y) would walking north initially change your height
y = ?
Explanation / Answer
B ) aking partial derivative = -6*x*deltax + 2*x*deltay+2*y*deltax - 2*y*deltay+2deltax +deltay
a) Putting delta x - 0.01 and deltay = 0.02 then
-6*2*0.01 + 2*2*0.02+2*3*0.01 - 2*3*0.02+2*0.01 + 0.02 = -0.06
For part 2
a) taking partial derivative of (-6x^2+3xy-2y^2+15x-42y+5000)/100
(-12*x*deltax+ 3*x*deltay+3*y*deltax - 4*y*deltay+15*deltax -42 *deltay)/100
Putting value of x and y
[Deltax (-24+9+15) + deltay ( 6- 12-43)]/100 = -0.49 delta y
Hence to go down he should go in negative y direction that will reduce the height.
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