If f(x,y) = xyey/x show that xfx + yfy = x2 fxx + 2xyfxy + y2 fyy Integrate the

Question

If f(x,y) = xyey/x show that xfx + yfy = x2 fxx + 2xyfxy + y2 fyy Integrate the function f(x, y) = xysin(xy2) over the bounded region with 0 le x le pi/2 and 0 le y le 1. Wheat production W, in tones, in a given year depends on the average temperature T and the annual rainfall R. Scientists estimate that, with global warming, the average temperature is rising at a rate of 0.15 Kelvins per year and rainfall is decreasing at a rate of 0.1 cm per year. They also estimate that, at current production levels . Write in words the meaning of = - 2. Estimate the current rate of change of wheat production. Why is this figure an estimate?

Explanation / Answer

a) fx,fy={(y e^(y/x) (x-y))/x, e^(y/x) (x+y)}

xfx +yfy =ye^(y/x) (x-y) + ye^(y/x) (x+y)

=ye^(y/x)2x

fxy=e^(y/x) (1-(y (x+y))/x^2)

fxx=(y^3 e^(y/x))/x^3

fyy=(e^(y/x) (2 x+y))/x

now,

x^2fxx + 2xyfxy + y^fyy

(y^3 e^(y/x))/x + 2ye^(y/x) (1-(y (x+y))/x + y^2(e^(y/x) (2 x+y))/x

ye^(y/x)[y^2/x+2(x-xy+y^2)/x+(2x+y)/x

=ye^(y/x)(2x)

b)integral_0^1 integral_0^(pi/2) x y sin(x y^2) dx dy

= 1/4 (pi-2)~~0.285398

c)

i) meaning of del/delT(W)=-2 is rate of change of wheat production w.r.t. temperature ,

as there is (-) sign so W is decreasing with increase in temperature.

ii) iii) d/dT(w)=-2

integrate it,

w=-2T+c(R) -----------------------(1)

where c(R) is function in R

then put value of w in d/dR(w)

then we get,

c'(R)=8,

again integrationg,

c(R)=8R+c , where c in const. independent of R and T,

put value of c(R) in W

we get,

W=-2T + 8R +c is current estimate production,

as c is a factor indepndent of R and T

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