Answer to Question #136787 in Calculus for Promise Omiponle

Question

Find the linearization of the function z=x√y at the point (-9, 25).
L(x,y)=

Answer 1

f(x,y)=f(Z)= xy^1/2

F_x of the function f(x,y) = y^1/2

F_y of the function f(x,y) = (xy^-1/2)/2

Now, linearization of a function f(x,y)=f(z) at (a,b) is given by

L(x,y) = f(a,b) + (x-a)f_x(a,b) + (y-b)f_y(a,b)

f(a,b) = f(-9,25)=(-9)(25)^1/2= (-45)

f_x(a,b) = f_x(-9,25)=y^1/2=(25)^1/2=5

fy(a,b)=f_y(-9,25)=(xy^-1/2)/2=-9(25^-1/2)/2

= -0.9

So, linearization of f(x,y) = xy^1/2 at (a,b)

= (-9,25) is given by

L(x,y) = f(-9,25)+(x-(-9))f_x(-9,25)+(y-25)

f_y(-9,25)

L(x,y) = -45+(x+9)(5)+(y-25)(-0.9)

L(x,y) = -45+5x+45+(-0.9y+22.5)

L(x,y) = -45+5x+45-0.9y+22.5

L(x,y) = 5x-0.9y+22.5

so, linearization of f(x,y)=xy^1/2 at (a,b)=

(-9,25) is given by

L(x,y) = 5x-0.9+22.5