The linear function that best aproximates z=x√y at (−7,64) is z=−56+8(x+7)−716(y−64)=28+8x−716y.
To get this result, we must first notice that z is a function of the two variables x and y. Let's write z=f(x,y). So, the best linear approximation Lr0(x,y) of f at r0=(x0,y0)=(−7,64) is given by
Lr0(x,y)=f(x0,y0)+−→∇f(x0,y0)⋅((x,y)−(x0,y0))
Where −→∇f is the gradient of f and ⋅ is the dot product.
Geometrically, this linear approxiamtion is the tangent plane of f at r0. The deduction of this equation is very similar to the deduction of the equation for the tangent line of a real function at a point, with the gradient −→∇f playing the role of the derivative.
Now we need to calculate the components of the equations for the linear aproximation. f(x0,y0) is simply the value of the function at (x0,y0):
f(x0,y0)=f(−7,64)=−7×√64=−56
The gradient −→∇f(x,y) of f is given by the expression
−→∇f(x,y)=(∂f∂x,∂f∂y)=(√y,x2√y)
So, −→∇f(x0,y0)=(√64,−72√64)=(8,−716)
Finally, we have:
Lr0(x,y)=−56+(8,−716)⋅((x,y)−(−7,64))=
=−56+(8,−716)⋅(x+7,y−64)=
=−56+8(x−7)−716(y−64)=28+8x−716y
