How do you find the linearization of the function z=xsqrt(y) at the point (-7, 64)?

1 Answer

The linear function that best aproximates z=x sqrt(y) at (-7, 64) is z = -56 + 8(x+7) - 7/16(y-64) = 28 + 8x - 7/16y.

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 L_(r_0)(x,y) of f at r_0 = (x_0,y_0) = (-7,64) is given by

L_(r_0) (x,y)= f(x_0, y_0) + vec(grad)f(x_0, y_0) * ((x,y)-(x_0,y_0))

Where vec(grad)f is the gradient of f and * is the dot product.

Geometrically, this linear approxiamtion is the tangent plane of f at r_0. 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 vec(grad)f playing the role of the derivative.

Now we need to calculate the components of the equations for the linear aproximation. f(x_0, y_0) is simply the value of the function at (x_0, y_0):

f(x_0, y_0) = f(-7, 64) = -7 times sqrt(64) = -56

The gradient vec(grad)f(x,y) of f is given by the expression

vec(grad)f(x, y) = ((del f)/(del x), (del f)/(del y)) = (sqrt(y), x/(2sqrt(y)))

So, vec(grad)f(x_0, y_0) = (sqrt(64), -7/(2sqrt(64))) = (8, -7/16)

Finally, we have:

L_(r_0) (x,y)= -56 + (8, -7/16) * ((x,y)-(-7,64)) =
= - 56 + (8, -7/16) * (x+7, y-64) =
= -56 + 8 (x - 7) - 7/16 (y - 64) =28 + 8x - 7/16y