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

Question

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

Isaiah Allen · Accepted 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#

Elijah Abram · Answer

undefined To find the linearization of the function $ z = x\sqrt{y} $ at the point $(-7, 64)$, follow these steps:

1. Calculate the partial derivatives of $ z $ with respect to $ x $ and $ y $, denoted as $ \frac{\partial z}{\partial x} $ and $ \frac{\partial z}{\partial y} $, respectively.
2. Evaluate these partial derivatives at the given point $(-7, 64)$.
3. Use the point-slope form of the equation of a line to write the linearization.
4. Plug in the values of the point and the partial derivatives to obtain the linearization.

By following these steps, you can find the linearization of the function $ z = x\sqrt{y} $ at the point $(-7, 64)$.