How do you find the linearization at (2,9) of #f(x,y) = xsqrty#?

Answer 1

The local linearization in point #p_0=(2,9,6)# is given by
# -3 (x-2) + (9 - y)/3 + z - 6 = 0#

The tangent plane to the surface #S(x.y,z)=z-x sqrt(y) = 0# in the point #p_0=(2,9,2sqrt(9))=(2,9,6)# is obtained as follows.
The first step is the normal vector to #S(2,9,6)#. The vector #vec v_0 # is obtained with the calculation of #grad S(x,y,z) = ((partial S(x,y,z))/(partial x),(partial S(x,y,z))/(partial y),(partial S(x,y,z))/(partial z))#
in #p_0# giving
#vec v_0 = (-sqrt[y], -(x/(2 sqrt[y])), 1)_0 = (-3, -(1/3), 1)#
Now, the tangent plane or the so called local linearization is given by
#(p - p_0).vec v_0 = 0 -> -3 (x-2) + (9 - y)/3 + z - 6 = 0#

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Answer 2

To find the linearization at the point ((2,9)) of the function (f(x,y) = x\sqrt{y}), follow these steps:

  1. Compute the partial derivatives of (f) with respect to (x) and (y), denoted as (f_x) and (f_y), respectively.
  2. Evaluate (f) and its partial derivatives at the given point ((2,9)).
  3. Use the formula for the linearization to construct the linear approximation.

Let's go through these steps:

  1. Compute the partial derivatives: [ f_x = \frac{\partial f}{\partial x} = \sqrt{y} ] [ f_y = \frac{\partial f}{\partial y} = \frac{x}{2\sqrt{y}} ]

  2. Evaluate (f) and its partial derivatives at ((2,9)): [ f(2,9) = 2\sqrt{9} = 6 ] [ f_x(2,9) = \sqrt{9} = 3 ] [ f_y(2,9) = \frac{2}{2\sqrt{9}} = \frac{1}{3} ]

  3. Use the linearization formula: [ L(x,y) = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b) ]

Substitute (a = 2), (b = 9), (f(2,9) = 6), (f_x(2,9) = 3), (f_y(2,9) = \frac{1}{3}): [ L(x,y) = 6 + 3(x-2) + \frac{1}{3}(y-9) ]

This is the linearization at the point ((2,9)) of the function (f(x,y) = x\sqrt{y}).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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