How do you find the linearization at #(5,16)# of #f(x,y) = x sqrt(y)#?

Answer 1

#z = 5/8 (88 - 8 x - y)#

Consider the surface

#S(x,y,z) = f(x,y)-z=0#.
The linearization of #f(x,y)# at the point #x_0,y_0# is equivalent to determining the tangent plane to #S# at the point #{x_0,y_0,f(x_0,y_0)}#
The tangent plane to #S# is
#Pi_t -> << p-p_0, vec n >> = 0# where
#p_0 ={x_0,y_0,f(x_0,y_0)} ={5,16,20}# #p = {x,y,z}# #vec n = grad S = {f_x,f_y,f_z} = {x_0,1/2x_0/sqrt(y_0),1} = {5,5/8,1}#

so

#Pi_t-> (x-5)5+(y-16)5/8+(z-20)=0#

The linearization in the desired point is

#z = 5/8 (88 - 8 x - y)#
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Answer 2

To find the linearization of ( f(x, y) = x\sqrt{y} ) at the point ( (5, 16) ), we first compute the partial derivatives of ( f(x, y) ) with respect to ( x ) and ( y ). Then, we evaluate these derivatives at the given point ( (5, 16) ). The linearization is then given by the equation of the tangent plane to the surface at that point, which is:

[ L(x, y) = f(a, b) + \frac{\partial f}{\partial x}(a, b)(x - a) + \frac{\partial f}{\partial y}(a, b)(y - b) ]

where ( (a, b) ) is the point of tangency. Substituting the given point ( (5, 16) ) into the equation, and plugging in the computed partial derivatives, we obtain the linearization.

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Answer 3

To find the linearization of ( f(x,y) = x\sqrt{y} ) at the point (5,16), follow these steps:

  1. Find the partial derivatives of ( f(x,y) ) with respect to ( x ) and ( y ). [ f_x = \sqrt{y} ] [ f_y = \frac{x}{2\sqrt{y}} ]

  2. Evaluate the partial derivatives at the given point (5,16). [ f_x(5,16) = \sqrt{16} = 4 ] [ f_y(5,16) = \frac{5}{2\sqrt{16}} = \frac{5}{8} ]

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

  4. Substitute the values into the formula: [ L(x,y) = f(5,16) + f_x(5,16)(x-5) + f_y(5,16)(y-16) ] [ L(x,y) = 5\sqrt{16} + 4(x-5) + \frac{5}{8}(y-16) ] [ L(x,y) = 20 + 4(x-5) + \frac{5}{8}(y-16) ]

Therefore, the linearization of ( f(x,y) = x\sqrt{y} ) at the point (5,16) is: [ L(x,y) = 20 + 4(x-5) + \frac{5}{8}(y-16) ]

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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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