How do you find the linearization at (2,9) of #f(x,y) = xsqrty#?
The local linearization in point
The tangent plane to the surface
The first step is the normal vector to
in
Now, the tangent plane or the so called local linearization is given by
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To find the linearization at the point ((2,9)) of the function (f(x,y) = x\sqrt{y}), follow these steps:
 Compute the partial derivatives of (f) with respect to (x) and (y), denoted as (f_x) and (f_y), respectively.
 Evaluate (f) and its partial derivatives at the given point ((2,9)).
 Use the formula for the linearization to construct the linear approximation.
Let's go through these steps:

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}} ]

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} ]

Use the linearization formula: [ L(x,y) = f(a,b) + f_x(a,b)(xa) + f_y(a,b)(yb) ]
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(x2) + \frac{1}{3}(y9) ]
This is the linearization at the point ((2,9)) of the function (f(x,y) = x\sqrt{y}).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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