How do you find the equations for the tangent plane to the surface #f(x,y)=22/3xy# through (3,1,1)?
The normal vector at any point is:
Plane is in form:
Then take the vector product of these tangent vectors:
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The tangent plane to the plane
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To find the equation for the tangent plane to the surface f(x, y) = 2  (2/3)x  y through the point (3, 1, 1), we can follow these steps:
 Calculate the partial derivatives of f(x, y) with respect to x and y.
 Evaluate the partial derivatives at the given point (3, 1).
 Use the values obtained in step 2 to find the normal vector to the tangent plane.
 Write the equation of the tangent plane using the normal vector and the given point.
Let's go through these steps:

The partial derivative of f(x, y) with respect to x is 2/3, and with respect to y is 1.

Evaluating the partial derivatives at (3, 1): f_x(3, 1) = 2/3 f_y(3, 1) = 1

The normal vector to the tangent plane is given by the coefficients of the partial derivatives, which are (2/3, 1, 1).

Using the point (3, 1, 1) and the normal vector (2/3, 1, 1), we can write the equation of the tangent plane as: 2/3(x  3)  (y + 1) + (z  1) = 0
Simplifying the equation, we get: 2/3x  y + z + 2/3 + 1  1 = 0 2/3x  y + z + 2/3 = 0
Therefore, the equation for the tangent plane to the surface f(x, y) = 2  (2/3)x  y through the point (3, 1, 1) is 2/3x  y + z + 2/3 = 0.
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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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