# Given the surface #f(x,y,z)=y^2 + 3 x^2 + z^2 - 4=0# and the points #p_1=(2,1,1)# and #p_2=(3,0,1)# determine the tangent plane to #f(x,y,z)=0# containing the points #p_1# and #p_2#?

See below.

Calling

and considering

Now, the vector

The

or

Solving for

and also

the corresponding normal surface vectors.

Note:

a) In (1,2) we consider only a vector component. The choice is one of

b) The sign

c) Here

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To determine the tangent plane to the surface ( f(x,y,z) = y^2 + 3x^2 + z^2 - 4 = 0 ) containing the points ( p_1 = (2,1,1) ) and ( p_2 = (3,0,1) ), we need to follow these steps:

- Find the gradient vector of the surface function ( f(x,y,z) ).
- Evaluate the gradient vector at the points ( p_1 ) and ( p_2 ) to obtain the normal vectors to the surface at these points.
- Use the normal vectors and the points ( p_1 ) and ( p_2 ) to find the equations of the tangent planes at these points.
- Equate the equations of the tangent planes to find the common tangent plane passing through both ( p_1 ) and ( p_2 ).

First, let's find the gradient vector of ( f(x,y,z) ):

[ \nabla f(x,y,z) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) ]

[ \nabla f(x,y,z) = \left( 6x, 2y, 2z \right) ]

Evaluate the gradient vector at ( p_1 ) and ( p_2 ):

At ( p_1 = (2,1,1) ):

[ \nabla f(2,1,1) = \left( 12, 2, 2 \right) ]

At ( p_2 = (3,0,1) ):

[ \nabla f(3,0,1) = \left( 18, 0, 2 \right) ]

The normal vector to the surface at ( p_1 ) is ( \left( 12, 2, 2 \right) ), and at ( p_2 ) is ( \left( 18, 0, 2 \right) ).

Now, we can use the point-normal form of the equation of a plane to find the equations of the tangent planes at ( p_1 ) and ( p_2 ):

At ( p_1 = (2,1,1) ), the equation of the tangent plane is:

[ 12(x - 2) + 2(y - 1) + 2(z - 1) = 0 ]

At ( p_2 = (3,0,1) ), the equation of the tangent plane is:

[ 18(x - 3) + 0(y - 0) + 2(z - 1) = 0 ]

Now, equate the equations of the tangent planes to find the common tangent plane passing through both ( p_1 ) and ( p_2 ).

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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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