# Calculus 3 : Tangent Planes and Linear Approximations

## Example Questions

### Example Question #1 : Tangent Planes And Linear Approximations

Find the linear approximation to  at .

Explanation:

The question is really asking for a tangent plane, so lets first find partial derivatives and then plug in the point.

Remember that we need to build the linear approximation general equation which is as follows.

### Example Question #2 : Tangent Planes And Linear Approximations

Find the tangent plane to the function  at the point .

Explanation:

To find the equation of the tangent plane, we use the formula

.

Taking partial derivatives, we have

Substituting our values into these, we get

Substituting our point into , and partial derivative values in the formula we get

.

### Example Question #2 : Tangent Planes And Linear Approximations

Find the Linear Approximation to  at .

None of the Above

Explanation:

We are just asking for the equation of the tangent plane:

Step 1: Find

Step 2: Take the partial derivative of  with respect with (x,y):

Step 3: Evaluate the partial derivative of x at

Step 4: Take the partial derivative of  with respect to :

Step 5: Evaluate the partial derivative at

.

Step 6: Convert (x,y) back into binomials:

Step 7: Write the equation of the tangent line:

### Example Question #4 : Tangent Planes And Linear Approximations

Find the equation of the plane tangent to  at the point .

Explanation:

To find the equation of the tangent plane, we find:  and evaluate  at the point given. , and . Evaluating  at the point  gets us . We then plug these values into the formula for the tangent plane: . We then get . The equation of the plane then becomes, through algebra,

### Example Question #3 : Tangent Planes And Linear Approximations

Find the equation of the plane tangent to  at the point

Explanation:

To find the equation of the tangent plane, we find:  and evaluate  at the point given. , and . Evaluating  at the point  gets us . We then plug these values into the formula for the tangent plane: . We then get . The equation of the plane then becomes, through algebra,

### Example Question #6 : Tangent Planes And Linear Approximations

Find the equation of the tangent plane to  at the point

Explanation:

To find the equation of the tangent plane, we need 5 things:

Using the equation of the tangent plane

, we get

Through algebraic manipulation to get z by itself, we get