Calculus 3 : Tangent Planes and Linear Approximations

Study concepts, example questions & explanations for Calculus 3

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Example Questions

Example Question #1 : Tangent Planes And Linear Approximations

Find the linear approximation to  at .

Possible Answers:

Correct answer:

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 .

Possible Answers:

Correct answer:

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 .

Possible Answers:

None of the Above

Correct answer:

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 .

Possible Answers:

Correct answer:

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 

Possible Answers:

Correct answer:

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 

Possible Answers:

Correct answer:

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

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