Calculus 3 : Applications of Partial Derivatives

Study concepts, example questions & explanations for Calculus 3

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Example Questions

Example Question #81 : Applications Of Partial Derivatives

Find the equation of the tangent plane to the given function at :

Possible Answers:

Correct answer:

Explanation:

The equation of the tangent plane is given by

So, we must find the partial derivatives for the function evaluated at the point given. To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

The derivatives evaluated at the given point are

Plugging all of this into the above formula, we get

which simplifies to

Example Question #82 : Applications Of Partial Derivatives

Find  of the following function:

Possible Answers:

Correct answer:

Explanation:

The gradient of the function is given by

To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

Example Question #83 : Applications Of Partial Derivatives

Find the equation of the plane tangent to the following function at :

Possible Answers:

Correct answer:

Explanation:

The equation of the tangent plane is given by

So, we must find the partial derivatives for the function evaluated at the point given. To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

The partial derivatives evaluated at the given point are

Plugging this into the formula above, we get

which simplifies to

Example Question #84 : Applications Of Partial Derivatives

Write the equation of the plane tangent to the given function at the point :

Possible Answers:

Correct answer:

Explanation:

The equation of the tangent plane is given by

So, we must find the partial derivatives for the function evaluated at the point given. To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

The partial derivatives evaluated at the given point are

Plugging this into the above formula, we get

which simplified becomes

 

Example Question #85 : Applications Of Partial Derivatives

Find  of the following function:

Possible Answers:

Correct answer:

Explanation:

The gradient of a function is given by

To find the given partial derivative of the function, we must treat the other variable(s) as constants. 

The partial derivatives are

Example Question #86 : Applications Of Partial Derivatives

Write the equation of the plane tangent to the given function at the point :

Possible Answers:

Correct answer:

Explanation:

The equation of the tangent plane is given by

So, we must find the partial derivatives for the function evaluated at the point given. To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

The partial derivatives evaluated at the given point are

Plugging this into the above formula, we get

which simplified becomes

Example Question #87 : Applications Of Partial Derivatives

Find the equation of the plane tangent to the following function at the point :

Possible Answers:

Correct answer:

Explanation:

The equation of the tangent plane is given by

So, we must find the partial derivatives for the function evaluated at the point given. To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

The partial derivatives evaluated at the given point are

Now, plug in the information into the formula above:

which simplified becomes

 

Example Question #88 : Applications Of Partial Derivatives

Find  of the following function:

Possible Answers:

Correct answer:

Explanation:

The gradient of the function is given by

To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

Example Question #89 : Applications Of Partial Derivatives

Find the equation of the plane tangent to the following function at the point :

Possible Answers:

Correct answer:

Explanation:

The equation of the tangent plane is given by

So, we must find the partial derivatives for the function evaluated at the point given. To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

The partial derivatives evaluated at the given point are

Plugging this into the formula above, we get

which simplified becomes

Example Question #90 : Applications Of Partial Derivatives

Find  for the following function:

Possible Answers:

Correct answer:

Explanation:

The gradient of the function is given by

To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

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