Calculus 3 : Gradient Vector, Tangent Planes, and Normal Lines

Study concepts, example questions & explanations for Calculus 3

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Example Questions

Example Question #91 : Applications Of Partial Derivatives

Find the equation to the tangent plane 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

Evaluated at the given point, the partial derivatives are

Plugging this into the formula above, we get

which simplifies to

Example Question #71 : Gradient Vector, Tangent Planes, And Normal Lines

Find  for the following function:

Possible Answers:

Correct answer:

Explanation:

The gradient vector of a function is given by

written in standard form as 

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

The partial derivatives are

Example Question #92 : Applications Of Partial Derivatives

Find the gradient vector of the following function:

Possible Answers:

Correct answer:

Explanation:

The gradient vector 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 #74 : Gradient Vector, Tangent Planes, And Normal Lines

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 

Evaluated at the given point, the partial derivatives are

Plugging these into the formula above, we get

which simplifies to

 

Example Question #75 : Gradient Vector, Tangent Planes, And Normal Lines

Find the equation of the tangent plane 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

Evaluated at the given point, the partial derivatives are

Putting this into the equation above, we get

which simplified becomes

Example Question #72 : Gradient Vector, Tangent Planes, And Normal Lines

Find the gradient vector for the following function:

Possible Answers:

Correct answer:

Explanation:

The gradient vector 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 #71 : Gradient Vector, Tangent Planes, And Normal Lines

Find the equation of the plane that passes through the points  and 

Possible Answers:

Correct answer:

Explanation:

Step 1:

Let 

Using these three points we will find two vectors  and . [You can find PQ and QR too]

Step 2:

We are required to find a perpendicular (normal) vector to both  and . So we need to take their cross product

We have found our normal vector.

 

Step 3: We will use the following formula to find the final answer

Example Question #71 : Gradient Vector, Tangent Planes, And Normal Lines

Find the gradient vector for

Possible Answers:

Correct answer:

Explanation:

Suppose that

then

taking the respective partial derivatives and putting them into order as stated in the formula above yields

Example Question #73 : Gradient Vector, Tangent Planes, And Normal Lines

Find the gradient vector for the following function:

Possible Answers:

Correct answer:

Explanation:

The gradient vector 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 #74 : Gradient Vector, Tangent Planes, And Normal Lines

Find the gradient vector of the following function:

Possible Answers:

Correct answer:

Explanation:

The gradient vector 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

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