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

Study concepts, example questions & explanations for Calculus 3

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Example Questions

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Example Question #1 : Gradient Vector, Tangent Planes, And Normal Lines

Find the equation of the tangent plane to  at .

Possible Answers:

Correct answer:

Explanation:

First, we need to find the partial derivatives in respect to , and , and plug in .

 

Remember that the general equation for a tangent plane is as follows:

Now lets apply this to our problem

Example Question #1 : Gradient Vector, Tangent Planes, And Normal Lines

Find the slope of the function  at the point 

Possible Answers:

Correct answer:

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

It is essentially the slope of a multi-dimensional function at any given point

Knowledge of the following derivative rules will be necessary:

Trigonometric derivative: 

Note that u may represent large functions, and not just individual variables!

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Looking at  at the point 

x:

y:

z:

 

Example Question #3 : Gradient Vector, Tangent Planes, And Normal Lines

Find the slope of the function  at the point 

Possible Answers:

Correct answer:

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

It is essentially the slope of a multi-dimensional function at any given point

Knowledge of the following derivative rules will be necessary:

Trigonometric derivative: 

Note that u may represent large functions, and not just individual variables!

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Looking at  at the point 

x:

y:

z:

 

Example Question #4 : Gradient Vector, Tangent Planes, And Normal Lines

Find the slope of the function  at the point 

Possible Answers:

Correct answer:

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

It is essentially the slope of a multi-dimensional function at any given point

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Looking at  at the point 

x:

y:

z:

 

Example Question #2 : Gradient Vector, Tangent Planes, And Normal Lines

Find the slope of the function  at the point 

Possible Answers:

Correct answer:

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

It is essentially the slope of a multi-dimensional function at any given point.

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Looking at  at the point 

x:

y:

z:

 

Example Question #6 : Gradient Vector, Tangent Planes, And Normal Lines

Find the slope of the function  at the point 

Possible Answers:

Correct answer:

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

It is essentially the slope of a multi-dimensional function at any given point.

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Looking at  at the point 

x:

y:

z:

 

Example Question #7 : Gradient Vector, Tangent Planes, And Normal Lines

Find the slope of the function  at the point 

Possible Answers:

Correct answer:

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

It is essentially the slope of a multi-dimensional function at any given point.

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Looking at  at the point 

x:

y:

z:

 

Example Question #3 : Gradient Vector, Tangent Planes, And Normal Lines

Find the slope of the function  at the point 

Possible Answers:

Correct answer:

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

It is essentially the slope of a multi-dimensional function at any given point.

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Looking at  at the point 

x:

y:

Example Question #9 : Gradient Vector, Tangent Planes, And Normal Lines

Find the slope of the function  at the point 

Possible Answers:

Correct answer:

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

It is essentially the slope of a multi-dimensional function at any given point.

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Looking at  at the point 

x:

y:

Example Question #10 : Gradient Vector, Tangent Planes, And Normal Lines

Find the slope of the function  at the point 

Possible Answers:

Correct answer:

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

It is essentially the slope of a multi-dimensional function at any given point.

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Looking at  at the point 

x: 

y:

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