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 #31 : Applications Of Partial Derivatives

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 #11 : 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:

Example Question #12 : 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:

Example Question #13 : Gradient Vector, Tangent Planes, And Normal Lines

Find the equation of the tangent plane to  at .

Possible Answers:

Correct answer:

Explanation:

The definition of a tangent plane of a surface  is given by 

.

Therefore, we need to first find , given below as

Noting that 

,

we can write our complete expression for the tangent line as

.

Example Question #15 : Gradient Vector, Tangent Planes, And Normal Lines

Calculate  if  and 

Possible Answers:

Correct answer:

Explanation:

By definition .  Therefore, 

, so we will need to find the partial derivatives of , shown below as

Therefore, 

Example Question #41 : Applications Of Partial Derivatives

Calculate  given 

Possible Answers:

Correct answer:

Explanation:

By definition, 

, where  are the respective  components of .

Therefore, we need to calculate the above terms, shown as

Therefore,

.

Example Question #17 : Gradient Vector, Tangent Planes, And Normal Lines

Find , where 

Possible Answers:

Correct answer:

Explanation:

The gradient vector of f, , is equal to 

So, we must find the partial derivatives of the function with respect to x, y, and z, keeping the other variables constant for each partial derivative:

The derivatives were found using the following rules:

Plugging this in to a vector, we get

Example Question #18 : Gradient Vector, Tangent Planes, And Normal Lines

Find , where f is 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.

Now, we find the partial derivatives:

The derivatives were found using the following rules:

, , ,

Example Question #14 : Gradient Vector, Tangent Planes, And Normal Lines

Find  of the following function:

Possible Answers:

Correct answer:

Explanation:

To find the given partial derivative of the function, we must treat the other variable(s) as constants. For higher order partial derivatives, we work from left to right for the given variables.

To start, we must find the partial derivative with respect to y:

The following rules were used to find the derivative:

Next, we find the derivative of the above function above with respect to x:

The rule used is stated above.

Finally, find the partial derivative of the above function with respect to z:

The rule used is already stated above.

 

Example Question #20 : Gradient Vector, Tangent Planes, And Normal Lines

Find  of the given function:

Possible Answers:

Correct answer:

Explanation:

The gradient of a function is given by

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

The derivatives were found using the following rules:

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