Calculus 3 : Applications of Partial Derivatives

Study concepts, example questions & explanations for Calculus 3

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Example Questions

Example Question #61 : Applications Of Partial Derivatives

Find  of the function 

Possible Answers:

Correct answer:

Explanation:

To find the gradient vector, you use the following definition: . Taking the partial derivatives with respect to x, y, and z, we get , and . Putting these into the vector produces the right answer. 

Example Question #62 : Applications Of Partial Derivatives

Find  of the function 

Possible Answers:

Correct answer:

Explanation:

To find the gradient vector for a function , we use the definition

.

 

Putting these answers into the vector gets us the correct answer

Example Question #63 : Applications Of Partial Derivatives

Find  of the function 

Possible Answers:

Correct answer:

Explanation:

To find the gradient vector for a function , we use the definition

.

 

Putting these answers into the vector gets us the correct answer

Example Question #64 : Applications Of Partial Derivatives

Find  of the function 

Possible Answers:

Correct answer:

Explanation:

To find the gradient vector for a function , we use the definition

.

 

Putting these answers into the vector gets us the correct answer

Example Question #65 : Applications Of Partial Derivatives

Compute the gradient of the following scalar function:

Possible Answers:

Correct answer:

Explanation:

The gradient of a function is defined as:

For our function:

Thus, the gradient is:

Example Question #66 : Applications Of Partial Derivatives

Compute the gradient of the following scalar function:

Possible Answers:

Correct answer:

Explanation:

The gradient of a function is defined as:

For our function:

Thus, the gradient is:

Example Question #67 : 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 #68 : 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 

The rules used to find the derivatives are

Example Question #69 : Applications Of Partial Derivatives

Find  of the function 

Possible Answers:

Correct answer:

Explanation:

To find the gradient vector, you use the formula , where . Using the function given and the rules for partial differentiation, we e. Plugging these values into vector notation gets you the correct answer.

Example Question #70 : Applications Of Partial Derivatives

Find  of the function 

Possible Answers:

Correct answer:

Explanation:

To find the gradient vector, you use the formula , where . Using the function given and the rules for partial differentiation, we e. Plugging these values into vector notation gets you the correct answer.

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