Calculus 3 : Applications of Partial Derivatives

Study concepts, example questions & explanations for Calculus 3

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Example Questions

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Example Question #131 : Applications Of Partial Derivatives

Find the relative maxima and minima of .

Possible Answers:

  and  are relative minima

 

  and  are saddle points

 

  is a relative minima and  is a relative maxima

 

  is a relative maxima and  is a relative minima

 

Correct answer:

  and  are saddle points

 

Explanation:

To find the relative maxima and minima, we must find all the first order and second order partial derivatives.  We will use the first order partial derivative to find the critical points, then use the equation

to classify the critical points.  

If  and , then there is a relative minimum at this point.

If  and , then there is a relative maximum at this point.

If , then this point is a saddle point.

If , then this point cannot be classified.

 

 The first order partial derivatives are

The second order partial derivatives are

 

To find the critical points, we will set the first derivatives equal to 

 

The critical points are   and .  We need to determine if the critical point is a maximum or minimum using  and .  

 

 At ,

Since  is a saddle point.

 

At ,

Since  is a saddle point.

 

 

Example Question #132 : Applications Of Partial Derivatives

Find the relative maxima and minima of .

Possible Answers:

 is a relative minima

 is a saddle point

 is a relative minima

 is a saddle point

Correct answer:

 is a saddle point

Explanation:

To find the relative maxima and minima, we must find all the first order and second order partial derivatives.  We will use the first order partial derivative to find the critical points, then use the equation

to classify the critical points.  

If  and , then there is a relative minimum at this point.

If  and , then there is a relative maximum at this point.

If , then this point is a saddle point.

If , then this point cannot be classified.

 

 The first order partial derivatives are

The second order partial derivatives are

 

To find the critical points, we will set the first derivatives equal to 

The exponential part of each expression cannot equal , so each derivative is  only when  and .  That is  and .

 

The critical points are  and .  We need to determine if the critical points are maxima or minima using  and .  

 

 At ,

 

Since  is a saddle point.

Example Question #133 : Applications Of Partial Derivatives

Find the relative maxima and minima of .

Possible Answers:

 is a saddle point

 is a relative maximum

 is a relative maximum

 is a saddle point

Correct answer:

 is a saddle point

Explanation:

To find the relative maxima and minima, we must find all the first order and second order partial derivatives.  We will use the first order partial derivative to find the critical points, then use the equation

to classify the critical points.  

If  and , then there is a relative minimum at this point.

If  and , then there is a relative maximum at this point.

If , then this point is a saddle point.

If , then this point cannot be classified.

 

 The first order partial derivatives are

The second order partial derivatives are

 

 

To find the critical points, we will set the first derivatives equal to 

The exponential part of each expression cannot equal , so each derivative is  only when  and .  That is  and .

 

The critical point is .  We need to determine if the critical points are maxima or minima using  and .  

 

 At ,

 

 

Since  is a saddle point.

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