Multivariable Functions

[mahowm10]

I have this assignment for my Calculus III class and I have no idea where to even start. Any help is appreciated! Thanks!!

Let f be a multivariable function defined by f(x, y) = x³y – x²y² where x and y are real numbers. Choose the same point to use as you work to complete parts A through D of the task.

A. Explain how to find the direction of maximum increase for f at your chosen point, showing all required work.

B. Explain how to find the direction of maximum decrease for f at your chosen point, showing all required work.

C. Explain how to find the equation of the tangent plane to f at your chosen point, showing all required work.

D. Explain how to find the equation of the normal line to f at your chosen point, showing all required work.

E. Demonstrate that the second derivative test for local extreme values of f is inconclusive for all points on the y-axis.

 

[Deleted member 4993]

I have this assignment for my Calculus III class and I have no idea where to even start. Any help is appreciated! Thanks!!

Let f be a multivariable function defined by f(x, y) = x³y – x²y² where x and y are real numbers. Choose the same point to use as you work to complete parts A through D of the task.

A. Explain how to find the direction of maximum increase for f at your chosen point, showing all required work.

B. Explain how to find the direction of maximum decrease for f at your chosen point, showing all required work.

C. Explain how to find the equation of the tangent plane to f at your chosen point, showing all required work.

D. Explain how to find the equation of the normal line to f at your chosen point, showing all required work.

E. Demonstrate that the second derivative test for local extreme values of f is inconclusive for all points on the y-axis.

start with finding $\displaystyle \dfrac{\partial f}{\partial x}$ and $\displaystyle \dfrac{\partial f}{\partial y}$ and from there to gradient of the function.....

 

[HallsofIvy]

I am puzzled by this. You have posted a pretty standard "Calculus of several variables" problem but seem to be saying you have no idea how to do a problem like this. Are you not taking a "Calculus of several variables" course? Do you not have a textbook?

I do not want to sound judgmental but you do understand that you are asking about basic definitions, which certainly should be in your textbook, don't you? For example, on basic property of the "gradient vector", sometimes given as the definition, is that it is a vector, pointing in the direction of greatest increase of a function, whose length is equal to the rate of increase in that direction. Did you not know that?

 

[mahowm10]

I am puzzled by this. You have posted a pretty standard "Calculus of several variables" problem but seem to be saying you have no idea how to do a problem like this. Are you not taking a "Calculus of several variables" course? Do you not have a textbook?

I do not want to sound judgmental but you do understand that you are asking about basic definitions, which certainly should be in your textbook, don't you? For example, on basic property of the "gradient vector", sometimes given as the definition, is that it is a vector, pointing in the direction of greatest increase of a function, whose length is equal to the rate of increase in that direction. Did you not know that?

Yes, of course I have a textbook, and obviously I tried that first; however, that made absolutely no sense to me. Not everyone learns the same way and can understand things straight out of a textbook. Sometimes other people have a way of explaining that makes things easier to understand, which was all I was looking for here. Maybe instead of trying "not" to be judgmental, you could have tried to legitimately help.