Differentials in Calculus

[swimmergrl85]

Here's the link, since it's hard to type the entire question on here, considering the mathematics-related symbols. I just want to make sure you all understand what I am asking.

http://i1056.photobucket.com/albums/t377/swimmergrl85/Mathematics/cal3dash5dash55.png

Anyway, what I did was derive the function f(x,y)= 3x^2 + 7y^2, which resulted in the answer E_1= 6 delta x, E_2= 14 delta y. However, the answer was declared wrong. I do not know what I did wrong. Was I supposed to derive the function? Could you please help explain this to me? Could you give me an example related to this problem? Please do what you are allowed to do to help me. I understand the need to figure things out for myself.

Thank you so much!

 

[HallsofIvy]

Well, you will need to differentiate- but that's not all the problem asks you to do. It asks you to find how a change in the value of the function differs from the derivative. What you need to look at is \(\displaystyle 3(x+ \Delta x)^2+ 7(y+ \Delta y)^2= 3x^2+ 6x\Delta x+ 3(\Delta x)^2+ 7y^2+ 14y\Delta y+ 7(\Delta x)^2\) so that \(\displaystyle \Delta z= 6x\Delta x+ 3(\Delta x)^2+ 14y\Delta y+ 7(\Delta y)^2\).

Now, \(\displaystyle 6x\Delta x\) and \(\displaystyle 14y\Delta y\), that you give correspond to the "\(\displaystyle f_x\Delta x\)" and "\(\displaystyle f_y\Delta y\)" but \(\displaystyle \epsilon_1\Delta x\) and \(\displaystyle \epsilon_2\Delta y\), that you want, are the rest of that.

 

[swimmergrl85]

Thank you.

Thank you for your help.