My question is how am I supposed to determine what function that is? I think I could handle the partial derivatives after that..
I couldn't find an example showing how to identify this graph anywhere. maybe my graph identification skills aren't that great. I don't know what else to try..
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$\begingroup$This is not the intention of the exercise. They do not want you to indentify the graph. To explain how they want you to do it, I will do 5(a).
It asks to find $f_x(1,2)$, i.e. how $f$ changes as you move in the $x$ direction from the point $(1,2)$. If you go in the positive $x$ direction from the pink spot which represents $(1,2)$, then you can see that the $z$ co-ordinate (i.e. $f$ value) increases. So the sign of $f_x$ is positive at this point.
Similarly you can do 5(b), 6(a) and 6(b).
When it asks to find $f_{xx}(-1,2)$, this means "the rate at which $f_x$ is changing as you advance in the $x$ direction at this point". At that point, $f_x$ is negative, since the graph is going downwards as you advance in the positive $x$ direction. But $f_x$ is increasing, since the "negative slope" is becoming less steep, like going from gradient of $-2$ to $-1$. So $f_{xx}$ is positive.
Similarly you can do 7(b).
When the two letters are different, you need to find "the rate at which $f_x$ changes as you move in the positive $y$ direction". This, combined with the method for question 7, is how to do 8(a) and 8(b).
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