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1.  Find \(\frac{\partial^2 f}{\partial x^2}\) where \( f(x) = \arctan(\frac{y}{x})\).

a.  \(\sec^2(\frac{y}{x})\)

b.  \(-\sec^2(\frac{y}{x})\frac{y}{x^2}\)

c.  \((y\sin(\frac{y}{x})/2\ (x^2 cos^3(\frac{y}{x}))\)

d.  \(\frac{2xy}{(x^2+y^2)^2}\)

e.  \(-\frac{2xy}{(x^2+y^2)^2}\)

2.  If \(f_{xy}(1,2) \neq f_{yx}(1,2)\), what can you conclude?

a.  Neither of the mixed partials exist at the point \((1,2)\).

b.  At least one of the mixed partials doesn't exist at the point \((1,2)\)

c.  There is some open disk containing the point \((1,2)\) where at least one of the mixed partials is not continuous.

d.  There is some open disk containing the point \((1,2)\) where neither mixed partial is continuous.

e.  There is some open disk containing the point \((1,2)\) where neither mixed partial exists.

3.  Consider the surface \(z = f(x,y) = \sqrt[3]{x}\). This surface is a cylinder, using the definition of cylinder we talked about in chapter 12. I am telling you that this surface is smooth everywhere. However, there is/are places where the function is not differentiable. Where?

a.  Just at \((0,0)\)

b.  On the y-axis.

c.  On the x-axis

d.  Everywhere

e.  \(f\) is not a function.