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For questions 1 and 2, consider the function \( f(x, y) = \frac{x}{y} + e^{xy^2}\)

1.  \(f_x(x,y) = \)

a.  \(\frac{1}{y} + y^2 e^{xy^2}\)

b.  \(\frac{x}{y^2} + 2xy e^{xy^2}\)

c.  \(\frac{1}{y} + y^2 e^{y^2}\)

d.  \(\frac{x}{y^2} + 2xy e^{2xy}\)

e.  \(\frac{1}{y} + 2y e^{2xy}\)

2.  \(f_y(x,y) = \)

a.  \(\frac{1}{y} + y^2 e^{xy^2}\)

b.  \(\frac{x}{y^2} + 2xy e^{xy^2}\)

c.  \(\frac{1}{y} + y^2 e^{y^2}\)

d.  \(\frac{x}{y^2} + 2xy e^{2xy}\)

e.  \(\frac{1}{y} + 2y e^{2xy}\)

For questions 3 through 7, consider the limit $$ \lim_{(x,y)\rightarrow(0,0)}\frac{x^2y}{x^4+y^2}$$

3.  Evaluate the limit along the x-axis

a.  \(0\)

b.  \(\frac{1}{4}\)

c.  \(\frac{1}{2}\)

d.  \(1\)

e.  DNE

4.  Evaluate the limit along the y-axis

a.  \(0\)

b.  \(\frac{1}{4}\)

c.  \(\frac{1}{2}\)

d.  \(1\)

e.  DNE

5.  Evaluate the limit along the line \(y = x\)

a.  \(0\)

b.  \(\frac{1}{4}\)

c.  \(\frac{1}{2}\)

d.  \(1\)

e.  DNE

6.  Evaluate the limit along the parabola \(y = x^2\)

a.  \(0\)

b.  \(\frac{1}{4}\)

c.  \(\frac{1}{2}\)

d.  \(1\)

e.  DNE

7.  What is the value of the original limit?

a.  \(0\)

b.  \(\frac{1}{4}\)

c.  \(\frac{1}{2}\)

d.  \(1\)

e.  DNE

8.   Describe the level set for the function \(z = f(x, y) = xy\) at height \(z = 0\).

a.  A hyperbola opening in quadrants 1 and 3.

b.  A hyperbola opening in quadrants 2 and 4.

c.  The x and the y axes.

d.  A circle of radius 1 centered at the origin.

e.  A parabola opening on the positive y-axis.

9.   Describe the level set for the function \(z = f(x, y) = xy\) at height \(z = 1\).

a.  A hyperbola opening in quadrants 1 and 3.

b.  A hyperbola opening in quadrants 2 and 4.

c.  The x and the y axes.

d.  A circle of radius 1 centered at the origin.

e.  A parabola opening on the positive y-axis.