Find the level curves f(x, y) = c for the given functions and values of c. Describe their shapes.

$$y = -x + 1 + c$$

These are equations of parallel lines with a slope of $-1$.

• For $c = -3$: $y = -x + 1 - 3 \Rightarrow \boxed{y = -x - 2}$
• For $c = -2$: $y = -x + 1 - 2 \Rightarrow \boxed{y = -x - 1}$
• For $c = -1$: $y = -x + 1 - 1 \Rightarrow \boxed{y = -x}$
• For $c = 0$: $y = -x + 1 + 0 \Rightarrow \boxed{y = -x + 1}$
• For $c = 1$: $y = -x + 1 + 1 \Rightarrow \boxed{y = -x + 2}$
• For $c = 2$: $y = -x + 1 + 2 \Rightarrow \boxed{y = -x + 3}$
• For $c = 3$: $y = -x + 1 + 3 \Rightarrow \boxed{y = -x + 4}$

14. $f(x, y) = x^2 + 4y^2$, $c = 0, 1, 4, 9, 16, 25$

$$x^2 + 4y^2 = c$$

These are equations of ellipses centered at the origin, or a point if $c=0$.

• For $c = 0$: $x^2 + 4y^2 = 0 \Rightarrow \boxed{x=0, y=0}$ (a single point at the origin)
• For $c = 1$: $x^2 + 4y^2 = 1 \Rightarrow \boxed{\frac{x^2}{1} + \frac{y^2}{1/4} = 1}$ (an ellipse with semi-axes $a=1, b=1/2$)
• For $c = 4$: $x^2 + 4y^2 = 4 \Rightarrow \boxed{\frac{x^2}{4} + \frac{y^2}{1} = 1}$ (an ellipse with semi-axes $a=2, b=1$)
• For $c = 9$: $x^2 + 4y^2 = 9 \Rightarrow \boxed{\frac{x^2}{9} + \frac{y^2}{9/4} = 1}$ (an ellipse with semi-axes $a=3, b=3/2$)
• For $c = 16$: $x^2 + 4y^2 = 16 \Rightarrow \boxed{\frac{x^2}{16} + \frac{y^2}{4} = 1}$ (an ellipse with semi-axes $a=4, b=2$)
• For $c = 25$: $x^2 + 4y^2 = 25 \Rightarrow \boxed{\frac{x^2}{25} + \frac{y^2}{25/4} = 1}$ (an ellipse with semi-axes $a=5, b=5/2$)

15. $f(x, y) = xy$, $c = -3, -2, -1, 0, 1, 2, 3$

$$xy = c$$

These are equations of hyperbolas, except for $c=0$.

• For $c = -3$: $\boxed{xy = -3}$ (a hyperbola in the second and fourth quadrants)
• For $c = -2$: $\boxed{xy = -2}$ (a hyperbola in the second and fourth quadrants)
• For $c = -1$: $\boxed{xy = -1}$ (a hyperbola in the second and fourth quadrants)
• For $c = 0$: $\boxed{xy = 0}$ (the x-axis and y-axis)
• For $c = 1$: $\boxed{xy = 1}$ (a hyperbola in the first and third quadrants)
• For $c = 2$: $\boxed{xy = 2}$ (a hyperbola in the first and third quadrants)
• For $c = 3$: $\boxed{xy = 3}$ (a hyperbola in the first and third quadrants)

16. $f(x, y) = \sqrt{25 - x^2 - y^2}$, $c = 0, 1, 2, 3, 4$

Note that $c$ must be non-negative.

$$25 - x^2 - y^2 = c^2$$

$$x^2 + y^2 = 25 - c^2$$

These are equations of concentric circles centered at the origin.

• For $c = 0$: $x^2 + y^2 = 25 - 0^2 \Rightarrow \boxed{x^2 + y^2 = 25}$ (a circle with radius 5)
• For $c = 1$: $x^2 + y^2 = 25 - 1^2 \Rightarrow \boxed{x^2 + y^2 = 24}$ (a circle with radius $\sqrt{24} = 2\sqrt{6}$)
• For $c = 2$: $x^2 + y^2 = 25 - 2^2 \Rightarrow \boxed{x^2 + y^2 = 21}$ (a circle with radius $\sqrt{21}$)
• For $c = 3$: $x^2 + y^2 = 25 - 3^2 \Rightarrow \boxed{x^2 + y^2 = 16}$ (a circle with radius 4)
• For $c = 4$: $x^2 + y^2 = 25 - 4^2 \Rightarrow \boxed{x^2 + y^2 = 9}$ (a circle with radius 3)