1. The problem asks to sketch the level curves of the function $f(x,y)$ for given values of $k$. A level curve is the set of points $(x,y)$ where $f(x,y) = k$. 2. For each part, we write the equation $f(x,y) = k$ and analyze the shape: **a)** $f(x,y) = 3x - y$, $k=0.3$. The level curve is: $$3x - y = 0.3$$ This is a linear equation representing a straight line. **b)** $f(x,y) = 3 - x^2 - y^2$, $k=2$ and $k=-1$. Set equal to $k$: $$3 - x^2 - y^2 = k \\ x^2 + y^2 = 3 - k$$ For $k=2$, $$x^2 + y^2 = 1$$ which is a circle of radius 1. For $k=-1$, $$x^2 + y^2 = 4$$ which is a circle of radius 2. **c)** $f(x,y) = x^2 - y$, $k=2$ and $k=-4$. Set equal to $k$: $$x^2 - y = k \\ y = x^2 - k$$ For $k=2$, $$y = x^2 - 2$$ For $k=-4$, $$y = x^2 + 4$$ These are parabolas shifted vertically. **d)** $f(x,y) = \sqrt{4x^2 + 9y^2}$, $k=0$ and $k=5$. Set equal to $k$: $$\sqrt{4x^2 + 9y^2} = k \\ 4x^2 + 9y^2 = k^2$$ For $k=0$, $$4x^2 + 9y^2 = 0$$ which is only the point $(0,0)$. For $k=5$, $$4x^2 + 9y^2 = 25$$ which is an ellipse. 3. Summary: - a) Line: $3x - y = 0.3$ - b) Circles: $x^2 + y^2 = 1$ and $x^2 + y^2 = 4$ - c) Parabolas: $y = x^2 - 2$ and $y = x^2 + 4$ - d) Ellipse and point: $4x^2 + 9y^2 = 25$ and $(0,0)$ These equations can be graphed to visualize the level curves.