Introduction: In the previous section we saw the graphs of functions of two variables. Since we have to deal with two input variables and one output variable, we need three axis to draw these graphs. This can be become rather difficult and cumbersome. By the use of level curves we can represent these graphs in two dimensions.
Definition: If $z(x,y)$ is a function of two variables $x$ and $y$, then the curve in an $(x,y)$-coordinate system with points such that the $x$- and $y$- coordinate satisfy the equation
$$z(x,y)=k,$$
is called the level curve with a (function) value equal to $k$.
Remark 1: Level curves of a function of two variables can be drawn in an $(x,y)$-coordinate system; the graph itself is drawn in an
$(x,y,z)$-coordinate system.
Remark 2: Level curves of the same function with different values cannot intersect.
Remark 3: Level curves of utility functions are called indifference curves. Level curves of production functions are called isoquants. Level curves of cost functions are called isocost lines.
Definition: If $z(x,y)$ is a function of two variables $x$ and $y$, then the curve in an $(x,y)$-coordinate system with points such that the $x$- and $y$- coordinate satisfy the equation
$$z(x,y)=k,$$
is called the level curve with a (function) value equal to $k$.
Remark 1: Level curves of a function of two variables can be drawn in an $(x,y)$-coordinate system; the graph itself is drawn in an
$(x,y,z)$-coordinate system.
Remark 2: Level curves of the same function with different values cannot intersect.
Remark 3: Level curves of utility functions are called indifference curves. Level curves of production functions are called isoquants. Level curves of cost functions are called isocost lines.