Q. 43. - Calculus

Question

In Exercises 43–52, sketch the level curves$c=-3,-2,-1,0,1,2,3$

 if they exist for the specified function

$f(x, y)=x+y$

Step-by-Step Solution

Answer

They all represent straight lines.

Step 1: Given information

The given function is $f(x, y)=x+y$

Step 2: The objective is to sketch the level curves for c = - 3 , - 2 , - 1 , 0 , 1 , 2 , 3  

Consider the function $f(x, y)$in two variables, this function is in ${\mathrm{ℝ}}^2$

This function's graph will be in ${\mathrm{ℝ}}^3$

Let the third variable is $z$

The equation of the graph is given as $z=f(x, y)$

The level curves of this function's graph are the graphs of the function for a constant value of $z$

Let $z=c$The level curve of the graph at ' $c$ 'is defined as the graph of the equation $f(x, y)=c$

Step 3: Use the above definition to determine the equation of the different required level curves.

$$\begin{gathered} x+y=-3 \\ x+y=-2 \\ x+y=-1 \\ x+y=0 \\ x+y=1 \\ x+y=2 \\ x+y=3 \end{gathered}$$

All of these are linear equations.

Therefore, they are all straight lines.

Plot these equations on same $xy$-plane