Question

Draw the level curves for f of values c.

(a) f(x,y)=x3-y, c=-1,0,1
(b) f(x,y)=y-2 log x, c=-3,0,3
(c) f(x,y)=y csc x, c=0,1,2
(d) f(x,y)=x/(x2+y2), c=-2,0,4

Answer 1

To draw the level curves for the given functions, we need to find the values of x and y that satisfy the equation f(x,y) = c for the given values of c.

Step-by-step explanation:

In order to draw the level curves for the given functions, we need to find the values of x and y that satisfy the equation f(x,y) = c for the given values of c.

(a) For f(x,y) = x^3 - y and c = -1, 0, 1:

When c = -1, we get x^3 - y = -1, which simplifies to y = x^3 + 1. Similarly, when c = 0, we get y = x^3, and when c = 1, we get y = x^3 - 1. These equations represent the level curves for the given values of c.

(b) For f(x,y) = y - 2log(x) and c = -3, 0, 3:

When c = -3, we get y - 2log(x) = -3, which simplifies to y = 2log(x) - 3. Similarly, when c = 0, we get y = 2log(x), and when c = 3, we get y = 2log(x) + 3. These equations represent the level curves for the given values of c.

(c) For f(x,y) = y csc(x) and c = 0, 1, 2:

When c = 0, we get y csc(x) = 0, which implies y = 0. Similarly, when c = 1, we get y csc(x) = 1, and when c = 2, we get y csc(x) = 2. These equations represent the level curves for the given values of c.

(d) For f(x,y) = x/(x^2 + y^2) and c = -2, 0, 4:

When c = -2, we get x/(x^2 + y^2) = -2, which simplifies to y^2 = -x^2/2 + 1. Similarly, when c = 0, we get y^2 = 1 - x^2/2, and when c = 4, we get y^2 = 5 - x^2/2. These equations represent the level curves for the given values of c.