hi... I am new to this topic and frustrated. I have a curve f(x,y)= -3y/(x^{2} +y^{2} + 1) I was asked to draw a level curve of this and I'm not getting anywhere with it. If anyone has any pointers, or can help me with solving this question I would be gretfull. The only other thing this question asks is to describe it at the orgin or at (0,3) ( which is steeper). thanks for any help.
Level curves are curves where the function is constant, and equals lets say to a number c. So you get an equation [tex]\frac{-3y}{x^{2}+y^{2}+1}=c[/tex] Rearranging this gives you [tex]cx^{2}+cy^{2}-3y=-c[/tex] [tex]cx^{2}+cy^{2}-2c\frac{3}{2c}y=-c[/tex] [tex]cx^{2}+cy^{2}-2c\frac{3}{2c}y+\frac{9}{4c^{2}}=\frac{9}{4c^{2}}-c[/tex] [tex]cx^{2}+(cy-\frac{3}{2c})^{2}=\frac{9}{4c^{2}}-c[/tex] Now you just have to investigate different values of c.
That should be [itex]c(y- 3/(2c))^2[/itex]. That is, that leading "c" should be outside the parentheses.