Sketch z = arctan(x^2 + y^2)

• Feb 15th 2011, 07:00 AM
SyNtHeSiS
Sketch z = arctan(x^2 + y^2)
Sketch the level curves for $z = arctan(x^2 + y^2)$. What does the graph of the function look like?

Attempt:

Let x = k:

1. $z = arctan(k^2 + y^2)$

Let y = k:

2. $z = arctan(x^2 + k^2)$

Let z = k:

3. $k = arctan(x^2 + y^2)$

Now what I am not sure of is how would you draw 1-3? Would you have to use all the steps for curve sketching e.g. first derivative, second derivative etc?
• Feb 15th 2011, 07:38 AM
FernandoRevilla
Take $\tan$ in booth sides. For example in 3) : $x^2+y^2=\tan k$ (circumference) .

Fernando Revilla
• Feb 15th 2011, 09:07 AM
SyNtHeSiS
How would you plot $tanz = x^2 + k^2$ from 2)?
• Feb 15th 2011, 09:40 AM
TheEmptySet
The level curves occur when the function

$a=f(x,y)=k$ a constant. This gives

$k=\tan^{-1}(x^2+y^2)$ As noted above this gives

$\tan(k)=x^2+y^2$ This is the equation of a circle centered at the origin with radius $r=\sqrt{\tan(k)}$
• Feb 15th 2011, 11:57 AM
SyNtHeSiS
Oh so by level curves the question actually means you must just let z = k (not x = k and y = k) and hence draw that graph?