Math 2060 - Section 14.1

1514 days ago by jimlb

Section 14.1

Worksheet by Jim Brown

For this worksheet we graph some functions $z = f(x,y)$ and some level curves. 

var('x,y,z') 
       
(x, y, z)
(x, y, z)

Example 1:  The first example we graph is the function $z = 5 \cos (x - y)$. 

plot3d(5*cos(x-y), (x,-2*pi,2*pi), (y,-2*pi,2*pi)); 
       

Example 2:  The next we graph is $z = \frac{5\cos(x)\sin(y)}{2 + \tan(xy)}$.

plot3d(5*cos(x)*sin(y)/(2 + tan(x*y)), (x,-20*pi, 20*pi), (y,-20*pi, 20*pi), aspect_ratio=true); 
       

Example 3:  We now graph $z = |x|$. 

plot3d(abs(x), (x,-10,10), (y,-10,10)); 
       

Example 4:  Here we graph the level curves of the function $z = y - \ln(x)$.  First we graph the function in blue with the curves with constant $z$ value.  The second graph is the level curves in the $xy$-plane.

P=plot3d(y - ln(x), (x,0.1,5),(y,-10,10)); for k in (-5..5): P+=parametric_plot3d([x,ln(x)+k,k],(x,0.1,5),color='red', thickness=3); P.show(); 
       
P=plot(ln(x), (x,0.1,5), color='red'); for k in (1..15): P+=plot(ln(x) +k, (x,0.1,5), color='red'); P.show() 
       

Example 5:  Here we graph the function and then the level curves of $z = \frac{1}{2 + x^2 + y^2}$. Note how the level curves get closer together as the graph gets steeper.

P=plot3d(1/(2 + x^2 + y^2), (x,-10,10), (y,-10,10)); P.show() 
       
P=implicit_plot(x^2 + y^2 -1, (x,-3,3),(y,-3,3)); for k in (1..5): P+=implicit_plot(x^2 + y^2 - k, (x,-3,3),(y,-3,3)); for k in (1..25): P+=implicit_plot(x^2 + y^2 - 1/k,(x,-3,3),(y,-3,3)); P.show()