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));
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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);
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Example 3: We now graph $z = |x|$.
plot3d(abs(x), (x,-10,10), (y,-10,10));
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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();
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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()
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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()
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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()
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