Math 241: Matlab project 1
due Thursday, April 30 in discussion session
First download the m-files from the course web page and put them in the same directory as your
m-files. First use the command format long g so that all digits of numerical answers will be be shown.
Use the command nice3d after all 3D plotting commands.
Note that the all answers must be prepared as m-files (not as commands entered on the command
line), otherwise there will be no credit. You then run the m-file and hand in the original m-file, the output
of the m-file, and the generated graphs. All results and plots must be clearly labeled.
Remember that you can work in teams of up to 3 students. Sharing of material between different teams
is not permitted.
1. Consider the curve given by the parametrization r(t) = (cos t, sin t, t2/100) for t∈[0,6π].
(a) Use Matlab to find symbolic expressions for velocity and speed. Use the int command to find
the length of the curve as a symbolic expression, then use double to find a numerical value.
(b) Consider t0= 11π/2. First compute numerical values of a(t0),v(t0). Using this find numerical
values for the unit tangent vector T(t0), the normal vector N(t0), the binormal vector B(t0)
and the curvature κ(t0).
(c) Plot the curve using tubeplot3. In the same plot draw the vectors T,N,Bfrom (b) originating
at the point r(t0). Use circle3tube to plot the osculating circle at r(t0) in the same plot.
Hint: The radius of the osculating circle is κ−1, its center is r(t0) + κ−1N.
2. Consider the function f(x, y) = −x4−y4−x2y2+ 6x2+ 6y2/10.
(a) Use ezsurf and plane to plot the graph of the function ffor x∈[−2,2], y∈[−2,2] together
with the tangent plane at the point (x0, y0) = (1.5,1.6). Use nice3d after the plot commands.
Use ezcontourc(...,25);axis equal to plot 25 level curves of the function ffor x∈[−2,2],
y∈[−2,2].
(b) Use solve to find the critical points and classify their type, as in the example on the web page.
Does Matlab find all critical points?
3. For the following problem use the symbolic integration command int and give the results V, ¯x, ¯y, ¯z
as symbolic expressions. Then use double() to find numerical values.
(a) Consider the cylinder consisting of points (x, y, z)∈R3satisfying x2+z2≤1. Let Ddenote
the part of this cylinder with |y| ≤ x,z≥0.
Plot the top surface of the region Dusing ezsurfvs.
Find the volume Vof Dand the center of mass (¯x, ¯y, ¯z), assuming density f(x, y, z) = 1.
(b) In cylindrical coordinates (r, θ, z) a torus is described by (r−2)2+z2≤1. Let Ddenote the
part of this torus with x≥0, y≥0, z≥0.
Plot the top surface of the region using ezsurfpol.
Find the volume Vof Dand the center of mass (¯x, ¯y, ¯z), assuming density f(x, y, z) = 1.