Exercises

Exercise 1

Consider the vector-valued function

r(t) = (2 Cos[t], 2 Sin[t], t)

This equation describes a helix currve in 3-space.

(a) Graph a portion of this curve using an interval for t which contains 0 and which is large enough to illustrate the behaviour of the curve.

(b) Add unit tangent and unit normal vectors to the curve in part (a) at three or more points along the curve. [Refer to the twisted cubic rotation or twisted cubic animation notebook if you need help with this.]

Exercise 2

Human DNA molecules are modeled using a double helix. Each helix of this double helix has diameter approximately 20 Angstroms and pitch approximately 34 Angstroms (an Angstrom is 10^-8 cm). There are about 2.9*10^8 turns in each helix. How long would such a helix be if it could be unraveled and pulled taught to achieve its full length?

Exercise 3

A Moebius band can be constructed by taking a long rectangular piece of paper and identifying an opposite pair of edges after a half twist is placed in the paper.

(a) Graph the surface which is constructed by taking such a rectangular piece of paper and identifying an opposite pair of edges after a full twist (i.e., two half twists) is placed in the paper.

(b) Do the same problem as in (a), except use 3 half twists.

Exercise 4

(a) Graph the equation 4x2 + 4y2 - 9z2 = 1 using cylindrical coordinatess.

(b) The catenoid is the surface obtained by rotating the catenary given by x = Cosh[z] in the xz-plane about the z-axis. Graph this catenoid.

Exercise 5

Mathematica defaults to placing axes along three sides of the bounding box which contains a graphic in 3-space. Override this default, and use the Line graphics primitive to place the x-, y- and z-axes where they belong -- through the origin in 3-space. In order to do this, choose any graphic of a surface you like, and place the axes accordingly.

Exercise 6

The following self-intersecting surface was obtained by making all the horizontal cross-sections pieces of limacons: the lower ones from limacons without loops; one of the middle ones from a cardiod; and the top ones from limacons with loops. Try to recreate such a folded surface using ParametricPlot3D.

Exercise 7

This exercise deals with the function given by

(a) Use Mathematica to show that the limit of f(x,y) as (x,y) approaches the origin along any straight line through the origin is 0.

(b) Use Mathematica to show that the limit of f(x,y) as (x,y) approaches the origin along the parabola given by x=y^2 is 1/2. Since this value disagrees with the values of limits obtained in part (a), conclude that the limit of f(x,y) fails to exist at the origin.

(c) Graph f(x,y). Adjust the PlotPoints and ViewPoint options so that an instructive view of the surface is obtained.

(d) Use the SpinShow function from the Graphics package to produce an animation of the surface rotating in space.

Exercise 8

Earlier in this lab, a cylindrical coordinate plot of the function whose rectangular equation is

was produced. Study this graph further by rotating it in space. You can use the SpinShow function from the Graphics package to do this. However, you will want to take full advantage of the symmetry of the surface to reduce the number of animation frames that need to be computed. In order to do this, look at the options to SpinShow. Reduce the range of the spin using the SpinRange option, and reduce the number of frames using the Frames option (but use at least 8 frames). If you do this right, you should be able to compute all the needed frames to obtain what appears to be a continuous rotation of the surface. Rotations of this type usually look better without axes and bounding boxes.

Exercise 9

Let f(x,y) = (x2 - 3xy - y2) exp(-x2 - .1x - y2) .

(a) Graph the function.

(b) Produce a contour plot of the function.

(c) Estimate the coordinates of and the values of the relative extrema of the function from the contour plot in (b). The following technique should be helpful: click on the contour graphic; while holding down the command key (the one with the apple on it), the approximate coordinates of the cursor appear in the lower left of the window.