**NOTE:** *Each exercise is worth 10 points and can be
turned in at any time before its ``expiration date''. You can work on
any number of the homework problems (none to all).
However, at the end of the semester, I will expect you to have
turned in at least 3/5 of the exercises assigned (approximately
10-15). If you do more, we will pick your best grades. If you do
less, the missing grades will be counted as zeros. This will determine
20% of your final grade for the class.*

**#01 (exp. 01/31)**- Go to the directory mat331/Alpha in our system (mathlab.sunysb.edu)
and find the file
*assignment.txt*in it. Read the file to learn what to do next. **#02 (exp. 2/7)**- Use Tailor expansion for the function
near zero at the point
to compute
with fifty decimal digits accuracy.
**#03 (exp. 2/14)**- Render (i.e., make a picture of) the Moebius strip - a one-sided nonorientable surface obtained by cutting a closed band into a single strip, giving one of the two ends thus produced a half twist, and then re-attaching the two ends. Your picture should look somewhat like this.
**#04 (exp. 2/28)**- Write a procedure
`firsttwodigits`that returns the first two digits of its argument (which is a natural number). Use the function to compute the*frequencies*of each possible pair of digits that occur as the first two digits of a power of 3 (how many of the powers to consider is up to you). Finally, plot the frequencies. **#05 (exp. 3/5)**- Let
and
be two random variables uniformly distributed in the interval [0,1]. Using Maple, plot the density function
**the product**of these two variables, i.e. Is this product itself a uniformly distributed random variable? [*Hint: First, produce the list of values (say, a thousand of them) of the product variable - for that, you might want to employ*]`random[uniform]`(twice). Then, use the procedure`mycount`from the file*distribs.mws*to obtain the density function out of the list of values. **#06 (exp. 3/5)**- [
*This problem is designed partly to frustrate you and partly to show that Maple can give you trouble, even when you do everything correctly. Hints are scattered throughout the text.*] Define*h*(*x*) = (1+*x*)*e*^{x}. Use Maple to evalute the following quantities: and*h*''(*x*)-*h*(*x*). Then solve the differential equation

[*It might take some time to find out how to use*] If there are no mistakes, the solution should coincide with`dsolve`to solve a differential equation with**initial conditions**.*h*(*x*), but it doesn't look that way, even if you`simplify`it! At any rate, tell Maple to call*f*(*x*) this solution. [*Cool Maple users try to never re-type what the computer just churned out; in this case*] Plot`rhs`and`unapply`can help.*f*(*x*)-*h*(*x*), between--say--0 and 20. What happens? Can you explain why? **#07 (exp. 4/1)**- Data in the file mat331/Worksheets1/data.dat are coordinates of points in the plane, in the following format:
A researcher has his reason to believe that the data may be approximated well by a function of the form
*y*=*ae*^{bx}. Write Maple code that determines the best fit values for the parameters*a*,*b*. Plot the data and the function to demonstrate how well the approximation is. **#08 (exp. 4/1)**- Find a value of the parameter
*c*such that the dynamical system has an**attracting**orbit of period 6. Show the orbit, and a point whose orbit is asymptotic (is being attracted to) this periodic oribt of period 6. **#09 (exp. 4/12)**- Find all the solutions to the differential
equation

Among them, single out the one for which*x*(0)=3. **#10 (exp. 4/12)**- Have Maple solve the following system of
differential equations,

with initial conditions: . Let us denote the solutions by (since they depend on the parameter*k*). For*k*taking all integer values from -10 to 10, and , plot on the same graph the functions*y*_{k}, using a color, and the functions*z*_{k}, using a different color. [*This is a case when you don't want to retype the functions that Maple finds.*] **#11 (exp. 4/12)**- Find all the fixed points of the system

a fixed point being a solution for which*both*stay constant. For each of these points, describe the behavior of the solutions that have initial conditions nearby.*x*(*t*) and*y*(*t*) **#12 (exp. 4/12)**- Consider the differential equation
,
where the vector
**z**(*t*) =(*x*(*t*),*y*(*t*)) and the field**F**(*x*,*y*) = (-*y*,*x*-*y*). Plot a few solutions. What happens to them when ? Give a ``Maple-proof'' that this is a general fact for*every*solution. [*A ``Maple-proof'' is an argument that is rigorous once we accept Maple results as incontrovertibly true.*] **#13 (exp. 5/01)**- Suppose that the turtle is moving with constant
velocity 1 unit/sec. The turtle is told, every second, to steer right
by an amount equal to
*t*^{2}degrees, where*t*is the time (in secs). (For example, after the first step, it turns right 1 degree, then after the second, turn right by 4 degrees, and so on.) Draw the curve the tutle describes after 10 and after 100 seconds. **#14 (exp. 5/01)**- By using only
`TurtleCmd`, draw a random walk of*n*steps. (In a random walk the turtle takes a step forward, backwards, to the right, to the left, with equal probabilities, and then repeats the process.) [*Check*.]`rand` **#15 (exp. 5/01)**- Sierpinski Carpet is a fractal which is constructed analogously to the Sierpinski Sieve, but using squares instead of triangles and whose first four approximations look like this. Write a procedure that draws the
*n*-th approximation of the Sierpinski Carpet and compute its box dimension. **#16 (exp. 5/01)**- Find a bulb in the Mandelbrot set of the map
*f*_{c}(*x*)=*x*^{2}+*c*such that the orbit of zero for this map is attracted to a cycle of period three, whenever*c*is within the bulb. Convince yourself that this is so by trying a few different values of*c*from the bulb. [*Some stuff from bifurcation.mws may be recycled*.]