MAT 331 Homework Exercises. Week 1 (Sep 21, 99).
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 problems per homework sheet (none to all, usually
5). However, at the end of the semester, we will expect you to have
turned in at least 3/5 of the exercises assigned (approximately
20-25). 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. 9/28)
- Factor the polynomial
x4 -2 x3 -19x2 +70x
-50, then find all its complex roots. [Hint: See factor
and/or solve.]
- #02 (exp. 9/28)
- Plot the function
,
for
.
Find all the zeros of the function with an accuracy
of 20 decimal digits. [Hint: See Digits, fsolve.]
- #03 (exp. 9/28)
- Consider the planar curve
defined by
x2 y3 + y2 + y -2 ex =0. Using only Maple, find the slope
of the tangent line to the curve at (0,1). Then plot the curve and
the tangent line on the same graph. [Hint: implicitplot
belongs to the library plots.]
- #04 (exp. 9/28)
- Define a Maple function g that, given
,
yields the sum of the first k primes. What is k such that
but
g(k+1) > 90,000? Write the solution as text in the Maple worksheet. [sum, ithprime]
- #05 (exp. 10/5)
- For
,
write a list of
pairs [i,d(i)], where d(i) is the ith decimal digit of
.
Use this list to plot a polyline joining the points (i,d(i))sequencially. [seq, floor and one or more of
evalf, mod, modp]
- #06 (exp. 10/5)
- [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) ex. 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 dsolve to
solve a differential equation with initial conditions.] If
there are no mistakes, the solution should coincide with 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
rhs and unapply can help.] Plot f(x)-h(x),
between--say--0 and 20. What happens? Can you explain why?
MAT 331
1999-09-22