Math 331, Fall 2002: Problems 1-6

NOTE: Each exercise is worth 10 points and can be turned in at any time before its ``expiration date''. At the end of the semester, I will expect you to have turned in at least 2/5 of the exercises assigned. If you do more, I will pick your best grades. If you do less, the missing grades will be counted as zeros. Altogether, these will count the same as one project.

1. (expires 9/23)     Use Maple to write $x^5 - 2x^4 - 10x^3 +20x^2 -16x +32$ as a product of exact linear factors. By exact, I mean you should leave any non-rational factors expressed as radicals; do not approximate terms like $\sqrt{3}$ as 1.73205, etc.

2. (expires 9/23)     Draw a graph showing both $\cos(x)$ and its fifth Taylor polynomial (that is, $1-\frac{1}{2!}x^2+\frac{1}{4!}x^4$ ) for $x$ between $-4$ and $4$. What degree of Taylor polynomial seems to be needed to get good agreement in this range'' Hint: use a variation of the command convert(taylor(cos(x),x,5),polynom) to make this work. Think of a suitable way to demonstrate that the approximation you have taken is ``good''- what is a good definition of ``good'' here?

3. (expires 9/30)    Consider the planar curve $\gamma$ defined by $x^2 y^3 + y^2 + y -2 e^x =0$. 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: you might want to use implicitplot from the library plots. You might find implicitdiff helpful, too.

4. (expires 9/30)     Plot the function $f(x) = 2\sin x - x^3 - 1/5$, for $x\in [-4,4]$. Find all the zeros of the function with an accuracy of 20 decimal digits. Hint: See Digits, fsolve.

5. (expires 9/30)     Define a Maple function $g$ that, given a positive integer $k$ yields the sum of the first $k$ primes. What is $k$ such that $g(k) \le 100,000$ but $g(k+1) > 100,000$? You might find sum and ithprime helpful.

6. (expires 9/30)     Use the Taylor expansion of $\arctan x$ near the point $x=1/\sqrt{3}$ to compute the value of $\pi$ to 30 places. How many terms are needed to compute the value to 50 places?