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.
*

- (
*expires 9/23*) Use`Maple`to write 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 as 1.73205, etc. - (
*expires 9/23*) Draw a graph showing both and its fifth Taylor polynomial (that is, ) for between and . 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? - (
*expires 9/30*) Consider the planar curve defined by . Using**only**`Maple`, find the slope of the tangent line to the curve at . 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. - (
*expires 9/30*) Plot the function , for . Find all the zeros of the function with an accuracy of 20 decimal digits.*Hint: See*`Digits`,`fsolve`. - (
*expires 9/30*) Define a`Maple`function that, given a positive integer yields the sum of the first primes. What is such that but ? You might find`sum`and`ithprime`helpful. - (
*expires 9/30*) Use the Taylor expansion of near the point to compute the value of to 30 places. How many terms are needed to compute the value to 50 places?

MAT 331 2002-09-03