Homework Problems Mat 331

Set no. 1 September 10, 2003

Due September 19, 2003

1. 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. 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. 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. Plot the function $f(x) = 2\sin x - x^3 - \frac{1}{5}$, for $x\in [-4,4]$. Find all the zeros of the function with an accuracy of 20 decimal digits.

   Hint: See Digits, fsolve.

   
   

5. 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 $g(k+1) > 100,000$?.

   Hint: You might find sum and ithprime helpful.

   
   

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