Exercises

1. Evaluate $\pi^{\sqrt{2}}_{}$ to 30 decimal places.
2. Input the following as a Maple function:
   f (x) = $${\frac{\sqrt{x}(-116x^3+8x^4+558x^2-891x)}{2x^5-29x^4+140x^3-225x^2}}$$ .

   a)

   Write this in a simpler form (that is, factor and reduce it).

   b)

   Draw the graph of the function for x between 0 and 1). Adjust the vertical range so that some detail can be seen.

   c)

   Compute the area of the part of the curve that lies above the x-axis and between x = 0 and x = 5 (that is, integrate the function over the appropriate range of x values). Give your answer both in actual form and as a decimal approximation to about 10 places.

   d)

   What is the value of the integral if you remove the factor of $\sqrt{x}$ from the numerator?

   e)

   Use the derivative of f (x) to determine for what real value x the function has a local maximum. Use an approximation to about 8 decimal places.

3. Use the commands seq and ithprime to generate a list of the first 20 primes. Compute the sum of these 20 primes, and give its integer factors.
4. Find the solutions of the system of equations {x² - y² = 4, x - 2y = 2}.
5. Draw a graph showing both cosx and its fifth Taylor polynomial (that is, 1 - ${\frac{1}{2!}}$x² + ${\frac{1}{4!}}$x⁴) for x between -4 and 4. How many terms do you seem to need 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.