Exercises

1.
Define a = 5. Define b = a2. Evaluate b. Now Define a = $\sqrt{{2}}$. Guess the value of b and check your answer by evaluation.

1.
Define f (x) = x2 + 3x + 2. Evaluate

$\displaystyle {\frac{{f(x+h)-f(x)}}{{h}}}$

and Simplify the result. Do computations in place to show intermediate steps in the simplification.

2.
Rewrite the function f (x) = max$\left(\vphantom{ x^{2}-1,7-x^{2}}\right.$x2 -1, 7 - x2$\left.\vphantom{ x^{2}-1,7-x^{2}}\right)$ as a piecewise-defined function.

3.
Experiment with the Euler phi function $\varphi$(n), which counts the number of positive integers kn such that gcd(k, n) = 1. Use Define + Define Maple Name to open a dialog box. Type phi(n) as the Maple name, $\varphi$(n) as the Scientific Notebook name, and check the Maple Library Numtheory box. Test the statement ``If gcd(n, m) = 1 then $\varphi$(nm) = $\varphi$(n)$\varphi$(m)'' for several specific choices of n and m.

4.
Define d (n) by typing divisors(n) as the Maple name, d (n) as the Scientific Notebook name, and check the Maple Library Numtheory box. Explain what the function d (n) produces. This is an example of a set-valued function, since the function values are sets instead of numbers.