Problem Set 1 - For practice

```1. Why do the following functions NOT possess Taylor series expansions at x=0 ?
a. f(x) = SQRT(x)
b. f(x) = |x|
f. f(x) = xπ	(x raised to pi)

2. For each expression below, describe the operations ( + - * / )
that could cause negligible addition, error magnification,
or subtractive cancelation, and for what sort of x values
(e.g. x near 2 , x >> 1 , etc) such errors might occur:
a. SQRT( x2 + 1 ) - x
b. 1 + cos x
c. x / ( x + 1 ) - 1

3. A real number x is represented approximately by 0.6032,
and we are told that the relative error is 0.1%. What is x ?

4. What is the relative error involved in rounding 4.9997 to 5.0 ?

5. Count the number of operations involved in evaluating a 5th degree polynomial
as commonly written and by using nested multiplication.

6. How can these polynomials be evaluated efficiently ?
a. p(x) = x12
b. p(x) = 6(x+2)3 + 9(x+2)7 + 3(x+2)15 - (x+2)31

7. The value of π (pi) can be generated to near machine precision by
pi = 4.0 atan(1.d0)
Suggest at least two other ways to compute pi using basic functions