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 

 A real number x is represented approximately by 0.6032,
 and we are told that the (absolute) relative error is 0.1% .  What is x ?
 Note: There are two answers.

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

 Consider the numbers a=123.45 and b=0.06543 on a 4-digit decimal machine that rounds. Do BY HAND:
 a. Write down their normalized floating point representations fl(a), fl(b) on this machine.
 b. Find the exact d = a − b (at full precision) and D = fl(a) − fl(b). Do fl(d) and fl(D) match?
 c. What value will the machine produce for the difference d ?
 d. Find the error |d − fl(D)|. To how many digits is fl(D) accurate?

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

 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

 The value of π (pi) can be generated to near machine precision by  pi = 4.0 atan(1.e0).
 Suggest at least one other way to compute pi using basic functions on your computer system.

 Determine the first two nonzero terms of the series expansion about zero of the function  ecos x

 How many terms are needed in the series expansion of  ex
 to compute ex for |x|<1/2 accurate to 12 decimals ?

 Find an upper bound for the quantity  |cos(x) − 1 + x2/2| when  |x|<1/2. 
 This is the remainder of the 2-term Taylor expansion of cos(x).