**1.3****a**, where machine epsilon , is the base of the system and is lowest possible value of exponent.**b**with being maximum value of exponent,**c****1.8****a****b**Here we divided numerator and denominator by to avoid overflow.

**c****d**Here we divided numerator and denominator by to avoid overflow.

**e**Calculate

Since , and , and , and using the fact that even number raised to the power of 10 is even, and odd number raised to teh power of 10 is odd, we obtain

**1.9**Homer states**a**If Horner right,**b**If Horner is right,**c**If Horner is right,**d**If Horner is right,**e**The product of two digit numbers may have up to digits. Therefore may have up to 48 digits, which is longer then Matlab's 16 digits. Therefore we can not use doubles in Matlab to either prove or disprove this conjecture.Note that can be explicitly calculated on a computer with inifinite precision (check out vpa command in matlab), because it is an integer. Using the “sym” command, or Mathematica allows us to get the answer:

**1.11**Here we deal with- 1.13
Here we study for small the function
**1.17****a**Use Taylorâ€™s theorem to show that

Denoting and using**b**Why**c**Since and , we get . More precisely, . Use this together with (32) and (33) to get

(33) **d**This question implies that

If this property is not satisfied, then the result can not be obtained.To obtain this result, write

Author's note: in (d) the right hand side shoule be and in (e) the right hand side should be .

**e**The RHS can be rewritten as**f**Using we get . Picture shows that accuracy is obtained at , i.e. . In other words the estimate is optimistic.