7. We have seen two approaches to approximate a function. These are a Taylor polynomial and Lagrange interpolating polynomial. No calcula- tions are required for this problem. (a) We used the Lagrange interpolating polynomial to derive the 3-point and 5-point formulas for approximating the derivative. Could we have derived similar formulas making use of a Taylor polynomial? Why or why not? (b) If I want a good approximation of a function f near a particular value of x, should I approximate f with a Taylor polynomial or a Lagrange polynomial? (c) Why might you want to use a Lagrange interpolating polynomial to approximate a function f instead of a Taylor polynomial? (d) As means to approximate functions, Taylor polynomials and La- grange polynomials both share what desirable property? 8. Describe how natural numbers, integers, real-numbers, sets and alphanu- meric characters can be represented by bits (i.e., collections of O's and I's). Be sure to note the limitations of each representation. No calculations are required for this problem. 7 3 F