Exercise 4B

(1) Write a function which takes a list of the coefficients of a polynomial

P(x) = ª₁ + ª₁x + a² + ... + a₂x²

of arbitrary degree n, and a value of x, and returns P(xo). You can use the function given in lectures,

ensuring you understand how it works.

(2) Use the function to evaluate

(a) P₁(x) = x³ + x² + 5x + 1 at x = 2.

(b) P₂(x) = 5 x² at x = √3.

Are these answers exact? Explain why or why not.

(Use a print statements to show the evaluation of your function, and answer the question in a comment.)

(3) The Maclaurin series for the natural logarithm ln(1 + x) is given by

=(−1)n+1 xn

n

In(1+x) :

=

n=1

=x-

x² x³

+

2 3

for all x. Use the first five terms in this series in the Horner evaluation function at a suitable value of x to give

an approximation of In 3/2.

(4) (a) Use your Horner's method function to evaluate the polynomial (x - 2)4 at the point x = 2.0001.

(b) Is this answer correct?

(c) Give brief reasoning for this answer.

(5) In week 3 we wrote a function to convert from binary to decimal. The efficiency of this function can be

improved using the same principle as Horner's method. Write such a function (horner_ternary_to_dec)

using the ideas of Horner's method which takes a list containing Os, 1s and 2s (representing a base-3

number) and returns the corresponding decimal integer (so the input [1,2,0] returns the integer 15).