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(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.

(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.

Database System Concepts
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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Python coding. Do number (2)

 

(1) Write a function which takes a list of the coefficients of a polynomial P(x) = @o+ @l® +a+…+an n of arbitrary degree n, and a value of xo, 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) = 5x² 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 ln(1 + x) = Σ(−1)¹+¹ 5 n n=1 x 2 OR + 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)¹ 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).
Transcribed Image Text:(1) Write a function which takes a list of the coefficients of a polynomial P(x) = @o+ @l® +a+…+an n of arbitrary degree n, and a value of xo, 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) = 5x² 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 ln(1 + x) = Σ(−1)¹+¹ 5 n n=1 x 2 OR + 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)¹ 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).
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