Exercise 4B (1) Write a function which takes a list of the coefficients of a polynomial P(x) = ª₁ + ª₁x + a² + ... + anxn 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.
Exercise 4B (1) Write a function which takes a list of the coefficients of a polynomial P(x) = ª₁ + ª₁x + a² + ... + anxn 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.
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![Exercise 4B
(1) Write a function which takes a list of the coefficients of a polynomial
P(x) = ª₁ + ª₁x + a² + …..
+anxn
of arbitrary degree n, and a value of x, and returns P(x). You can use the function given in lectures,
ensuring you understand how it works.
(2) Use the function to evaluate
(a) P₁(x) = 4x4 + 3x² + 2 at x = 2.
(b) P₂(x) = 2 — ¹x¹ at x = √√/2.
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 power series for the sine function sin(x) is given by
∞
x2n+1
(2n + 1)!
sin(x) = Σ(−1)”.
n=0
= x
6
+
x5
120
for all x. Use the first four terms in this series in the Horner evaluation function at a suitable value of a to
give an approximation of sin(π/4).
(4) (a) Use your Horner's method function to evaluate the polynomial (x − 1)³ at the point
x = 1.000001.
(b) Is this answer correct?
(c) Briefly explain why, or why not.
(5) In week 3 we wrote a function to convert from binary to decimal. This can be adapted to other
bases, here will explore octal (base 8). The efficiency of the function we wrote can be improved using the
same principle as Horner's method. Write such a function (horner_octal_to_dec) using the ideas of
Horner's method which takes a list whose entries are integers between 0 and 7, where the list represents
a base-8 number, and returns the corresponding decimal integer (so the input [2,5,1] returns the
integer 169).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fed029406-a1c1-473f-a3a0-6fd0fbd8e89d%2Ffb803bb2-8cd8-43d5-864c-57f826136fa7%2Fvhkxucu_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Exercise 4B
(1) Write a function which takes a list of the coefficients of a polynomial
P(x) = ª₁ + ª₁x + a² + …..
+anxn
of arbitrary degree n, and a value of x, and returns P(x). You can use the function given in lectures,
ensuring you understand how it works.
(2) Use the function to evaluate
(a) P₁(x) = 4x4 + 3x² + 2 at x = 2.
(b) P₂(x) = 2 — ¹x¹ at x = √√/2.
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 power series for the sine function sin(x) is given by
∞
x2n+1
(2n + 1)!
sin(x) = Σ(−1)”.
n=0
= x
6
+
x5
120
for all x. Use the first four terms in this series in the Horner evaluation function at a suitable value of a to
give an approximation of sin(π/4).
(4) (a) Use your Horner's method function to evaluate the polynomial (x − 1)³ at the point
x = 1.000001.
(b) Is this answer correct?
(c) Briefly explain why, or why not.
(5) In week 3 we wrote a function to convert from binary to decimal. This can be adapted to other
bases, here will explore octal (base 8). The efficiency of the function we wrote can be improved using the
same principle as Horner's method. Write such a function (horner_octal_to_dec) using the ideas of
Horner's method which takes a list whose entries are integers between 0 and 7, where the list represents
a base-8 number, and returns the corresponding decimal integer (so the input [2,5,1] returns the
integer 169).
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