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![Exercise 4A Write functions to perform the following tasks efficiently. Do not change the function names in the template file, and do not import any packages besides those already indicated in the template. (1) Take x and return 2x^3. (2) Take two inputs x and y and return their product and the absolute value of their difference, in that order. E.g. inputs 3,4 should return 12,1. (3) The geometric mean of n numbers is the nth root of their product. The mid-range is the ordinary (arithmetic) mean of the maximum and minimum elements. Write a function which takes a list of numbers as input and returns True if the geometric mean is less than or equal to the midrange, and False if the geometric mean is greater than the midrange. For example, the input [3,8,9] should return True and [3,6,7,5] should return False. (4) Take an integer n and return a list of the first n terms of the Padovan sequence pn. (This is a variant of the Fibonacci numbers, defined by p[1] = p[2] = p[3] = 1, and p[n] = p[n-2] + p[n-3] for n>=4.) (5) Take a pair r0 and theta representing a point's polar coordinates and return a pair (x,y) representing its Cartesian coordinates. (6) Take a pair (x,y) representing a point's Cartesian coordinates and return a pair (r, theta) representing its polar coordinates, where theta is in the range [0,2π). Check that your function works for (x,y) in all quadrants.](https://gotit-pro.com/wp-content/uploads/2023/10/69b0fd8b-79ad-4a6a-ac52-c635ef316302.jpg)
![(1) Write a function which takes a list of the coefficients of a polynomial P(x) = a0 + a1x + a2x^2 + ... + anx^n of arbitrary degree n, and a value of x0, and returns P(x0). You can use the function given in lectures, ensuring you understand how it works. (2) Use the function to evaluate (a) P1(x) = 4x^4 + 3x^2 + 2 at x = 2. (b) P2(x) = 241x^4 at x = 2. Are these answers exact? Explain why or why not. (Use a print statement 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 sin(x) = Σ (1)^n * (2n+1)! * x^(2n+1) = x - x^3/3! + x^5/5! - x^7/7! + ..., for all x. Use the first four terms in this series in the Horner evaluation function at a suitable value of x 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 we 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).](https://gotit-pro.com/wp-content/uploads/2023/10/627bbfd5-a1bb-4d60-954c-b70c3537706b.jpg)
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