Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 3.2, Problem 4E
Program Plan Intro
To check whether the functions
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- Compute and plot the derivative of the function f(x) = x³ • Apply the above methodology to compute and plot the derivative of the function f(x) = x³, for x = [-n, n], for n = 5 and 8 In [39]: import matplotlib.pyplot as plt n = 5 r = list (range(-n, n+1)) cubes = [] ## generate the squares using a for Loop for value in r: s = value**3 cubes.append(s) slopes = [] first = b = ## generate plot points from existing lists plt.plot(r, cubes, label = 'function line') plt.plot(r, cubes, 'ro', label = 'function points') plt.plot(r, slopes, label = 'slope line') plt.plot(r, slopes, 'go', label = 'slope points') plt.plot(r, trueslope, label = 'true slope') # generate axes Lables plt.ylabel('cubes value') plt.xlabel('integer input') plt.title('cubes plot') plt.legend() # display plot plt.show()arrow_forwardcode required for python: For this question, you will be required to use the binary search to find the root of some function f(x)f(x) on the domain x∈[a,b]x∈[a,b] by continuously bisecting the domain. In our case, the root of the function can be defined as the x-values where the function will return 0, i.e. f(x)=0f(x)=0 For example, for the function: f(x)=sin2(x)x2−2f(x)=sin2(x)x2−2 on the domain [0,2][0,2], the root can be found at x≈1.43x≈1.43 Constraints Stopping criteria: ∣∣f(root)∣∣<0.0001|f(root)|<0.0001 or you reach a maximum of 1000 iterations. Round your answer to two decimal places. Function specifications Argument(s): f (function) →→ mathematical expression in the form of a lambda function. domain (tuple) →→ the domain of the function given a set of two integers. MAX (int) →→ the maximum number of iterations that will be performed by the function. Return: root (float) →→ return the root (rounded to two decimals) of the given function. my code below , however as…arrow_forwardL₁ = {abman: n ≥ 0, m>0} Use the rule of contradiction, prove that following language is regular or not.arrow_forward
- For Letter -> A | B Digit -> 0 | 1 Id -> letter (letter | digit)* Generate NFA and transfer it to DFA.arrow_forward8- Determine if each of the following recursive definition is a valid recursive definition of a function f from a set of non-negative integers. If f is well defined, find a formula for f(n) where n is non- negative and prove that your formula is valid. a. f(0) = 2,f(1) = 3, f(n) = f(n-1)-1 for n ≥ 2 b. f(0) = 1,f(1) = 2, f(n) = 2f (n-2) for n = 2arrow_forwardWhich of the following is the recursive definition for the function f(n)=2n with initial condition f(1) = 2? f(n) = 2n + f(n-1) f(n) = 2n - f(n-1) f(n) = 2n * f(n-1) f(n) = 2f(n − 1) Which of the following is the recursive definition for the function f(n)=5n+2 with initial condition f(1)=7? f(n) = 5n - f(n-1) f(n) = 5f(n-1) + 2 f(n) = 5n + f(n-1) f(n) = f(n-1) + 5arrow_forward
- What is the determinant of M?M = [1 0 2; 3 8 5; 1 1 3]arrow_forwardWrite the polynomial function, which takes a list of tuples and returns the polynomial function corre- sponding to that list. Each tuple in the input list consists of (i) the coefficient, and (ii) the exponent. # (* below is the polynomial function f(x) = 3x^3 2x + 5 *) # let f = polynomial [3, 3; -2, 1; 5, 0];; val f : int -> int = # f 2; ; : int = 25arrow_forward1.Implement a recursive function allCharsPerm( listOfChars ) the generates all unique permutations of the chars in listOfChars. You can assume listOfChars contains unique chars (no repeated characters in the list). 2.Solve the recurrence and state the time complexity using Big-O notation. This will be challenging. Hint: Try forward or backward iteration / substitution, and keep very organized with your parentheses.arrow_forward
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