Evaluation of Polynomial Function A polynomial of degree n is a function of the form f(x) = anx" + an-1x"-1 +...+ azx² + a1x + ao where the as are real numbers (sometimes called the coefficients of the polynomial). Although this general formula might look quite complicated, particular examples are much simpler. For example, f(x) = 4x3 - 3x² + 2 is a polynomial of degree 3, as 3 is the highest power of x in the formula. This is called a cubic polynomial, or just cubic.

Evaluation of Polynomial Function A polynomial of degree n is a function of the form f(x) = anx" + an-1x"-1 +...+ azx² + a1x + ao where the as are real numbers (sometimes called the coefficients of the polynomial). Although this general formula might look quite complicated, particular examples are much simpler. For example, f(x) = 4x3 - 3x² + 2 is a polynomial of degree 3, as 3 is the highest power of x in the formula. This is called a cubic polynomial, or just cubic.

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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Find simple, closed formula of the following sequence: F1, 2F2, 3F3, 4F4, ... (The F's are Fibonacci numbers)We have the answer (given by the textbook) : x*(d/dx){x/(1-x-x2)} = (x(1+x2))/((1-x-x2)2) And I'm just having trouble figuring out how to get to that point.
write a computer program that uses newtons method to find the root of a given function. and apply this program to find the root of the followig functions, using X0 as given . stop the iteration when the error as estimated by |Xn+1- Xn| is less than 10-6 . compare to your result for bisection.(a) f(x)=1-2xe-×/2 , X0=0; (b) f(X)=5-x-1, x0=1/4; (c) f(x)=x3-2x-5, x0=2; (d) f(x)=ex-2, x0=1; (e) f(x)=x-ex, x0=1; (f) f(x)=x6 - x -1, x0=1; (g) f(x)=x2 -sin x, x0=1/2; (h) f(x)=x3 - 2, x0=1; (i) f(x)=x+ tan x, x0=3; (j) f(x)=2 - x-1lnx , x0=1/3;
Given a real number x and a positive integer k, determinethe number of multiplications used to find x^(2^k)startingwith x and successively squaring (to find x^2, x^4, and soon). Is this a more efficient way to find x^(2^k)than by multiplying x by itself the appropriate number of times?
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What is the effect in the time required to solve a problem when you increase the size of the input from n ton + 1, assuming that the number of milliseconds the algorithm uses to solve the problem with input size n iseach of these functions? [Express your answer in the simplest form possible, either as a ratio or a difference. Youranswer may be a function of n or a constant.]a) log n b) 100n c) n^2 d) 2^n e) n!
Consider the problem of making change for n cents using the fewest number of coins. Assume that we live in a country where coins come in k dierent denominations c1, c2, . . . , ck, such that the coin values are positive integers, k ≥ 1, and c1 = 1, i.e., there are pennies, so there is a solution for every value of n. For example, in case of the US coins, k = 4, c1 = 1, c2 = 5, c3 = 10, c4 = 25, i.e., there are pennies, nickels, dimes, and quarters. To give optimal change in the US for n cents, it is sufficient to pick as many quarters as possible, then as many dimes as possible, then as many nickels as possible, and nally give the rest in pennies. Design a bottom-up (non-recursive) O(nk)-time algorithm that makes change for any set of k different coin denominations. Write down the pseudocode and analyze its running time. Argue why your choice of the array and the order in which you fill in the values is the correct one. Notice how it is a lot easier to analyze the running time of…
Consider the problem of making change for n cents using the fewest number of coins. Assume that we live in a country where coins come in k dierent denominations c1, c2, . . . , ck, such that the coin values are positive integers, k ≥ 1, and c1 = 1, i.e., there are pennies, so there is a solution for every value of n. For example, in case of the US coins, k = 4, c1 = 1, c2 = 5, c3 = 10, c4 = 25, i.e., there are pennies, nickels, dimes, and quarters. To give optimal change in the US for n cents, it is sufficient to pick as many quarters as possible, then as many dimes as possible, then as many nickels as possible, and nally give the rest in pennies. Design a bottom-up (non-recursive) O(nk)-time algorithm that makes change for any set of k different coin denominations. Write down the pseudocode and analyze its running time. Argue why your choice of the array and the order in which you ll in the values is the correct one.
Consider the problem of making change for n cents using the fewest number of coins. Assume that we live in a country where coins come in k dierent denominations c1, c2, . . . , ck, such that the coin values are positive integers, k ≥ 1, and c1 = 1, i.e., there are pennies, so there is a solution for every value of n. For example, in case of the US coins, k = 4, c1 = 1, c2 = 5, c3 = 10, c4 = 25, i.e., there are pennies, nickels, dimes, and quarters. To give optimal change in the US for n cents, it is sufficient to pick as many quarters as possible, then as many dimes as possible, then as many nickels as possible, and nally give the rest in pennies. Prove that the coin changing problem exhibits optimal substructure. Design a recursive backtracking (brute-force) algorithm that returns the minimum number of coins needed to make change for n cents for any set of k different coin denominations. Write down the pseudocode and prove that your algorithm is correct.
Note that for this question, you can in addition use ``land'' for the symbol ∧ ``lor'' for the symbol ∨ ``lnot'' for the symbol ¬. Given the following three sentences:A) Every mathematician is married to an engineer.B) A bachelor is not married to anyone.C) If George is a mathematician, then he is not a bachelor. a) Convert A,B,C into three FOL sentences, whereMn(x): x is a mathematician.Er(x): x is an engineer.Md(x,y): x is married to y.Br(x): x is a bachelor.george: George is a constant. b) Show that A does-not-entail C. (Hint: Consider defining an interpretation I such that I models A, but does-not-model C.)c) Show that {A,B} entails C. (Hint: For a given interpretation I, consider two difference cases, the case where Mn(george) is true, and the case Mn(george) is false. For both cases, argue that it is always that I models C).d) Convert A,B, lnot C into a set of clausal forms, number your clauses. (Note that C is negated here!) e) Derive the empty clause from the set of clauses…
Prove that the following algorithmfor the multiplication of natural numbers is correct, for all integer constants c ≥ 2. function multiply(y, z) comment Return yz, where y, z ∈ IN 1. x := 0; 2. while z > 0 do 3. x := x + y · (z mod c); 4. y := c · y; z := z/c ; 5. return(x)
The linear programming model is solved by the Simplex Method Min Z = 8 X1 - 12 X2 + 4 X3 Hold it: X1 + X2 + X3> = 30 2 X1 + 3X2 + X3> = 12 X1> = 0; X2> = 0; X3> = 0 Select one:1.True2.Fake