Chapter 3.1 Derivatives of Polynomials and Exponential Functions. I don’t understand how to get the answers provided. Please explain this step by step. Differentiate each function. Y = X2 + 4X +3 V(t)=t2 – 1 Z= A + BeY √ (X) 4√(t3) Y10 Answer Y’=3
A geologist you spoke with is concerned about the rate of land erosion around the base of a dam. Another geologist is studying the magma activity within the earth in an area of New Zealand known for its volcanic activity. One of the shortcuts they apply when doing calculations in the field is to use synthetic division. After t
(See attached file for full problem description) The problem is solved by integration and evaluation between the limits and algebraic manipulation
There are problems to test the commutativity, distributive property, definition of natural numbers. Please find the attachment for the problems.
Discuss an element of drinking water where you see a public health gap or need that should be addressed. You may bring up a specific case relevant to your community if you have one, or bring up a global issue.
Show that there are infinitely many algebraic numbers such that . If and are distinct algebraic numbers with = , then what are the possible values of ? Illustrate each value in your list by giving a specific choice of and . Please see the attached file for the fully formatted problems.
Advanced Math : Let p be a poistive prime integer. Show that there is no rational number r such that p = r^2.
1. Let p be a poistive prime integer. Show that there is no rational number r such that p = r^2. Please see the attached file for the fully formatted problems. ---
Need help in proving the following. May need above assertion to prove. (See attached file for full problem description)
The number n = 15744539 is a product of two prime numbers. Find these two prime numbers if it is also given that φ(n) = 15736560. You may only use elementary functions on your calculator (adding, subtracting, dividing, multiplying, taking the square root). Show your work.
Consider the function f(x)=x/(ex-1)+x/2. (a) f has a so-called "removeable singularity" at x=0, where it is (so far) undefined. What value should we assign to f(0) to make f continuous at x=0? (b) With this taken care of, f actually has a Taylor series about x=0. Find the first 10 terms or so of this Taylor series (use C
Complete Residue Systems : Suppose that p is a prime number ≥ 3. We can write 1 + 1/2 + 1/3 + 1/(p-1) = a/b with (a,b) =1. Prove that p divides a.
Suppose that p is a prime number ≥ 3. We can write 1 + 1/2 + 1/3 + 1/(p-1) = a/b with (a,b) =1. Prove that p divides a. Please see the attached file for the fully formatted problem.
Complete Residue System : Suppose that p is a prime number. Show that 0, (p-1)!/2, (p-1)!/3, ..., (p-1)!/(p-1) is a complete residue system modulo p.
Suppose that p is a prime number. Show that 0, (p-1)!/2, (p-1)!/3, ..., (p-1)!/(p-1) is a complete residue system modulo p.
One to One Map : Suppose that p is a prime number ≥ 3 and r1, r2...rp is a complete residue system of modulo p. Prove that r1+r2+...+rp is divisible by p.
Suppose that p is a prime number ≥ 3 and r1, r2...rp is a complete residue system of modulo p. Prove that r1+r2+...+rp is divisible by p.
Topics usually include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity and quadratic fields. (See attached file for full problem description)
Problem 6. Suppose that f(s) = adxd+ad-1xd... is a polynomial with integral coefficients (so a0, a1 . . ,ad E Z). Show that f(n)?f(m) is divisible by n ? m for all distinct integers n and m.
Find Prime factorization (by hand) of 2006 and 2007.
2. (Taylor polynomials) (a) Write down the Taylor polynomials Pn(x) of degree n = 0, 1, 2,3 for the function f(x) = ln x about the point x = 1. (h) Plot the polynomials Pn(x) and the function f(x) on the interval [0, 3] using Matlab. Describe how the error |f(x) ? Pn(x)| behaves with respect to the point x and the degree n.
List 4 problems dealt with by cryptography & give real world examples of each. 2 paragraphs please.
(See attached file for full problem description) --- Suppose that a(x) and m(x) are relatively prime polynomials in F[x]. Bézout's Theorem guarantees the existence of polynomials u(x) and v(x) in F[x] such that... --- (See attached file for full problem description)
6301 is prime. If x, y, and z are integers that are not divisible by 6301, which of the following is equal to x^6299.y^12600.z^18903 mod 6301 ? (a) xyz (b) yz2/x2 (c) z3/x (d) 1/(x2 y2) (e) none of the above
Which of the following is true. a) 16 is a non-trivial square root of 1 modulo 51; hence 51 is composite b) 7 is a non-trivial square root of 1 modulo 47; hence 47 is composite c) 8 is a non-trivial square root of 1 modulo 55; hence 55 is composite d) all of the above e) None of the above
Q.1 For a number x, with 1< x < p, the number x^n mod p can be computed with at most 2log2 n modulo p multiplications. Asymptotic notation questions Q.2 2^(2n) = O(2^n) Q.3 log*n = O(log*(log n)) Q.4 The sqrt n th Fibonacci number can be computed and written in O(log n) time Please see the attac
1. Find the prime factorization of 54. ____a. 188.8.131.52 _x__b. 184.108.40.206 ____c. 2.2.17 ____d. 220.127.116.11 2. Determine which of the following is divisible by 3. ____a. 3106 ____b. 2251 _x__c. 1239 ____d. 1172 3. Determine which of the following is divisible by 8. __x_a. 1336 ____b. 1473 ____c. 1534 ____d.
Does a prime number multiplied by a prime number ever result in a prime - Why? Does a nonprime multiplied by a nonprime ever result in a prime - why? Is it possible for an extremely large prime to be expressed as a large integer raised to a very large power? Explain. Are there infinitely many natural numbers that are not pri
1) Let e = ... be an RSA enciphering exponent. Prove that, for any... Please see attached.
The Mobius Inversion Formula 1) Prove that ∑ μ(d) φ(d) = Π (2 - p) d/n p/n Primitive roots modulo p 2) Find all primitive roots modulo 5, modulo 9, modulo 11, modulo 13 and modulo 15. Prime Numbers 3) The Fermat numbers are numbers of the form 2^(2^n) + 1 = φn . Prove that if n < m, then φn| φm - 2 4) Prove that if n ≠ m, then gcd (φn, φm) = 1 5) Use the above exercise to give a proof that there exist infinitely many primes.
Theory of Numbers Mobius Theorem
5. A polynomial is said to be monic is its leading coefficient is perpendicular... Please see attached.
Problem 1. Let n be a natural number and a1.... ,an > 0 be pairwise different positive real numbers. Show that if λ1...λn are such real numbers that the equality ... holds true for all x E R then .... Problem 2. Show that there are infinitely many real numbers x in the interval [0, pi/2] such that both sinx and co
Proofs 1. Let F be a field and p(x) and irreducible polynomial over F. Prove that (p(x)) is a prime ideal of F[x]. 2. If R has no divisors of zero, then neither does R[x].
I hope you will be able to help me with this problem, I'm really stuck. 10-(k+5) = 3(k+2) I would also appreciate any explanation.