3. а) (and assume there exists a polynomial p(x) of degree at most n + 1 such thatp(Tk)is unique.Let xo, ..., xn be n + 1 distinct points. Consider a function f(x)f(xk) for k = 0, ..., n, and p'(xn) = f'(xn). Show that this polynomial••. )

To prove p(x) is unique, we will assume another polynomial q(x) which also satisfies the conditions.…

Answered: Let xo, ..., xn be n + 1 distinct…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter4: Polynomial And Rational Functions

Section4.1: Polynomial Functions Of Degree Greater Than

Problem 36E

See similar textbooks

Similar questions

Determine if the statemment is true or false. If the statement is false, then correct it and make it true. If the function f increases on the interval -,x1 and decreases on the interval x1,, then fx1 is a local minimum value.
Determine if the statemment is true or false. If the statement is false, then correct it and make it true. If the function f decreases on the interval -,x1 and increases on the interval x1,, then fx1 is a local maximum value.
Please help me
3. а) Let xo, . . . , xn be n + 1 distinct points. Consider a function f(x) and assume there exists a polynomial p(x) of degree at most n + 1 such that p(xk) = f(xk) for k = 0,..., n, and p'(xn) = f'(xn). Show that this polynomial is unique. b) satisfies p(k) = 0 for k = 0,.., n, and p'(n) = 1. Use divided differences to find the polynomial p(x) of degree n+1 which || || .. .
4.) For each of the following polynomial functions, find the value of k such that a.) f(x) = x3 - kx2 + kx +2 has the factor x-r2. b - b.) f(x) = x* kx' + kx² +1 has the factor ofx+2.
2.33 Let p(r) = x4 + x³ -- 2x2 + x + 1. Prove that the polynomial equation p(x) = 0 has a root in (-1,0).
2. Sketch a graph of a quartic polynomial function y = (x) such that: € f) > 0 when - 2 1
Which of the following are true for all polynomial functions? (1) If f'(x) = g'(x), then f(x) = g(x) %3D (2) If f(x) = g(x), then f'(x) = g'(x) %3D Select one: O a. Only (2) O b. Neither (1) or (2) O c. Only (1) O d. Both (1) and (2)
+ aaz", as + 0 be a polynomial with real coefficients 3. Let d 2 2 and let P(z) = ao + aịz + such that Ja;| < 1 for j = 0,1,...,d. The purpose of this exercise is to show the existence of a polynomial f(z) = co + c1z + ...+ caz“, where c; = ±1 for j = 0,1,...,d and f well approximates ... P, i.e., max |P(z) – f(z)| = 0( Vd log d), by employing the probabilistic method based on the next steps. (a) Show that P satisfies |P(z) – P(w)| < d²]z – w[ for all z, w e [-1, 1]. Hint; you may show that |P(z) < d² for all |z| < 1. (b) Show that there exist independent random signs fo. E1....Ea such that EF;] = a; for j = 0, 1, ..., d.
Determine the poles and residue of
Let p be a polynomial function. For some k EN suppose p(0) = p' (0) = p"(0) = ...= p(*) (0) = 0. Prove that p(z) = x*+'q(x) where q is another polynomial.
a) Find a second degree polynomial approximating the function f(x)=ln.x, and passing from the points (1, f(1)),(2, F(2)) and (4, f(4)). b) Approximate f(2.5) using the polynomial in part (a) E„(x) = (x-x))(x-x,)(x-x,)...(x-xx) (N+1)! c) use the error formula to estimate the error at x-2.5 d) Obtain a bound for the uniform error E2(x)

SEE MORE QUESTIONS

Recommended textbooks for you

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

Trigonometry (MindTap Course List)

Trigonometry

ISBN:

9781337278461

Author:

Ron Larson

Publisher:

Cengage Learning

College Algebra (MindTap Course List)

Algebra

ISBN:

9781305652231

Author:

R. David Gustafson, Jeff Hughes

Publisher:

Cengage Learning

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

Trigonometry (MindTap Course List)

Trigonometry

ISBN:

9781337278461

Author:

Ron Larson

Publisher:

Cengage Learning

College Algebra (MindTap Course List)

Algebra

ISBN:

9781305652231

Author:

R. David Gustafson, Jeff Hughes

Publisher:

Cengage Learning

College Algebra

Algebra

ISBN:

9781938168383

Author:

Jay Abramson

Publisher:

OpenStax

Algebra: Structure And Method, Book 1

Algebra

ISBN:

9780395977224

Author:

Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Publisher:

McDougal Littell

College Algebra

Algebra

ISBN:

9781305115545

Author:

James Stewart, Lothar Redlin, Saleem Watson

Publisher:

Cengage Learning

SEE MORE TEXTBOOKS

Let xo, ..., xn be n + 1 distinct points. Consider a function f(x) and assume there exists a polynomial p(x) of degree at most n + 1 such that p(Tk) = f(xk) for k = 0, ...,n, and p'(xn) = f'(xn). Show that this polynomial is unique. %3D