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Solutions for Chapter 4.6: Polynomial and Rational Inequalities

Precalculus Enhanced with Graphing Utilities | 6th Edition | ISBN: 9780132854351 | Authors: Michael Sullivan

Full solutions for Precalculus Enhanced with Graphing Utilities | 6th Edition

ISBN: 9780132854351

Precalculus Enhanced with Graphing Utilities | 6th Edition | ISBN: 9780132854351 | Authors: Michael Sullivan

Solutions for Chapter 4.6: Polynomial and Rational Inequalities

Solutions for Chapter 4.6
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Textbook: Precalculus Enhanced with Graphing Utilities
Edition: 6
Author: Michael Sullivan
ISBN: 9780132854351

Chapter 4.6: Polynomial and Rational Inequalities includes 85 full step-by-step solutions. Since 85 problems in chapter 4.6: Polynomial and Rational Inequalities have been answered, more than 53684 students have viewed full step-by-step solutions from this chapter. Precalculus Enhanced with Graphing Utilities was written by and is associated to the ISBN: 9780132854351. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Precalculus Enhanced with Graphing Utilities, edition: 6.

Key Math Terms and definitions covered in this textbook
  • Affine transformation

    Tv = Av + Vo = linear transformation plus shift.

  • Associative Law (AB)C = A(BC).

    Parentheses can be removed to leave ABC.

  • Basis for V.

    Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

  • Big formula for n by n determinants.

    Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.

  • Circulant matrix C.

    Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.

  • Condition number

    cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

  • Conjugate Gradient Method.

    A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.

  • Cramer's Rule for Ax = b.

    B j has b replacing column j of A; x j = det B j I det A

  • Echelon matrix U.

    The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

  • Hankel matrix H.

    Constant along each antidiagonal; hij depends on i + j.

  • Nullspace N (A)

    = All solutions to Ax = O. Dimension n - r = (# columns) - rank.

  • Pascal matrix

    Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

  • Positive definite matrix A.

    Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

  • Schur complement S, D - C A -} B.

    Appears in block elimination on [~ g ].

  • Schwarz inequality

    Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

  • Solvable system Ax = b.

    The right side b is in the column space of A.

  • Spectrum of A = the set of eigenvalues {A I, ... , An}.

    Spectral radius = max of IAi I.

  • Symmetric factorizations A = LDLT and A = QAQT.

    Signs in A = signs in D.

  • Triangle inequality II u + v II < II u II + II v II.

    For matrix norms II A + B II < II A II + II B II·

  • Vector addition.

    v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

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