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Solutions for Chapter 6.8: Exponential Growth and Decay Models; Newtons Law; Logistic Growth and Decay Models

Full solutions for College Algebra | 9th Edition

ISBN: 9780321716811

Solutions for Chapter 6.8: Exponential Growth and Decay Models; Newtons Law; Logistic Growth and Decay Models

Solutions for Chapter 6.8
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Textbook: College Algebra
Edition: 9
Author: Michael Sullivan
ISBN: 9780321716811

This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: College Algebra, edition: 9. Since 25 problems in chapter 6.8: Exponential Growth and Decay Models; Newtons Law; Logistic Growth and Decay Models have been answered, more than 36114 students have viewed full step-by-step solutions from this chapter. College Algebra was written by and is associated to the ISBN: 9780321716811. Chapter 6.8: Exponential Growth and Decay Models; Newtons Law; Logistic Growth and Decay Models includes 25 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
  • Block matrix.

    A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

  • Cholesky factorization

    A = CTC = (L.J]))(L.J]))T for positive definite A.

  • Cofactor Cij.

    Remove row i and column j; multiply the determinant by (-I)i + j •

  • Commuting matrices AB = BA.

    If diagonalizable, they share n eigenvectors.

  • Diagonalization

    A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.

  • Exponential eAt = I + At + (At)2 12! + ...

    has derivative AeAt; eAt u(O) solves u' = Au.

  • Free variable Xi.

    Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

  • Graph G.

    Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.

  • Inverse matrix A-I.

    Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.

  • Lucas numbers

    Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

  • Markov matrix M.

    All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

  • Nullspace N (A)

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

  • Orthogonal subspaces.

    Every v in V is orthogonal to every w in W.

  • Particular solution x p.

    Any solution to Ax = b; often x p has free variables = o.

  • Pivot.

    The diagonal entry (first nonzero) at the time when a row is used in elimination.

  • Right inverse A+.

    If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.

  • Rotation matrix

    R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().

  • Similar matrices A and B.

    Every B = M-I AM has the same eigenvalues as A.

  • Skew-symmetric matrix K.

    The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

  • Tridiagonal matrix T: tij = 0 if Ii - j I > 1.

    T- 1 has rank 1 above and below diagonal.

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