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Solutions for Chapter 6.9: Building Exponential, Logarithmic, and Logistic Models from Data

Full solutions for College Algebra | 9th Edition

ISBN: 9780321716811

Solutions for Chapter 6.9: Building Exponential, Logarithmic, and Logistic Models from Data

College Algebra was written by and is associated to the ISBN: 9780321716811. This textbook survival guide was created for the textbook: College Algebra, edition: 9. Chapter 6.9: Building Exponential, Logarithmic, and Logistic Models from Data includes 13 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 13 problems in chapter 6.9: Building Exponential, Logarithmic, and Logistic Models from Data have been answered, more than 18017 students have viewed full step-by-step solutions from this chapter.

Key Math Terms and definitions covered in this textbook
  • 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.

  • Fourier matrix F.

    Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

  • 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.

  • Identity matrix I (or In).

    Diagonal entries = 1, off-diagonal entries = 0.

  • Krylov subspace Kj(A, b).

    The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

  • Least squares solution X.

    The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.

  • Linearly dependent VI, ... , Vn.

    A combination other than all Ci = 0 gives L Ci Vi = O.

  • Matrix multiplication AB.

    The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

  • Multiplication Ax

    = Xl (column 1) + ... + xn(column n) = combination of columns.

  • Nullspace matrix N.

    The columns of N are the n - r special solutions to As = O.

  • Outer product uv T

    = column times row = rank one matrix.

  • 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.

  • Reflection matrix (Householder) Q = I -2uuT.

    Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.

  • Row picture of Ax = b.

    Each equation gives a plane in Rn; the planes intersect at x.

  • Stiffness matrix

    If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

  • 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·

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

    T- 1 has rank 1 above and below diagonal.

  • Vandermonde matrix V.

    V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.

  • Vector addition.

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

  • Volume of box.

    The rows (or the columns) of A generate a box with volume I det(A) I.

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