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Solutions for Chapter 3: Derivatives

Full solutions for Biocalculus: Calculus for Life Sciences | 1st Edition

ISBN: 9781133109631

Solutions for Chapter 3: Derivatives

Solutions for Chapter 3
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ISBN: 9781133109631

Chapter 3: Derivatives includes 94 full step-by-step solutions. Biocalculus: Calculus for Life Sciences was written by and is associated to the ISBN: 9781133109631. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Biocalculus: Calculus for Life Sciences , edition: 1. Since 94 problems in chapter 3: Derivatives have been answered, more than 26317 students have viewed full step-by-step solutions from this chapter.

Key Math Terms and definitions covered in this textbook
• Cayley-Hamilton Theorem.

peA) = det(A - AI) has peA) = zero matrix.

• 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.

• Complete solution x = x p + Xn to Ax = b.

(Particular x p) + (x n in nullspace).

• Cross product u xv in R3:

Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

• Dimension of vector space

dim(V) = number of vectors in any basis for V.

• Elimination.

A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

• Fibonacci numbers

0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

• Identity matrix I (or In).

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

• 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.

• 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.

• Multiplication Ax

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

• Orthonormal vectors q 1 , ... , q n·

Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

• Partial pivoting.

In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

• Polar decomposition A = Q H.

Orthogonal Q times positive (semi)definite H.

• Projection matrix P onto subspace S.

Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.

• Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.

Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

• Row picture of Ax = b.

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

• Vector v in Rn.

Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

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