 6.3.1: Animal survival and renewal An animal population currently has 7400...
 6.3.2: City population A city currently has 36,000 residents and is adding...
 6.3.3: Insect survival and renewal A population of insects currently numbe...
 6.3.4: Animal survival and renewal There are currently 3800 birds of a par...
 6.3.5: Drug concentration A drug is administered intravenously to a patien...
 6.3.6: Drug concentration A patient receives a drug at a constant rate of ...
 6.3.7: Water pollution A contaminant is leaking into a lake at a rate of R...
 6.3.8: Insect survival and renewal Sterile fruit flies are used in an expe...
 6.3.9: Blood flow Use Poiseuilles Law to calculate the rate of flow in a s...
 6.3.10: Blood flow High blood pressure results from constriction of the art...
 6.3.11: Cardiac output The dye dilution method is used to measure cardiac o...
 6.3.12: Cardiac output After an 8mg injection of dye, the readings of dye ...
 6.3.13: Cardiac output The graph of the concentration function cstd is show...
 6.3.14: Drug administration A patient is continually receiving a drug. If t...
Solutions for Chapter 6.3: Further Applications to Biology
Full solutions for Biocalculus: Calculus for Life Sciences  1st Edition
ISBN: 9781133109631
Solutions for Chapter 6.3: Further Applications to Biology
Get Full SolutionsChapter 6.3: Further Applications to Biology includes 14 full stepbystep solutions. This textbook survival guide was created for the textbook: Biocalculus: Calculus for Life Sciences , edition: 1. Since 14 problems in chapter 6.3: Further Applications to Biology have been answered, more than 26077 students have viewed full stepbystep solutions from this chapter. 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.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

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

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

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

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

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