- 6.8.1: Growth of an Insect Population The size P of a certain insect popul...
- 6.8.2: Growth of Bacteria The number N of bacteria present in a culture at...
- 6.8.3: Radioactive Decay Strontium 90 is a radioactive material that decay...
- 6.8.4: Radioactive Decay Iodine 131 is a radioactive material that decays ...
- 6.8.5: Growth of a Colony of Mosquitoes The population of a colony of mosq...
- 6.8.6: Bacterial Growth A culture of bacteria obeys the law of uninhibited...
- 6.8.7: Population Growth The population of a southern city follows the exp...
- 6.8.8: Population Decline The population of a midwestern city follows the ...
- 6.8.9: Radioactive Decay The half-life of radium is 1690 years. If 10 gram...
- 6.8.10: Radioactive Decay The half-life of radioactive potassium is 1.3 bil...
- 6.8.11: Estimating the Age of a Tree A piece of charcoal is found to contai...
- 6.8.12: Estimating the Age of a Fossil A fossilized leaf contains 70% of it...
- 6.8.13: Cooling Time of a Pizza Pan A pizza pan is removed at 5:00 PM from ...
- 6.8.14: Newtons Law of Cooling A thermometer reading 72F is placed in a ref...
- 6.8.15: Newtons Law of Heating A thermometer reading 8C is brought into a r...
- 6.8.16: Warming Time of a Beer Stein A beer stein has a temperature of 28F....
- 6.8.17: Decomposition of Chlorine in a Pool Under certain water conditions,...
- 6.8.18: Decomposition of Dinitrogen Pentoxide At 45C, dinitrogen pentoxide ...
- 6.8.19: Decomposition of Sucrose Reacting with water in an acidic solution ...
- 6.8.20: Decomposition of Salt in Water Salt (NaCl) decomposes in water into...
- 6.8.21: Radioactivity from Chernobyl After the release of radioactive mater...
- 6.8.22: Pig Roasts The hotel Bora-Bora is having a pig roast. At noon, the ...
- 6.8.23: Population of a Bacteria Culture The logistic growth model represen...
- 6.8.24: Population of an Endangered Species Often environmentalists capture...
- 6.8.25: The Challenger Disaster After the Challenger disaster in 1986, a st...
Solutions for Chapter 6.8: Exponential Growth and Decay Models; Newtons Law; Logistic Growth and Decay Models
Full solutions for College Algebra | 9th Edition
Solutions for Chapter 6.8: Exponential Growth and Decay Models; Newtons Law; Logistic Growth and Decay ModelsGet Full Solutions
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.
A = CTC = (L.J]))(L.J]))T for positive definite A.
Remove row i and column j; multiply the determinant by (-I)i + j •
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
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!).
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.
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.
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.
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.
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.