- 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.
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.)
Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
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.
Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).
Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.
Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.
The diagonal entry (first nonzero) at the time when a row is used in elimination.
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.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.
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