 12.12.1.593: In Exercises 14, set up a table of coordinates for each function. S...
 12.12.1.594: In Exercises 14, set up a table of coordinates for each function. S...
 12.12.1.595: In Exercises 14, set up a table of coordinates for each function. S...
 12.12.1.596: In Exercises 14, set up a table of coordinates for each function. S...
 12.12.1.597: In Exercises 58, graph functions f and g in the same rectangular co...
 12.12.1.598: In Exercises 58, graph functions f and g in the same rectangular co...
 12.12.1.599: In Exercises 58, graph functions f and g in the same rectangular co...
 12.12.1.600: In Exercises 58, graph functions f and g in the same rectangular co...
 12.12.1.601: Use the compound interest formulas A P1 r n nt and A Pert to solve ...
 12.12.1.602: Use the compound interest formulas A P1 r n nt and A Pert to solve ...
 12.12.1.603: A cup of coffee is taken out of a microwave oven and placed in a ro...
 12.12.1.604: In Exercises 1214, write each equation in its equivalent exponentia...
 12.12.1.605: In Exercises 1214, write each equation in its equivalent exponentia...
 12.12.1.606: In Exercises 1214, write each equation in its equivalent exponentia...
 12.12.1.607: In Exercises 1517, write each equation in its equivalent logarithmi...
 12.12.1.608: In Exercises 1517, write each equation in its equivalent logarithmi...
 12.12.1.609: In Exercises 1517, write each equation in its equivalent logarithmi...
 12.12.1.610: In Exercises 1828, evaluate each expression without using a calcula...
 12.12.1.611: In Exercises 1828, evaluate each expression without using a calcula...
 12.12.1.612: In Exercises 1828, evaluate each expression without using a calcula...
 12.12.1.613: In Exercises 1828, evaluate each expression without using a calcula...
 12.12.1.614: In Exercises 1828, evaluate each expression without using a calcula...
 12.12.1.615: In Exercises 1828, evaluate each expression without using a calcula...
 12.12.1.616: In Exercises 1828, evaluate each expression without using a calcula...
 12.12.1.617: In Exercises 1828, evaluate each expression without using a calcula...
 12.12.1.618: In Exercises 1828, evaluate each expression without using a calcula...
 12.12.1.619: In Exercises 1828, evaluate each expression without using a calcula...
 12.12.1.620: In Exercises 1828, evaluate each expression without using a calcula...
 12.12.1.621: Graph f (x) 2x and g(x) log2 x in the same rectangular coordinate s...
 12.12.1.622: Graph f (x) 13 x and g(x) log1 3 x in the same rectangular coordina...
 12.12.1.623: In Exercises 3133, find the domain of each logarithmic function. f ...
 12.12.1.624: In Exercises 3133, find the domain of each logarithmic function. f ...
 12.12.1.625: In Exercises 3133, find the domain of each logarithmic function. f ...
 12.12.1.626: In Exercises 3436, simplify each expression ln e6x
 12.12.1.627: In Exercises 3436, simplify each expression elnx
 12.12.1.628: In Exercises 3436, simplify each expression 10log 4x2
 12.12.1.629: On the Richter scale, the magnitude, R, of an earthquake of intensi...
 12.12.1.630: Students in a psychology class took a final examination. As part of...
 12.12.1.631: The formula t 1 c ln A A N describes the time, t, in weeks, that it...
 12.12.1.632: In Exercises 4043, use properties of logarithms to expand each log...
 12.12.1.633: In Exercises 4043, use properties of logarithms to expand each log...
 12.12.1.634: In Exercises 4043, use properties of logarithms to expand each log...
 12.12.1.635: In Exercises 4043, use properties of logarithms to expand each log...
 12.12.1.636: In Exercises 4447, use properties of logarithms to condense each lo...
 12.12.1.637: In Exercises 4447, use properties of logarithms to condense each lo...
 12.12.1.638: In Exercises 4447, use properties of logarithms to condense each lo...
 12.12.1.639: In Exercises 4447, use properties of logarithms to condense each lo...
 12.12.1.640: In Exercises 4849, use common logarithms or natural logarithms and ...
 12.12.1.641: In Exercises 4849, use common logarithms or natural logarithms and ...
 12.12.1.642: In Exercises 5053, determine whether each equation is true or false...
 12.12.1.643: In Exercises 5053, determine whether each equation is true or false...
 12.12.1.644: In Exercises 5053, determine whether each equation is true or false...
 12.12.1.645: In Exercises 5053, determine whether each equation is true or false...
 12.12.1.646: In Exercises 5459, solve each exponential equation. Where necessary...
 12.12.1.647: In Exercises 5459, solve each exponential equation. Where necessary...
 12.12.1.648: In Exercises 5459, solve each exponential equation. Where necessary...
 12.12.1.649: In Exercises 5459, solve each exponential equation. Where necessary...
 12.12.1.650: In Exercises 5459, solve each exponential equation. Where necessary...
 12.12.1.651: In Exercises 5459, solve each exponential equation. Where necessary...
 12.12.1.652: In Exercises 6069, solve each logarithmic equation. log5 x 3
 12.12.1.653: In Exercises 6069, solve each logarithmic equation. log x 2
 12.12.1.654: In Exercises 6069, solve each logarithmic equation. log4(3x 5) 3
 12.12.1.655: In Exercises 6069, solve each logarithmic equation. ln x 1
 12.12.1.656: In Exercises 6069, solve each logarithmic equation. 3 4 ln(2x) 15
 12.12.1.657: In Exercises 6069, solve each logarithmic equation. log2(x 3) log2(...
 12.12.1.658: In Exercises 6069, solve each logarithmic equation. log3(x 1) log3(...
 12.12.1.659: In Exercises 6069, solve each logarithmic equation. log4(3x 5) log4 3
 12.12.1.660: In Exercises 6069, solve each logarithmic equation. ln(x 4) ln(x 1)...
 12.12.1.661: In Exercises 6069, solve each logarithmic equation. log6(2x 1) log6...
 12.12.1.662: The function P(x) 14.7e 0.21x models the average atmospheric pressu...
 12.12.1.663: The amount of carbon dioxide in the atmosphere, measured in parts p...
 12.12.1.664: The function W(x) 0.37 ln x 0.05 models the average walking speed, ...
 12.12.1.665: Use the compound interest formula A P1 r n nt to solve this problem...
 12.12.1.666: How long, to the nearest tenth of a year, will it take $50,000 to t...
 12.12.1.667: What interest rate is required for an investment subject to continu...
 12.12.1.668: According to the U.S. Bureau of the Census, in 1990 there were 22.4...
 12.12.1.669: Use the exponential decay model, A A0ekt, to solve this exercise. T...
 12.12.1.670: Exercises 7879 present data in the form of tables. For each data se...
 12.12.1.671: Exercises 7879 present data in the form of tables. For each data se...
 12.12.1.672: In Exercises 8081, rewrite the equation in terms of base e. Express...
 12.12.1.673: In Exercises 8081, rewrite the equation in terms of base e. Express...
 12.12.1.674: The figure shows world population projections through the year 2150...
 12.12.1.675: Graph f (x) 2x and g(x) 2x 1 in the same rectangular coordinate sys...
 12.12.1.676: Use A P1 r n nt and A Pert to solve this problem. Suppose you have ...
 12.12.1.677: Write in exponential form: log5 125 3.
 12.12.1.678: Write in logarithmic form: 36 6.
 12.12.1.679: Graph f (x) 3x and g(x) log3 x in the same rectangular coordinate s...
 12.12.1.680: In Exercises 68, simplify each expression ln e5x
 12.12.1.681: In Exercises 68, simplify each expression logb b
 12.12.1.682: log6 1
 12.12.1.683: Find the domain: f (x) log5 (x 7).
 12.12.1.684: On the decibel scale, the loudness of a sound, in decibels, is give...
 12.12.1.685: In Exercises 1112, use properties of logarithms to expand each loga...
 12.12.1.686: In Exercises 1112, use properties of logarithms to expand each loga...
 12.12.1.687: In Exercises 1314, write each expression as a single logarithm 6 lo...
 12.12.1.688: In Exercises 1314, write each expression as a single logarithm ln 7...
 12.12.1.689: Use a calculator to evaluate log15 71 to four decimal places.
 12.12.1.690: In Exercises 1623, solve each equation. 3x 2 81
 12.12.1.691: In Exercises 1623, solve each equation. 5x 1.4
 12.12.1.692: In Exercises 1623, solve each equation. 400e0.005x 1600
 12.12.1.693: In Exercises 1623, solve each equation. log25 x 1 2
 12.12.1.694: In Exercises 1623, solve each equation. log6(4x 1) 3
 12.12.1.695: In Exercises 1623, solve each equation. 2 ln(3x) 8
 12.12.1.696: In Exercises 1623, solve each equation. log x log(x 15) 2
 12.12.1.697: In Exercises 1623, solve each equation. ln(x 4) ln(x 1) ln 6
 12.12.1.698: The function A 82.4e 0.002t models the population of Germany, A, in...
 12.12.1.699: How long, to the nearest tenth of a year, will it take $4000 to gro...
 12.12.1.700: What interest rate is required for an investment subject to continu...
 12.12.1.701: The 1990 population of Europe was 509 million; in 2000, it was 729 ...
 12.12.1.702: Use the exponential decay model for carbon14, A A0e 0.000121t, to ...
 12.12.1.703: In Exercises 2932, determine whether the values in each table belon...
 12.12.1.704: In Exercises 2932, determine whether the values in each table belon...
 12.12.1.705: In Exercises 2932, determine whether the values in each table belon...
 12.12.1.706: In Exercises 2932, determine whether the values in each table belon...
 12.12.1.707: Rewrite y 96(0.38)x in terms of base e. Express the answer in terms...
Solutions for Chapter 12: CHAPTER 12 REVIEW EXERCISES
Full solutions for Introductory & Intermediate Algebra for College Students  4th Edition
ISBN: 9780321758941
Solutions for Chapter 12: CHAPTER 12 REVIEW EXERCISES
Get Full SolutionsChapter 12: CHAPTER 12 REVIEW EXERCISES includes 115 full stepbystep solutions. This textbook survival guide was created for the textbook: Introductory & Intermediate Algebra for College Students, edition: 4. Introductory & Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758941. Since 115 problems in chapter 12: CHAPTER 12 REVIEW EXERCISES have been answered, more than 71504 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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.)

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

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].

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

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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 .

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

Pascal matrix
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).

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).