 2.6.1: Let P = 1x, y2 be a point on the graph of y = x2  8. (a) Express t...
 2.6.2: Let P = 1x, y2 be a point on the graph of y = x2  8. (a) Express t...
 2.6.3: Let P = 1x, y2 be a point on the graph of y = 1x. (a) Express the d...
 2.6.4: Let P = 1x, y2 be a point on the graph of y = 1 x . (a) Express the...
 2.6.5: A right triangle has one vertex on the graph of y = x3 , x 7 0, at ...
 2.6.6: A right triangle has one vertex on the graph of y = 9  x2 , x 7 0,...
 2.6.7: A rectangle has one corner in quadrant I on the graph of y = 16  x...
 2.6.8: A rectangle is inscribed in a semicircle of radius 2. See the figur...
 2.6.9: A rectangle is inscribed in a circle of radius 2. See the figure. L...
 2.6.10: A circle of radius r is inscribed in a square. See the figure.
 2.6.11: A wire 10 meters long is to be cut into two pieces. One piece will ...
 2.6.12: A wire 10 meters long is to be cut into two pieces. One piece will ...
 2.6.13: A wire of length x is bent into the shape of a circle. (a) Express ...
 2.6.14: A wire of length x is bent into the shape of a square. (a) Express ...
 2.6.15: A semicircle of radius r is inscribed in a rectangle so that the di...
 2.6.16: An equilateral triangle is inscribed in a circle of radius r. See t...
 2.6.17: An equilateral triangle is inscribed in a circle of radius r. See t...
 2.6.18: An equilateral triangle is inscribed in a circle of radius r. See t...
 2.6.19: Two cars are approaching an intersection. One is 2 miles south of t...
 2.6.20: Inscribe a right circular cylinder of height h and radius r in a sp...
 2.6.21: nscribe a right circular cylinder of height h and radius r in a con...
 2.6.22: MetroMedia Cable is asked to provide service to a customer whose ho...
 2.6.23: An island is 2 miles from the nearest point P on a straight shoreli...
 2.6.24: Water is poured into a container in the shape of a right circular c...
 2.6.25: An open box with a square base is to be made from a square piece of...
 2.6.26: An open box with a square base is required to have a volume of 10 c...
Solutions for Chapter 2.6: Mathematical Models: Building Functions
Full solutions for Precalculus Enhanced with Graphing Utilities  6th Edition
ISBN: 9780132854351
Solutions for Chapter 2.6: Mathematical Models: Building Functions
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 2.6: Mathematical Models: Building Functions includes 26 full stepbystep solutions. Since 26 problems in chapter 2.6: Mathematical Models: Building Functions have been answered, more than 59914 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Precalculus Enhanced with Graphing Utilities, edition: 6. Precalculus Enhanced with Graphing Utilities was written by and is associated to the ISBN: 9780132854351.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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

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).

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

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).

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

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

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

Outer product uv T
= column times row = rank one matrix.

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

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

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.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.