- 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
Tv = Av + Vo = linear transformation plus shift.
Upper triangular systems are solved in reverse order Xn to Xl.
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
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 n-l - 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.
Invert A by row operations on [A I] to reach [I A-I].
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. 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.
lA-II = 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.
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.
Constant down each diagonal = time-invariant (shift-invariant) filter.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.