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# Solutions for Chapter 2.2: The Graph of a Function

## Full solutions for Precalculus Enhanced with Graphing Utilities | 6th Edition

ISBN: 9780132854351

Solutions for Chapter 2.2: The Graph of a Function

Solutions for Chapter 2.2
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##### ISBN: 9780132854351

This expansive textbook survival guide covers the following chapters and their solutions. Chapter 2.2: The Graph of a Function includes 48 full step-by-step solutions. Since 48 problems in chapter 2.2: The Graph of a Function have been answered, more than 59540 students have viewed full step-by-step solutions from this chapter. Precalculus Enhanced with Graphing Utilities was written by and is associated to the ISBN: 9780132854351. This textbook survival guide was created for the textbook: Precalculus Enhanced with Graphing Utilities, edition: 6.

Key Math Terms and definitions covered in this textbook
• Block matrix.

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.

• Characteristic equation det(A - AI) = O.

The n roots are the eigenvalues of A.

• Condition number

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

• Elimination.

A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

• Exponential eAt = I + At + (At)2 12! + ...

has derivative AeAt; eAt u(O) solves u' = Au.

• Free columns of A.

Columns without pivots; these are combinations of earlier columns.

• Full column rank r = n.

Independent columns, N(A) = {O}, no free variables.

• Fundamental Theorem.

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.

• Identity matrix I (or In).

Diagonal entries = 1, off-diagonal entries = 0.

• Least squares solution X.

The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.

• Nullspace matrix N.

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

• Partial pivoting.

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

• 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.

• 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.

• Skew-symmetric matrix K.

The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

• Solvable system Ax = b.

The right side b is in the column space of A.

• Spanning set.

Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

• Vector v in Rn.

Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

• Wavelets Wjk(t).

Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).

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