 14.5.1: The formula for the area A of a rectangle of length l and width w i...
 14.5.2: a 4 k=1 12k + 12 = . (pp. 803806)
 14.5.3: The integral from a to b of f1x2 is denoted by the symbol
 14.5.4: The area under the graph of f from a to b is denoted by the symbol .
 14.5.5: Approximate the area A choosing u as the left endpoint of each subi...
 14.5.6: Approximate the area A choosing u as the right endpoint of each sub...
 14.5.7: Approximate the area A choosing u as the left endpoint of each subi...
 14.5.8: Approximate the area A choosing u as the right endpoint of each sub...
 14.5.9: A Preview of Calculus: The Limit, Derivative, and Integral of a Fun...
 14.5.10: Repeat for f1x2 = 4x.
 14.5.11: The function f1x2 = 3x + 9 is defined on the interval 30, 34. (a) ...
 14.5.12: Repeat for f1x2 = 2x + 8. In 1322, a function f is defined over an...
 14.5.13: In 1322, a function f is defined over an interval 3a, b4. (a) Graph...
 14.5.14: In 1322, a function f is defined over an interval 3a, b4. (a) Graph...
 14.5.15: In 1322, a function f is defined over an interval 3a, b4. (a) Graph...
 14.5.16: In 1322, a function f is defined over an interval 3a, b4. (a) Graph...
 14.5.17: In 1322, a function f is defined over an interval 3a, b4. (a) Graph...
 14.5.18: In 1322, a function f is defined over an interval 3a, b4. (a) Graph...
 14.5.19: In 1322, a function f is defined over an interval 3a, b4. (a) Graph...
 14.5.20: In 1322, a function f is defined over an interval 3a, b4. (a) Graph...
 14.5.21: In 1322, a function f is defined over an interval 3a, b4. (a) Graph...
 14.5.22: In 1322, a function f is defined over an interval 3a, b4. (a) Graph...
 14.5.23: In 2330, an integral is given. (a) What area does the integral repr...
 14.5.24: In 2330, an integral is given. (a) What area does the integral repr...
 14.5.25: In 2330, an integral is given. (a) What area does the integral repr...
 14.5.26: In 2330, an integral is given. (a) What area does the integral repr...
 14.5.27: In 2330, an integral is given. (a) What area does the integral repr...
 14.5.28: In 2330, an integral is given. (a) What area does the integral repr...
 14.5.29: In 2330, an integral is given. (a) What area does the integral repr...
 14.5.30: In 2330, an integral is given. (a) What area does the integral repr...
 14.5.31: Confirm the entries in Table 6.
 14.5.32: Consider the function f1x2 = 21  x2 whose domain is the interval 3...
Solutions for Chapter 14.5: The Area Problem; The Integral
Full solutions for Precalculus Enhanced with Graphing Utilities  6th Edition
ISBN: 9780132854351
Solutions for Chapter 14.5: The Area Problem; The Integral
Get Full SolutionsChapter 14.5: The Area Problem; The Integral includes 32 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Precalculus Enhanced with Graphing Utilities, edition: 6. Since 32 problems in chapter 14.5: The Area Problem; The Integral have been answered, more than 54342 students have viewed full stepbystep solutions from this chapter. Precalculus Enhanced with Graphing Utilities was written by and is associated to the ISBN: 9780132854351.

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.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

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

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

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.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

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