 11.5.11.5.1: Fill in the blanks. ni1i
 11.5.11.5.2: Fill in the blanks. ni1i3
 11.5.11.5.3: Fill in the blanks. The exact _______ of a plane region is given by...
 11.5.11.5.4: In Exercises 18, evaluate the sum using the summation formulas and ...
 11.5.11.5.5: In Exercises 18, evaluate the sum using the summation formulas and ...
 11.5.11.5.6: In Exercises 18, evaluate the sum using the summation formulas and ...
 11.5.11.5.7: In Exercises 18, evaluate the sum using the summation formulas and ...
 11.5.11.5.8: In Exercises 18, evaluate the sum using the summation formulas and ...
 11.5.11.5.9: In Exercises 916, (a) rewrite the sum as a rational function (b) Us...
 11.5.11.5.10: In Exercises 916, (a) rewrite the sum as a rational function (b) Us...
 11.5.11.5.11: In Exercises 916, (a) rewrite the sum as a rational function (b) Us...
 11.5.11.5.12: In Exercises 916, (a) rewrite the sum as a rational function (b) Us...
 11.5.11.5.13: In Exercises 916, (a) rewrite the sum as a rational function (b) Us...
 11.5.11.5.14: In Exercises 916, (a) rewrite the sum as a rational function (b) Us...
 11.5.11.5.15: In Exercises 916, (a) rewrite the sum as a rational function (b) Us...
 11.5.11.5.16: In Exercises 916, (a) rewrite the sum as a rational function (b) Us...
 11.5.11.5.17: In Exercises 1720, approximate the area of the region using the ind...
 11.5.11.5.18: In Exercises 1720, approximate the area of the region using the ind...
 11.5.11.5.19: In Exercises 1720, approximate the area of the region using the ind...
 11.5.11.5.20: In Exercises 1720, approximate the area of the region using the ind...
 11.5.11.5.21: In Exercises 2124, complete the table showing the approximate area ...
 11.5.11.5.22: In Exercises 2124, complete the table showing the approximate area ...
 11.5.11.5.23: In Exercises 2124, complete the table showing the approximate area ...
 11.5.11.5.24: In Exercises 2124, complete the table showing the approximate area ...
 11.5.11.5.25: In Exercises 2532, complete the table using the function over the s...
 11.5.11.5.26: In Exercises 2532, complete the table using the function over the s...
 11.5.11.5.27: In Exercises 2532, complete the table using the function over the s...
 11.5.11.5.28: In Exercises 2532, complete the table using the function over the s...
 11.5.11.5.29: In Exercises 2532, complete the table using the function over the s...
 11.5.11.5.30: In Exercises 2532, complete the table using the function over the s...
 11.5.11.5.31: In Exercises 2532, complete the table using the function over the s...
 11.5.11.5.32: In Exercises 2532, complete the table using the function over the s...
 11.5.11.5.33: In Exercises 33 44, use the limit process to find the area of the r...
 11.5.11.5.34: In Exercises 33 44, use the limit process to find the area of the r...
 11.5.11.5.35: In Exercises 33 44, use the limit process to find the area of the r...
 11.5.11.5.36: In Exercises 33 44, use the limit process to find the area of the r...
 11.5.11.5.37: In Exercises 33 44, use the limit process to find the area of the r...
 11.5.11.5.38: In Exercises 33 44, use the limit process to find the area of the r...
 11.5.11.5.39: In Exercises 33 44, use the limit process to find the area of the r...
 11.5.11.5.40: In Exercises 33 44, use the limit process to find the area of the r...
 11.5.11.5.41: In Exercises 33 44, use the limit process to find the area of the r...
 11.5.11.5.42: In Exercises 33 44, use the limit process to find the area of the r...
 11.5.11.5.43: In Exercises 33 44, use the limit process to find the area of the r...
 11.5.11.5.44: In Exercises 33 44, use the limit process to find the area of the r...
 11.5.11.5.45: Geometry The boundaries of a parcel of land are two edges modeled b...
 11.5.11.5.46: Geometry The table shows the measurements (in feet) of a lot bounde...
 11.5.11.5.47: True or False? In Exercises 47 and 48, determine whether the statem...
 11.5.11.5.48: True or False? In Exercises 47 and 48, determine whether the statem...
 11.5.11.5.49: Writing Describe the process of finding the area of a region bounde...
 11.5.11.5.50: Think About It Determine which value best approximates the area of ...
 11.5.11.5.51: In Exercises 51 and 52, solve the equation. 2 tan x tan 2x c
 11.5.11.5.52: In Exercises 51 and 52, solve the equation. cos 2x 3 sin x 23
 11.5.11.5.53: In Exercises 5356, use the vectors and to find the indicated quantity
 11.5.11.5.54: In Exercises 5356, use the vectors and to find the indicated quantity
 11.5.11.5.55: In Exercises 5356, use the vectors and to find the indicated quantity
 11.5.11.5.56: In Exercises 5356, use the vectors and to find the indicated quantity
Solutions for Chapter 11.5: The Area Problem
Full solutions for Precalculus With Limits A Graphing Approach  5th Edition
ISBN: 9780618851522
Solutions for Chapter 11.5: The Area Problem
Get Full SolutionsSince 56 problems in chapter 11.5: The Area Problem have been answered, more than 33051 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 11.5: The Area Problem includes 56 full stepbystep solutions. This textbook survival guide was created for the textbook: Precalculus With Limits A Graphing Approach, edition: 5. Precalculus With Limits A Graphing Approach was written by and is associated to the ISBN: 9780618851522.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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 .

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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.

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

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

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.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.