 19.5.19.1.88: (Rectangular rule) Evaluate the integral in Example I by the rectan...
 19.5.19.1.89: Derive a formula for lower and upper bounds for the rectangular rul...
 19.5.19.1.90: Evaluate the integrals numelically as indicated and determine the e...
 19.5.19.1.91: Evaluate the integrals numelically as indicated and determine the e...
 19.5.19.1.92: Evaluate the integrals numelically as indicated and determine the e...
 19.5.19.1.93: Evaluate the integrals numelically as indicated and determine the e...
 19.5.19.1.94: Evaluate the integrals numelically as indicated and determine the e...
 19.5.19.1.95: Evaluate the integrals numelically as indicated and determine the e...
 19.5.19.1.96: Estimate the error by halving. In Prob. 5
 19.5.19.1.97: Estimate the error by halving.In Prob. 6
 19.5.19.1.98: Estimate the error by halving.In Prob. 7
 19.5.19.1.99: Estimate the error by halving.In Prob. 8
 19.5.19.1.100: The following integrals cannot be evaluated by the usual methods of...
 19.5.19.1.101: The following integrals cannot be evaluated by the usual methods of...
 19.5.19.1.102: The following integrals cannot be evaluated by the usual methods of...
 19.5.19.1.103: The following integrals cannot be evaluated by the usual methods of...
 19.5.19.1.104: The following integrals cannot be evaluated by the usual methods of...
 19.5.19.1.105: The following integrals cannot be evaluated by the usual methods of...
 19.5.19.1.106: The following integrals cannot be evaluated by the usual methods of...
 19.5.19.1.107: (Stability) Prove that the trapezoidal rule is stable with respect ...
 19.5.19.1.108: Integrate by (11) with II = 5:IIx from I to 3
 19.5.19.1.109: Integrate by (11) with II = 5:co~ x from 0 to!7T
 19.5.19.1.110: Integrate by (11) with II = 5: ex" from 0 to I
 19.5.19.1.111: Integrate by (11) with II = 5: sin <x2 ) from 0 to 1.25
 19.5.19.1.112: (Given TOL) Find the smallest 11 in computing the integral of 1Ix f...
 19.5.19.1.113: TEAM PROJECT. Romberg Integration (W. Romberg. Norske Videllskab. T...
 19.5.19.1.114: Consider f(x) = X4 for Xo = 0, Xl = 0.2, X2 = 0.4, X3 = 0.6, X4 = 0...
 19.5.19.1.115: A "fourpoint formula" for the derivative is Apply it to f(x) = X4 ...
 19.5.19.1.116: The derivative f' (x) can also be approximated in terms of firstor...
 19.5.19.1.117: Derive the formula in Prob. 29 from (14) in Sec. 19.3
Solutions for Chapter 19.5: Numeric Integration and Differentiation
Full solutions for Advanced Engineering Mathematics  9th Edition
ISBN: 9780471488859
Solutions for Chapter 19.5: Numeric Integration and Differentiation
Get Full SolutionsThis textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 19.5: Numeric Integration and Differentiation includes 30 full stepbystep solutions. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859. Since 30 problems in chapter 19.5: Numeric Integration and Differentiation have been answered, more than 46568 students have viewed full stepbystep solutions from this chapter.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

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.

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

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.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

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

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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