 11.4.1: Graph each function defined in 18.
 11.4.2: Graph each function defined in 18.
 11.4.3: Graph each function defined in 18.
 11.4.4: Graph each function defined in 18.
 11.4.5: Graph each function defined in 18.
 11.4.6: Graph each function defined in 18.
 11.4.7: Graph each function defined in 18.
 11.4.8: Graph each function defined in 18.
 11.4.9: The scale of the graph shown in Figure 11.4.1 is onefourth inch to...
 11.4.10: a. Use the definition of logarithm to show that logbbx = x for all ...
 11.4.11: Let b > 1. a. Use the fact that u = logb v v = bu to show that a po...
 11.4.12: Give a graphical interpretation for property (11.4.2) in Example 11...
 11.4.13: Suppose a positive real number x satisfies the inequality 10m x < 1...
 11.4.14: a. Prove that if x is a positive real number and k is a nonnegative...
 11.4.15: If n is an odd integer and n > 1, is log2(n 1) = log2(n)? Justify y...
 11.4.16: If n is an odd integer and n > 1, is log2(n + 1) = log2(n)? Justify...
 11.4.17: If n is an odd integer and n > 1, is log2(n + 1) = log2(n)? Justify...
 11.4.18: In 18 and 19, indicate how many binary digits are needed to represe...
 11.4.19: In 18 and 19, indicate how many binary digits are needed to represe...
 11.4.20: It was shown in the text that the number of binary digits needed to...
 11.4.21: In each of 21 and 22, a sequence is specified by a recurrence relat...
 11.4.22: In each of 21 and 22, a sequence is specified by a recurrence relat...
 11.4.23: Define a sequence c1, c2, c3,..., recursively as follows: c1 = 0, c...
 11.4.24: Define a sequence c1, c2, c3,..., recursively as follows: c1 = 0, c...
 11.4.25: Exercises 2528 refer to properties 11.4.9 and 11.4.10. To solve the...
 11.4.26: Exercises 2528 refer to properties 11.4.9 and 11.4.10. To solve the...
 11.4.27: Exercises 2528 refer to properties 11.4.9 and 11.4.10. To solve the...
 11.4.28: Exercises 2528 refer to properties 11.4.9 and 11.4.10. To solve the...
 11.4.29: Use notation to express the following statement: x 2 7x 2 + 3x ...
 11.4.30: Derive each statement in 3033.
 11.4.31: Derive each statement in 3033.
 11.4.32: Derive each statement in 3033.
 11.4.33: Derive each statement in 3033.
 11.4.34: Show that 4n is not O(2n ).
 11.4.35: Prove each of the statements in 3540, assuming n is an integer vari...
 11.4.36: Prove each of the statements in 3540, assuming n is an integer vari...
 11.4.37: Prove each of the statements in 3540, assuming n is an integer vari...
 11.4.38: Prove each of the statements in 3540, assuming n is an integer vari...
 11.4.39: Prove each of the statements in 3540, assuming n is an integer vari...
 11.4.40: Prove each of the statements in 3540, assuming n is an integer vari...
 11.4.41: Quantities of the form kn + kn log2 n for positive integers k1 k2, ...
 11.4.42: Calculate the values of the harmonic sums 1 + 1 2 + 1 3 ++ 1 n for ...
 11.4.43: Use part (d) of Example 11.4.7 to show that n + n 2 + n 3 ++ n n is...
 11.4.44: Use the fact that log2 x = 1 loge 2 loge x and loge x = ln x, for a...
 11.4.45: a. Show that log2 n is (log2 n). b. Show that log2 n + 1 is (log2 n...
 11.4.46: Prove by mathematical induction that n 10n for all integers n 1.
 11.4.47: Prove by mathematical induction that log2 n n for all integers n 1.
 11.4.48: Show that if n is a variable that takes positive integer values, th...
 11.4.49: Let n be a variable that takes positive integer values. a. Show tha...
 11.4.50: a. For all positive real numbers u, log2 u < u. Use this fact to sh...
 11.4.51: a. For all real numbers x, x < 2x . Use this fact to show that for ...
 11.4.52: For all positive real numbers u, log2 u < u. Use this fact and the ...
 11.4.53: Use the result of exercise 52 above to prove the following: For all...
 11.4.54: Exercises 54 and 55 use LHpitals rule from calculus.
 11.4.55: Exercises 54 and 55 use LHpitals rule from calculus.
 11.4.56: Complete the proof in Example 11.4.4.
Solutions for Chapter 11.4: Exponential and Logarithmic Functions: Graphs and Orders
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 11.4: Exponential and Logarithmic Functions: Graphs and Orders
Get Full SolutionsThis textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4. This expansive textbook survival guide covers the following chapters and their solutions. Since 56 problems in chapter 11.4: Exponential and Logarithmic Functions: Graphs and Orders have been answered, more than 45197 students have viewed full stepbystep solutions from this chapter. Chapter 11.4: Exponential and Logarithmic Functions: Graphs and Orders includes 56 full stepbystep solutions. Discrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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

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.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

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.

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

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

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

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

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.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

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