 Chapter 14.1: In 111, find the limit.
 Chapter 14.2: In 111, find the limit.
 Chapter 14.3: In 111, find the limit.
 Chapter 14.4: In 111, find the limit.
 Chapter 14.5: In 111, find the limit.
 Chapter 14.6: In 111, find the limit.
 Chapter 14.7: In 111, find the limit.
 Chapter 14.8: In 111, find the limit.
 Chapter 14.9: In 111, find the limit.
 Chapter 14.10: In 111, find the limit.
 Chapter 14.11: In 111, find the limit.
 Chapter 14.12: In 1215, determine whether f is continuous at c.
 Chapter 14.13: In 1215, determine whether f is continuous at c.
 Chapter 14.14: In 1215, determine whether f is continuous at c.
 Chapter 14.15: In 1215, determine whether f is continuous at c.
 Chapter 14.16: In 1627, use the accompanying graph of y = f 1x 2.
 Chapter 14.17: In 1627, use the accompanying graph of y = f 1x 2.
 Chapter 14.18: In 1627, use the accompanying graph of y = f 1x 2.
 Chapter 14.19: In 1627, use the accompanying graph of y = f 1x 2.
 Chapter 14.20: In 1627, use the accompanying graph of y = f 1x 2.
 Chapter 14.21: In 1627, use the accompanying graph of y = f 1x 2.
 Chapter 14.22: In 1627, use the accompanying graph of y = f 1x 2.
 Chapter 14.23: In 1627, use the accompanying graph of y = f 1x 2.
 Chapter 14.24: In 1627, use the accompanying graph of y = f 1x 2.
 Chapter 14.25: In 1627, use the accompanying graph of y = f 1x 2.
 Chapter 14.26: In 1627, use the accompanying graph of y = f 1x 2.
 Chapter 14.27: In 1627, use the accompanying graph of y = f 1x 2.
 Chapter 14.28: Discuss whether R1x 2 = x + 4 x 2  16 is continuous at c = 4 and ...
 Chapter 14.29: Determine where the rational function R1x 2 = x 3  2x 2 + 4 x  8 ...
 Chapter 14.30: In 30 32, find the slope of the tangent line to the graph of f at t...
 Chapter 14.31: In 30 32, find the slope of the tangent line to the graph of f at t...
 Chapter 14.32: In 30 32, find the slope of the tangent line to the graph of f at t...
 Chapter 14.33: 3335, find the derivative of each function at the number indicated
 Chapter 14.34: 3335, find the derivative of each function at the number indicated
 Chapter 14.35: A mosaic tile floor is designed in the shape of a trapezoid 30 feet...
 Chapter 14.36: In 36 and 37, find the derivative of each function at the number in...
 Chapter 14.37: In 36 and 37, find the derivative of each function at the number in...
 Chapter 14.38: In physics it is shown that the height s of a ball thrown straight ...
 Chapter 14.39: The following data represent the revenue R (in dollars) received fr...
 Chapter 14.40: The function f 1x 2 = 2x + 3 is defined on the interval 30, 4 4. (a...
 Chapter 14.41: In 41 and 42, a function f is defined over an interval 3a, b 4. (a)...
 Chapter 14.42: In 41 and 42, a function f is defined over an interval 3a, b 4. (a)...
 Chapter 14.43: In 43 and 44, an integral is given. (a) What area does the integral...
 Chapter 14.44: In 43 and 44, an integral is given. (a) What area does the integral...
Solutions for Chapter Chapter 14: Sequences; Induction; the Binomial Theorem
Full solutions for Precalculus Enhanced with Graphing Utilities  6th Edition
ISBN: 9780132854351
Solutions for Chapter Chapter 14: Sequences; Induction; the Binomial Theorem
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 44 problems in chapter Chapter 14: Sequences; Induction; the Binomial Theorem have been answered, more than 54371 students have viewed full stepbystep solutions from this chapter. Chapter Chapter 14: Sequences; Induction; the Binomial Theorem includes 44 full stepbystep solutions. This textbook survival guide was created for the textbook: Precalculus Enhanced with Graphing Utilities, edition: 6. Precalculus Enhanced with Graphing Utilities was written by and is associated to the ISBN: 9780132854351.

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.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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.

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.

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

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

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.

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).

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

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

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

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

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

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

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