- Chapter 1.1: The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing Equations
- Chapter 1.2: Intercepts; Symmetry; Graphing Key Equations
- Chapter 1.3: Solving Equations Using a Graphing Utility
- Chapter 1.4: Lines
- Chapter 1.5: Circles
- Chapter 10.1: Conics
- Chapter 10.2: The Parabola
- Chapter 10.3: The Ellipse
- Chapter 10.4: The Hyperbola
- Chapter 10.5: Rotation of Axes; General Form of a Conic
- Chapter 10.6: Polar Equations of Conics
- Chapter 10.7: Plane Curves and Parametric Equations
- Chapter 11.1: Systems of Linear Equations: Substitution and Elimination
- Chapter 11.2: Systems of Linear Equations: Matrices
- Chapter 11.3: Systems of Linear Equations: Determinants
- Chapter 11.4: Matrix Algebra
- Chapter 11.5: Partial Fraction Decomposition
- Chapter 11.6: Systems of Nonlinear Equations
- Chapter 11.7: Systems of Inequalities
- Chapter 11.8: Linear Programming
- Chapter 12.1: Sequences
- Chapter 12.2: Arithmetic Sequences
- Chapter 12.3: Geometric Sequences; Geometric Series
- Chapter 12.4: Mathematical Induction
- Chapter 12.5: The Binomial Theorem
- Chapter 13.1: Sequences; Induction; the Binomial Theorem
- Chapter 13.2: Probability
- Chapter 14.1: Finding Limits Using Tables and Graphs
- Chapter 14.2: Algebra Techniques for Finding Limits
- Chapter 14.3: One-sided Limits; Continuous Functions
- Chapter 14.4: The Tangent Problem; The Derivative
- Chapter 14.5: The Area Problem; The Integral
- Chapter 2.1: Functions
- Chapter 2.2: The Graph of a Function
- Chapter 2.3: Properties of Functions
- Chapter 2.4: Library of Functions; Piecewise-defined Functions
- Chapter 2.5: Graphing Techniques: Transformations
- Chapter 2.6: Mathematical Models: Building Functions
- Chapter 3.1: Linear Functions and Their Properties
- Chapter 3.2: Linear Models: Building Linear Functions from Data
- Chapter 3.3: Quadratic Functions and Their Properties
- Chapter 3.4: Build Quadratic Models from Verbal Descriptions and from Data
- Chapter 3.5: Inequalities Involving Quadratic Functions
- Chapter 4.1: Polynomial Functions and Models
- Chapter 4.2: The Real Zeros of a Polynomial Function
- Chapter 4.3: Complex Zeros; Fundamental Theorem of Algebra
- Chapter 4.4: Properties of Rational Functions
- Chapter 4.5: The Graph of a Rational Function
- Chapter 4.6: Polynomial and Rational Inequalities
- Chapter 4.7: Polynomial and Rational Functions
- Chapter 5.1: Composite Functions
- Chapter 5.2: One-to-One Functions; Inverse Functions
- Chapter 5.3: Exponential Functions
- Chapter 5.4: Logarithmic Functions
- Chapter 5.5: Properties of Logarithms
- Chapter 5.6: Logarithmic and Exponential Equations
- Chapter 5.7: Financial Models
- Chapter 5.70: Financial Models
- Chapter 5.8: Exponential Growth and Decay Models; Newtons Law; Logistic Growth and Decay Models
- Chapter 5.80: Exponential Growth and Decay Models; Newtons Law; Logistic Growth and Decay Models
- Chapter 5.9: Building Exponential, Logarithmic, and Logistic Models from Data
- Chapter 6.1: Angles and Their Measure
- Chapter 6.2: Trigonometric Functions: Unit Circle Approach
- Chapter 6.3: Properties of the Trigonometric Functions
- Chapter 6.4: Graphs of the Sine and Cosine Functions
- Chapter 6.5: Open the Period applet. On the screen you will see a slider. Move the point along the slider to see the role v plays in the graph of f1x2 = sin1vx2. Pay particular attention to the key points matched by color on each graph. For convenience the graph of g1
- Chapter 6.6: Phase Shift; Sinusoidal Curve Fitting 4
- Chapter 7.1: The Inverse Sine, Cosine, and Tangent Functions
- Chapter 7.2: The Inverse Trigonometric Functions (Continued)
- Chapter 7.3: Trigonometric Equations
- Chapter 7.4: Trigonometric Identities
- Chapter 7.5: Sum and Difference Formulas
- Chapter 7.6: Double-angle and Half-angle Formulas
- Chapter 7.7: Product-to-Sum and Sum-to-Product Formulas
- Chapter 8.1: Right Triangle Trigonometry; Applications
- Chapter 8.2: The Law of Sines
- Chapter 8.3: The Law of Cosines
- Chapter 8.4: Area of a Triangle
- Chapter 8.5: Simple Harmonic Motion; Damped Motion; Combining Waves
- Chapter 9.1: Polar Coordinates
- Chapter 9.2: Polar Equations and Graphs
- Chapter 9.3: The Complex Plane; De Moivres Theorem
- Chapter 9.4: Vectors
- Chapter 9.5: The Dot Product
- Chapter 9.6: Vectors in Space
- Chapter 9.7: The Cross Product
- Chapter A.1: Algebra Essentials
- Chapter A.10: nth Roots; Rational Exponents
- Chapter A.2: Geometry Essentials
- Chapter A.3: Polynomials
- Chapter A.4: Synthetic Division
- Chapter A.5: Rational Expressions
- Chapter A.6: Solving Equations
- Chapter A.7: Complex Numbers; Quadratic Equations in the Complex Number System
- Chapter A.8: Problem Solving: Interest, Mixture, Uniform Motion, Constant Rate Job Applications
- Chapter A.9: Interval Notation; Solving Inequalities
- Chapter Appendix B: The Limit of a Sequence; Infinite Series
- Chapter Chapter 1: Graphs
- Chapter Chapter 10: Analytic Geometry
- Chapter Chapter 11: Systems of Equations and Inequalities
- Chapter Chapter 12: Sequences; Induction; the Binomial Theorem
- Chapter Chapter 13: Sequences; Induction; the Binomial Theorem
- Chapter Chapter 14: Sequences; Induction; the Binomial Theorem
- Chapter Chapter 15: Sequences; Induction; the Binomial Theorem
- Chapter Chapter 16: Sequences; Induction; the Binomial Theorem
- Chapter Chapter 17: Sequences; Induction; the Binomial Theorem
- Chapter Chapter 2: Functions and Their Graphs
- Chapter Chapter 3: Linear and Quadratic Functions
- Chapter Chapter 4: Polynomial and Rational Functions
- Chapter Chapter 5: Exponential and Logarithmic Functions
- Chapter Chapter 6: Trigonometric Functions
- Chapter Chapter 7: Analytic Trigonometry
- Chapter Chapter 8: Applications of Trigonometric Functions
- Chapter Chapter 9: Polar Coordinates; Vectors
Precalculus Enhanced with Graphing Utilities 6th Edition - Solutions by Chapter
Full solutions for Precalculus Enhanced with Graphing Utilities | 6th Edition
Precalculus Enhanced with Graphing Utilities | 6th Edition - Solutions by ChapterGet Full Solutions
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.
peA) = det(A - AI) has peA) = zero matrix.
Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.
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).
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.
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.
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .
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 = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
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 •
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
The diagonal entry (first nonzero) at the time when a row is used in elimination.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
Pseudoinverse A+ (Moore-Penrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).
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.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.
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