- 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
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.
peA) = det(A - AI) has peA) = zero matrix.
Column space C (A) =
space of all combinations of the columns of A.
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.
Invert A by row operations on [A I] to reach [I A-I].
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = 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 B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
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).
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.
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.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
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).
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.