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Classes
Algebra
1. Preliminaries
1.1 Integer Exponents
1.2 Rational Exponents
1.3 Radicals
1.4 Polynomials
1.5 Factoring Polynomials
1.6 Rational Expressions
1.7 Complex Numbers
2. Solving Equations and Inequalities
2.1 Solutions and Solution Sets
2.2 Linear Equations
2.3 Applications of Linear Equations
2.4 Equations With More Than One Variable
2.5 Quadratic Equations - Part I
2.6 Quadratic Equations - Part II
2.7 Quadratic Equations : A Summary
2.8 Applications of Quadratic Equations
2.9 Equations Reducible to Quadratic in Form
2.10 Equations with Radicals
2.11 Linear Inequalities
2.12 Polynomial Inequalities
2.13 Rational Inequalities
2.14 Absolute Value Equations
2.15 Absolute Value Inequalities
3. Graphing and Functions
3.1 Graphing
3.2 Lines
3.3 Circles
3.4 The Definition of a Function
3.5 Graphing Functions
3.6 Combining Functions
3.7 Inverse Functions
4. Common Graphs
4.1 Lines, Circles and Piecewise Functions
4.2 Parabolas
4.3 Ellipses
4.4 Hyperbolas
4.5 Miscellaneous Functions
4.6 Transformations
4.7 Symmetry
4.8 Rational Functions
5. Polynomial Functions
5.1 Dividing Polynomials
5.2 Zeroes/Roots of Polynomials
5.3 Graphing Polynomials
5.4 Finding Zeroes of Polynomials
5.5 Partial Fractions
6. Exponential and Logarithm Functions
6.1 Exponential Functions
6.2 Logarithm Functions
6.3 Solving Exponential Equations
6.4 Solving Logarithm Equations
6.5 Applications
7. Systems of Equations
7.1 Linear Systems with Two Variables
7.2 Linear Systems with Three Variables
7.3 Augmented Matrices
7.4 More on the Augmented Matrix
7.5 Nonlinear Systems
Calculus I
1. Review
1.1 Functions
1.2 Inverse Functions
1.3 Trig Functions
1.4 Solving Trig Equations
1.5 Trig Equations with Calculators, Part I
1.6 Trig Equations with Calculators, Part II
1.7 Exponential Functions
1.8 Logarithm Functions
1.9 Exponential and Logarithm Equations
1.10 Common Graphs
2. Limits
2.1 Tangent Lines and Rates of Change
2.2 The Limit
2.3 One-Sided Limits
2.4 Limit Properties
2.5 Computing Limits
2.6 Infinite Limits
2.7 Limits At Infinity, Part I
2.8 Limits At Infinity, Part II
2.9 Continuity
2.10 The Definition of the Limit
3. Derivatives
3.1 The Definition of the Derivative
3.2 Interpretation of the Derivative
3.3 Differentiation Formulas
3.4 Product and Quotient Rule
3.5 Derivatives of Trig Functions
3.6 Derivatives of Exponential and Logarithm Functions
3.7 Derivatives of Inverse Trig Functions
3.8 Derivatives of Hyperbolic Functions
3.9 Chain Rule
3.10 Implicit Differentiation
3.11 Related Rates
3.12 Higher Order Derivatives
3.13 Logarithmic Differentiation
4. Applications of Derivatives
4.1 Rates of Change
4.2 Critical Points
4.3 Minimum and Maximum Values
4.4 Finding Absolute Extrema
4.5 The Shape of a Graph, Part I
4.6 The Shape of a Graph, Part II
4.7 The Mean Value Theorem
4.8 Optimization
4.9 More Optimization Problems
4.10 L'Hospital's Rule and Indeterminate Forms
4.11 Linear Approximations
4.12 Differentials
4.13 Newton's Method
4.14 Business Applications
5. Integrals
5.1 Indefinite Integrals
5.2 Computing Indefinite Integrals
5.3 Substitution Rule for Indefinite Integrals
5.4 More Substitution Rule
5.5 Area Problem
5.6 Definition of the Definite Integral
5.7 Computing Definite Integrals
5.8 Substitution Rule for Definite Integrals
6. Applications of Integrals
6.1 Average Function Value
6.2 Area Between Curves
6.3 Volumes of Solids of Revolution / Method of Rings
6.4 Volumes of Solids of Revolution/Method of Cylinders
6.5 More Volume Problems
6.6 Work
Appendix A. Extras
A.1 Proof of Various Limit Properties
A.2 Proof of Various Derivative Properties
A.3 Proof of Trig Limits
A.4 Proofs of Derivative Applications Facts
A.5 Proof of Various Integral Properties
A.6 Area and Volume Formulas
A.7 Types of Infinity
A.8 Summation Notation
A.9 Constant of Integration
Calculus II
7. Integration Techniques
7.1 Integration by Parts
7.2 Integrals Involving Trig Functions
7.3 Trig Substitutions
7.4 Partial Fractions
7.5 Integrals Involving Roots
7.6 Integrals Involving Quadratics
7.7 Integration Strategy
7.8 Improper Integrals
7.9 Comparison Test for Improper Integrals
7.10 Approximating Definite Integrals
8. Applications of Integrals
8.1 Arc Length
8.2 Surface Area
8.3 Center of Mass
8.4 Hydrostatic Pressure
8.5 Probability
9. Parametric Equations and Polar Coordinates
9.1 Parametric Equations and Curves
9.2 Tangents with Parametric Equations
9.3 Area with Parametric Equations
9.4 Arc Length with Parametric Equations
9.5 Surface Area with Parametric Equations
9.6 Polar Coordinates
9.7 Tangents with Polar Coordinates
9.8 Area with Polar Coordinates
9.9 Arc Length with Polar Coordinates
9.10 Surface Area with Polar Coordinates
9.11 Arc Length and Surface Area Revisited
10. Series & Sequences
10.1 Sequences
10.2 More on Sequences
10.3 Series - The Basics
10.4 Convergence/Divergence of Series
10.5 Special Series
10.6 Integral Test
10.7 Comparison Test/Limit Comparison Test
10.8 Alternating Series Test
10.9 Absolute Convergence
10.10 Ratio Test
10.11 Root Test
10.12 Strategy for Series
10.13 Estimating the Value of a Series
10.14 Power Series
10.15 Power Series and Functions
10.16 Taylor Series
10.17 Applications of Series
10.18 Binomial Series
11. Vectors
11.1 Vectors - The Basics
11.2 Vector Arithmetic
11.3 Dot Product
11.4 Cross Product
12. 3-Dimensional Space
12.1 The 3-D Coordinate System
12.2 Equations of Lines
12.3 Equations of Planes
12.4 Quadric Surfaces
12.5 Functions of Several Variables
12.6 Vector Functions
12.7 Calculus with Vector Functions
12.8 Tangent, Normal and Binormal Vectors
12.9 Arc Length with Vector Functions
12.10 Curvature
12.11 Velocity and Acceleration
12.12 Cylindrical Coordinates
12.13 Spherical Coordinates
Calculus III
12. 3-Dimensional Space
12.1 The 3-D Coordinate System
12.2 Equations of Lines
12.3 Equations of Planes
12.4 Quadric Surfaces
12.5 Functions of Several Variables
12.6 Vector Functions
12.7 Calculus with Vector Functions
12.8 Tangent, Normal and Binormal Vectors
12.9 Arc Length with Vector Functions
12.10 Curvature
12.11 Velocity and Acceleration
12.12 Cylindrical Coordinates
12.13 Spherical Coordinates
13. Partial Derivatives
13.1 Limits
13.2 Partial Derivatives
13.3 Interpretations of Partial Derivatives
13.4 Higher Order Partial Derivatives
13.5 Differentials
13.6 Chain Rule
13.7 Directional Derivatives
14. Applications of Partial Derivatives
14.1 Tangent Planes and Linear Approximations
14.2 Gradient Vector, Tangent Planes and Normal Lines
14.3 Relative Minimums and Maximums
14.4 Absolute Minimums and Maximums
14.5 Lagrange Multipliers
15. Multiple Integrals
15.1 Double Integrals
15.2 Iterated Integrals
15.3 Double Integrals over General Regions
15.4 Double Integrals in Polar Coordinates
15.5 Triple Integrals
15.6 Triple Integrals in Cylindrical Coordinates
15.7 Triple Integrals in Spherical Coordinates
15.8 Change of Variables
15.9 Surface Area
15.10 Area and Volume Revisited
16. Line Integrals
16.1 Vector Fields
16.2 Line Integrals - Part I
16.3 Line Integrals - Part II
16.4 Line Integrals of Vector Fields
16.5 Fundamental Theorem for Line Integrals
16.6 Conservative Vector Fields
16.7 Green's Theorem
17.Surface Integrals
17.1 Curl and Divergence
17.2 Parametric Surfaces
17.3 Surface Integrals
17.4 Surface Integrals of Vector Fields
17.5 Stokes' Theorem
17.6 Divergence Theorem
Differential Equations
1. Basic Concepts
1.1 Definitions
1.2 Direction Fields
1.3 Final Thoughts
2. First Order DE's
2.1 Linear Equations
2.2 Separable Equations
2.3 Exact Equations
2.4 Bernoulli Differential Equations
2.5 Substitutions
2.6 Intervals of Validity
2.7 Modeling with First Order DE's
2.8 Equilibrium Solutions
2.9 Euler's Method
3. Second Order DE's
3.1 Basic Concepts
3.2 Real & Distinct Roots
3.3 Complex Roots
3.4 Repeated Roots
3.5 Reduction of Order
3.6 Fundamental Sets of Solutions
3.7 More on the Wronskian
3.8 Nonhomogeneous Differential Equations
3.9 Undetermined Coefficients
3.10 Variation of Parameters
3.11 Mechanical Vibrations
4. Laplace Transforms
4.1 The Definition
4.2 Laplace Transforms
4.3 Inverse Laplace Transforms
4.4 Step Functions
4.5 Solving IVP's with Laplace Transforms
4.6 Nonconstant Coefficient IVP's
4.7 IVP's With Step Functions
4.8 Dirac Delta Function
4.9 Convolution Integrals
4.10 Table Of Laplace Transforms
5. Systems of DE's
5.1 Review : Systems of Equations
5.2 Review : Matrices & Vectors
5.3 Review : Eigenvalues & Eigenvectors
5.4 Systems of Differential Equations
5.5 Solutions to Systems
5.6 Phase Plane
5.7 Real Eigenvalues
5.8 Complex Eigenvalues
5.9 Repeated Eigenvalues
5.10 Nonhomogeneous Systems
5.11 Laplace Transforms
5.12 Modeling
6. Series Solutions to DE's
6.1 Review : Power Series
6.2 Review : Taylor Series
6.3 Series Solutions
6.4 Euler Equations
7. Higher Order Differential Equations
7.1 Basic Concepts for
n
th
Order Linear Equations
7.2 Linear Homogeneous Differential Equations
7.3 Undetermined Coefficients
7.4 Variation of Parameters
7.5 Laplace Transforms
7.6 Systems of Differential Equations
7.7 Series Solutions
8. Boundary Value Problems & Fourier Series
8.1 Boundary Value Problems
8.2 Eigenvalues and Eigenfunctions
8.3 Periodic Functions & Orthogonal Functions
8.4 Fourier Sine Series
8.5 Fourier Cosine Series
8.6 Fourier Series
8.7 Convergence of Fourier Series
9. Partial Differential Equations
9.1 The Heat Equation
9.2 The Wave Equation
9.3 Terminology
9.4 Separation of Variables
9.5 Solving the Heat Equation
9.6 Heat Equation with Non-Zero Temperature Boundaries
9.7 Laplace's Equation
9.8 Vibrating String
9.9 Summary of Separation of Variables
Extras
Algebra & Trig Review
1. Algebra
1.1 Exponents
1.2 Absolute Value
1.3 Radicals
1.4 Rationalizing
1.5 Functions
1.6 Multiplying Polynomials
1.7 Factoring
1.8 Simplifying Rational Expressions
1.9 Graphing and Common Graphs
1.10 Solving Equations, Part I
1.11 Solving Equations, Part II
1.12 Solving Systems of Equations
1.13 Solving Inequalities
1.14 Absolute Value Equations and Inequalities
2. Trigonometry
2.1 Trig Function Evaluation
2.2 Graphs of Trig Functions
2.3 Trig Formulas
2.4 Solving Trig Equations
2.5 Inverse Trig Functions
3. Exponentials & Logarithms
3.1 Basic Exponential Functions
3.2 Basic Logarithm Functions
3.3 Logarithm Properties
3.4 Simplifying Logarithms
3.5 Solving Exponential Equations
3.6 Solving Logarithm Equations
Common Math Errors
1. General Errors
2. Algebra Errors
3. Trig Errors
4. Common Errors
5. Calculus Errors
Complex Number Primer
1. The Definition
2. Arithmetic
3. Conjugate and Modulus
4. Polar and Exponential Forms
5. Powers and Roots
How To Study Math
1. General Tips
2. Taking Notes
3. Getting Help
4. Doing Homework
5. Problem Solving
6. Studying For an Exam
7. Taking an Exam
8. Learn From Your Errors
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Section 7.2 : Linear Systems with Three Variables
Find the solution to each of the following systems of equations.
\(\begin{align*}2x + 5y + 2z &= - 38\\ 3x - 2y + 4z &= 17\\ - 6x + y - 7z &= - 12\end{align*}\)
Solution
\(\begin{align*}3x - 9z &= 33\\ 7x - 4y - z &= - 15\\ 4x + 6y + 5z &= - 6\end{align*}\)
Solution