WELCOME TO COHTRAN MMPC-VN

MATHEMATICS - APPLIED MECHANICS

College Algebra





LECTURE 

1.1            FUNCTIONS , GRAPHS  

Topics : 

- Graphs , tables and equations that represent functions .

- Domain and range .

- Evaluating functions .

Applications :

- Stock market ; men in workforce .

Key Concepts :

* Data - Scatter plot – Inputs , Outputs - Function ( Graphically , Numerically , Analytically defined ) – Domain and Range - Functional Notation - The conditions of domains . 


LECTURE 

1.2            GRAPHS - MATHEMATICAL MODELS

Topics : 

- Graphing and evaluating functions with technology .

- Graphing and evaluating mathematical models .

Applications :

- Cost-Benefit , Bankruptcies .

Key Concepts :

* Point-Ploting methods – Graphing - Mathematical Models - Aligning Data - Evaluating functions with technology – Scatter Plots . 


LECTURE 

1.3             LINEAR FUNCTIONS 

Topics : 

- Linear functions .

- Intercepts and slopes .

- Constant rates of change .

Applications :

- Loan Balances , Temperature measurements .

Key concepts :

Linear functions – x and y-intercepts – Slope of the line - Linear Models - Slope-Intercept form – Rate of change – Special Functions .


LECTURE 

1.4             EQUATIONS OF LINES   

Topics : 

- Equations of lines .

- Constant and average rates of change .

Applications :

- Depreciation , Toyota sales .
Key concepts :

Equations of Lines – Slope-intercept form   - General form – Point -Slope form - Linear Models - Constant Rates of change – Average Rates of change  .


LECTURE 

1.5             SOLUTIONS OF LINEAR EQUATIONS   

Topics : 

- Solutions , zeros and x-intercept .

- Solving linear equations .

- Functional forms of equations in two variables .

Applications :

- Credit card debt , Simple interest .

Key concepts :

Solution of linear equations algebraically – graphically  – Solution of equations for a specified linear variable  .


LECTURE  

1.6            FITTING LINES  -  MODELING LINEAR FUNCTIONS  

Topics : 

- Linear Regression .

- Modeling and applying with linear equations .

Applications :

- SAT score , Health Service  .
Key concepts :

Fitting lines to data points – Linear Regression ( Least squares method ) – Applying Models  – Constant first Difference .



LECTURE  

1.7             MODELS IN BUSINESS AND ECONOMICS  

Topics : 

- Modeling Revenue , Cost and Profit .

- Break Even – Marginal Revenue , Cost and Profit , Supply and Demand .

- Market Equilibrium   .

Applications :

- Break Even in Production and Sales , Marginal Revenue and Cost , Market Equilibrium .

Key concepts :

Total Revenue – Total Cost – Profit – Marginal Profit – Marginal Revenue – Marginal Cost – Break Even


LECTURE  

1.8             SOLUTIONS OF LINEAR INEQUALITIES    

Topics : 

- Solving linear inequalities .

- Solving double (system) inequalities .

Applications :

- Profit , Body temperature .

Key concepts :

Linear inequalities – Algebraic solution of an inequality – Graphical solution of an inequality  - Double Inequalities .





LECTURE  

 

2.1            QUADRATIC FUNCTIONS  ,  PARABOLAS 

 

Topics : 

 

- Quadratic function.

- Vertex of Parabola .

- Zero of Quadratic function .

 

Applications :

- Maximum revenue from sales  .

 

Key Concepts :

*Quadratic function – Parabola – Zero of  Quadratic function. 

 

LECTURE  

 

 2.2   QUADRATIC EQUATIONS  

 

Topics : 

 

- Solving Quadratic equations by algebraic method .

- Solving Quadratic equations by graphic method .

 

Applications :

- Profit  , Break-Even .

 

Key Concepts :

* Solving Quadratic equations  by graphic    algebraic method .

 


LECTURE  

 

2.3   POWER FUNCTIONS  AND  TRANSFORMATIONS 

 

Topics : 

 

- Power functions and root functions .

- Increasing and decreasing functions .

- Symmetry .

- Transformations of graphs .

 

Applications :

- Magnetude of stimulus and response .

- Mutual funds  .

 

Key Concepts :

* Power functions –Increasing and decreasing – Root functions – Transformations ( Shift , Compression , Stretch , Reflection )

- Radical equations   .

 

LECTURE

 

2.4   QUADRATIC  AND  POWER  MODELS 

 

Topics : 

 

- Quadratic functions.

- Comparing linear and quadratic models  .

- Modeling with power functions .

- Comparing quadratic and power models.

 

Applications :

- Auto noise .

- Voting  .

 

Key Concepts :

* Modeling quadratic functions - power functions – Comparison of models – First and second differences - Comparison of quadratic and power models

 

LECTURE  

 

2.5  RECIPROCAL ,  ABSOLUTE VALUE – PIECEWISE -DEFINED FUNCTIONS  

 

Topics : 

 

- Reciprocal functions  .

- Absolute value and Piecewise-defined functions .

 

 

Applications :

- Cost - Benefit  .

- Average cost  .

- Residental power cost .

 

Key Concepts :

* Reciprocal functions – Average cost function – Piecewise-defined functions – Absolute value functions – Solving absolute value equations.

 

LECTURE   

 

2.6   INVERSE  FUNCTIONS   

 

Topics : 

 

- Inverse functions.

- Finding and graphing the inverse of a functions .

 

 

Applications :

- Loan balances   .

- Money conversion  .

 

Key Concepts :

* Inverse  functions – Horizontal line test  – Finding the inverse functions – Graphs of inverse functions.

 

LECTURE   

 

2.7   QUADRATIC AND POWER INEQUALITIES    

 

Topics : 

 

- Solving quadratic inequalities analytically and graphically .

- Solving power inequalities .

 

Applications :

- Internet users .

- Investment.

 

Key Concepts :

* Quadratic inequalities - Solving analytically –graphically – Power inequalities .

 


 

LECTURE  

3.1   EXPONENTIAL FUNCTIONS  

Topics : 

- Exponential functions . 

- Growth and decay .

- Transformation of graphs .

- Spreadsheet and the number e .

Applications :

- Federal funds.
- Sales decay .
Key Concepts :

* Exponential growth function – Horizontal asymptote – Transformation of graphs of exponential functions – The number e .



LECTURE  

3.2   LOGARITHMIC FUNCTIONS

Topics : 

- Logarithmic functions . 

- Common and natural logarithms .

- Change of base .
Applications :

- Richter scale .

- Doubling time for investments .

Key Concepts :

* Logarithmic functions – Base – logarithm – Exponent - Exponential and logarithmic forms - Common and natural logarithms - Change of base .  


LECTURE  

3.3   EXPONENTIAL EQUATIONS – PROPERTIES OF LOGARITHMS


Topics : 

- Logarithmic properties .

- Solving exponential and logarithmic equations . 

- Exponential inequalities .

Applications :

- Measuring earthquake intensity .

-Sales decay .
Key Concepts :

* Solving exponential equations–Properties of Logarithms .   



LECTURE  


3.4   EXPONENTIAL AND LOGARITHMIC MODELS  

Topics : 

- Modeling with exponential equations .

- Exponential and Logarithmic models .

- Comparison of models .

Applications :

-Sales decay .
- Inflation .
Key Concepts :

* Modeling with exponential equations – Comparing quadratic and exponential models - Logarithmic modeling .  


LECTURE 

3.5   EXPONENTIAL FUNCTIONS AND INVESTING  

Topics : 

- Compound interest .

- Continuous compounding . 

- Present value .

- Investment models .

Applications :

-Future value of an account .

- Continuous compounding .

- Present value of an investment .

- Mutual fund growth .

Key Concepts :

* Future value of investment – Interest compounded annually – Interest compounded k-times per year - Interest compounded continuous - Present value of an investment – Investment models .  


LECTURE  


3.6  ANNUITIES – LOAN REPAYMENT  


Topics : 

- Future value and Present value of an annuity .

- Amortization .
Applications :

- Future value of an ordinary annuity .

- Present value of an ordinary annuity .

- Home Mortage .
- Loan Repayment .
Key Concepts :

* Annuity – Ordinary Annuity – Present value of an investment - Present value of an ordinary annuity - Loan Repayment – Amortization .  


LECTURE 

3.7    LOGISTIC AND GOMPERTZ FUNCTIONS     

Topics : 

- Logistic growth and decay functions .

- Gompertz functions .

Applications :

- Company growth .
Key Concepts :

* Logistic functions  – Limiting value of logistic functions – Modeling logistic functions – Gompertz functions .  

 




LECTURE 

 

4.1    POLYNOMIAL  FUNCTIONS   

 

Topics : 

 

- Polynomial functions .

- Cubic functions . 

- Quartic functions .

 

 

Applications :

- United Nations debt .

- Foreign born population .

 

 

Key Concepts :

*  Polynomial functions –Cubic functions - Quartic functions – Local extrema ( local minimum , local maximum , absolute minimum, absolute maximum – Increasing and decreasing functions .  

 

LECTURE 

 

4.2    CUBIC AND  QUARTIC  FUNCTIONS     

 

Topics : 

 

- Modeling with cubic and quartic functions .

- Model comparison .

 

 

Applications :

- Sales of tobacco .

- US foreign born population .

 

Key Concepts :

*Cubic  models – Quartic  models  .

 

LECTURE 

 

4.3     POLYNOMIAL EQUATIONS     

 

Topics : 

 

- Solving polynomial equations .

- Estimating solutions with technology .

 

Applications :

- Cost .

- FV of an investment  .

- Stock prices  .

 

Key Concepts :

*  Solving polynomial equations – Estimating solutions with technology . 

 

LECTURE 

 

4.4                 SOLUTIONS OF POLYNOMIAL EQUATIONS 

 

Topics : 

 

 

- Solving cubic and quartic equations .

- Combining graphical and analytical methods .

 

Applications :

- Break-even revenue .

 

Key Concepts :

*  Polynomial division – Using division to solve cubic equations – Combining graphical and analytical methods  – Solving quartic equations  . 

 

LECTURE

 

4.5                 COMPLEX  SOLUTIONS  -  FUNDAMENTAL THEOREM   OF  ALGEBRA  

 

Topics : 

 

- Fundamental Theorem of Algebra .

- Imaginary and complex numbers .

- Operations with complex numbers .

- Quadratic and Polynomial equations with the complex solutions .

 

Applications :

- Fractal .

 

Key Concepts :

*  Imaginary and complex numbers – Operation with complex numbers – Quadratic equations with the complex solutions – Polynomial equations with complex solutions – Fundamental Theorem of Algebra  . 

 

 

LECTURE

 

4.6            RATIONAL FUNCTIONS  AND RATIONAL EQUATIONS

         

Topics : 

 

- Rational functions .

- Solutions of rational equations .

 

Applications :

- Average cost of production .

- Cost - Benefit .

- Advertising and Sales .

 

Key Concepts :

*  Graph of rational functions – Vertical asymptotes  – Horizontal asymptotes – Slant asymptotes – Solving the rational equations ( analytically and graphically )  . 

 

LECTURE

 

4.7            POLYNOMIAL AND RATIONAL INEQUALITIES .

         

Topics : 

 

- Polynomial inequalities .

- Rational inequalities .

 

Applications :

- Average cost of production .

 

Key Concepts :

*  Polynomial inequalities ( graphical solutions , analytical solutions )– Rational inequalities ( graphical solutions , analytical solutions )   .

 



LECTURE 

 

5.1            SYSTEMS OF LINEAR  EQUATIONS IN  2  VARIABLES

         

Topics : 

 

- Graphical solution of systems .

- Modeling systems of linear equations .

- Inconsistent and dependent systems .

 

Applications :

- Break-even .

- Market equilibrium .

- Investing .

 

Key Concepts :

*  Solution of systems of equations  (  by graphing , substitution , elimination )– Type of systems  ( unique solutions , dependent systems , inconsistent systems ) – Market equilibrium ( demand , supply ) – Modeling systems of linear equations .

 

 

LECTURE 

 

5.2            SYSTEMS OF LINEAR  EQUATIONS IN  3  VARIABLES  -  MATRIX  SOLUTION 

         

Topics : 

 

- Left-to-right elimination method .

- Matrix representation of systems of linear equations .

- Solving systems with matrices.

- Gauss-Jordan elimination method  .

 

 

Applications :

- Product transportation .

- Manufacturing .

- Investment .

 

Key Concepts :

*  Left-to-right elimination method – Matrix representation of systems of linear equations  ( matrix , dimensions , square matrix , indentity matrix , augmented matrix , coefficient matrix ) – Gauss-Jordan elimination method  – Solution with technology  .

 

 

LECTURE 

 

5.3            SYSTEMS  OF LINEAR  EQUATIONS  WITH  NON-UNIQUE  SOLUTIONS 

         

Topics : 

 

- Non-unique solutions .

- Dependent systems of linear equations .

- Inconsistent systems .

 

Applications :

- Purchashing .

- Investment .

- Transportation .

 

Key Concepts :

*  Non-unique solutions – Dependent systems – Inconsistent systems  .

 

 

LECTURE 

 

5.4            MATRICES – BASIC  OPERATIONS 

         

Topics : 

 

- Addition and substraction of matrices  .

- Multiplication of a matrix by a number   .

- Matrix multiplication .

- Multiplication with technology  .

 

Applications :

- Pricing .

- Trade Balance .

- Advertising .

- Manufacturing .

 

Key Concepts :

*  Matrix operations  ( Addition , Substraction , Multiplication by a constant – Matrix multiplication )  .

 

LECTURE 

 

5.5            INVERSE MATRICES – MATRIX  EQUATIONS 

         

Topics : 

 

- Inverse matrices  .

- Inverse and technology   .

- Solving matrix equations .

- Matrix equations with technology  .

 

Applications :

- Manufacturing .

- Investment .

 

 

Key Concepts :

*  Inverse matrices  -  Inverse and technology - Solving matrix equations .

 



LECTURE 

 

6.1              SYSTEMS OF LINEAR  INEQUALITIES 

         

Topics : 

 

- Linear inequalities in 2 variables  .

- Systems  of linear inequalities in 2 variables .

 

Applications :

- Auto purchases  .

- Car rental .

- Advertising .

 

Key Concepts :

*  Linear inequalities in 2 variables  ( Solution region , test point ) – Systems  of linear inequalities in 2 variables ( border , corner )  .

 

 

 

LECTURE 

 

6.2  LINEAR  PROGRAMMING  -  GRAPHICAL METHODS 

         

Topics : 

 

- Linear programming  .

- Solutions with technology .

 

Applications :

- Maximizing Car rental profit  .

- Cost Minimization .

- Profit  .

 

Key Concepts :

*  Linear programming   – Constraints  – Objective function  – Feasible solution  – Feasible region – Optimum value ( Maximum , minimum )   .

 

 

LECTURE 

 

6.3              SEQUENCES   -  DISCRETE  FUNCTIONS  

         

Topics : 

 

- Sequences  .

- Arithmetic Sequences .

- Geometric Sequences .

 

Applications :

- Investment  .

- Depreciation .

- Compound Interest  .

 

Key Concepts :

*  Sequences   – Arithmetic Sequences  – Geometric Sequences   .

 

 

LECTURE 

 

6.4            SERIES  

         

Topics : 

 

-  Finite and Infinite Series  .

- Arithmetic and Geometric Series .

- Infinite Geometric Series

 

Applications :

- Profit .

- Depreciation .

- Annuities  .

 

Key Concepts :

*  Finite Series   – Infinite Series -  Sigma Notation  -  Arithmetic Series  – Geometric Series - Infinite Geometric Series   .

 

 

LECTURE 

 

6.5            PREPARING  FOR  CALCULUS  

         

Topics : 

 

- Simplifying expressions and fractions .

- Evaluating Functions .

- Rewriting Radical  .

- Decomposition of Functions  .

- Rewriting Expressions using negative and positive exponents .

- Using logarithmic properties .

- Reducing Fractions .

 

 

Applications :

 

- The equation of tangent line .

 

Key Concepts :

* Chapter 1 Skills    – Chapter 2 Skills  – Chapter 3 Skills – Chapter 4 Skills – Chapter 5 Skills  .

 



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