OMPT-D Syllabus

Chapter 1: Numbers [Prerequisite Chapter] (17 topics)

1. Integers (3 topics)

1. Calculating with integers
2. Integers
3. Division of integers

2. Negative numbers (1 topic)

1. Absolute value

3. Fractions (7 topics)

1. Fractions
2. Equivalent fractions
3. Simplifying fractions
4. Addition and subtraction of fractions
5. Multiplication and division of fractions
6. Integer powers of fractions
7. Decimal numbers

4. Powers and roots (6 topics)

1. Exponents
2. Calculating with powers
3. Roots of integers
4. Roots of fractions
5. Standard notation of higher roots
6. Order of operations for powers and roots

Chapter 2: Algebra [Prerequisite Chapter] (28 topics)

5. Variables (6 topics)

1. Variables
2. Sum and product of variables
3. Substitution
4. Simplification of multiplications
5. Simplification of additions and subtractions
6. Simplification with algebraic rules

6. Calculating with exponents and roots (7 topics)

1. Integer powers
2. Rules for calculating with integer exponents
3. Square roots
4. Rules for calculating with square roots
5. Higher degree roots
6. Calculating with fractional exponents
7. Order of operations

7. Expanding brackets (2 topics)

1. Expanding single brackets
2. Expanding double brackets

8. Factorization (2 topics)

1. Factorization
2. Product sum method

9. Notable products (2 topics)

1. The square sum of a sum or a difference
2. The difference of two squares

10. Adding and subtracting fractions (9 topics)

1. Fractions with variables
2. Similar fractions
3. Simplifying fractions
4. Addition and subtraction of like fractions
5. Making fractions similar
6. Addition and subtraction of fractions
7. Multiplication of fractions
8. Division of fractions
9. Fraction decomposition

Chapter 3: Linear formulas and equations [Prerequisite Chapter] (17 topics)

11. Formulas (3 topics)

1. Formulas
2. Dependent and independent variables
3. Graph of a formula

12. Linear functions (6 topics)

1. Linear formulas
2. Slope
3. Determining the slope from two points
4. Intercept
5. Composing a linear formula
6. Parallel and intersecting linear formulas

13. Linear equations and inequalities (8 topics)

1. Linear equations
2. Equivalent linear equations
3. The general solution of a linear equation (rules of reduction)
4. Intersection points of linear formulas with the axes
5. Intersection point of two linear formulas
6. Linear inequalities
7. Equivalent inequalities
8. General solution of a linear inequality

Chapter 4: System of linear equations [Prerequisite Chapter] (10 topics)

14. An equation of a line (4 topics)

1. A linear equation with two unknowns
2. Solution of linear equations with two unknowns (rules of reduction)
3. The equation of a line
4. Composing the equation of a line

15. Two equations with two unknowns (6 topics)

1. Systems of linear equations
2. Solving systems of linear equations by substitution
3. Solving systems of equations by elimination
4. Consistent systems of linear equations
5. Inconsistent systems of linear equations
6. Dependent systems of linear equations

Chapter 5: Quadratic equations (21 topics)

16. Parabola (3 topics)

1. Quadratics
2. Parabola
3. Plotting the graph of the quadratic

17. Solving quadratic equations (7 topics)

1. Quadratic equations with two solutions
2. Quadratic equations with one solution
3. Quadratic equations with no solutions
4. Solving quadratic equations by factorization
5. Solving quadratic equations by completing the square
6. The quadratic formula and the discriminant
7. Solving quadratic equations using the quadratic formula

18. Drawing parabolas (7 topics)

1. Intersection of parabolas with the axes
2. Vertex of a parabola
3. Determining the vertex by completing the square
4. Determining the vertex by substitution
5. Drawing of parabolas
6. Transformations of parabolas
7. Multiple transformations of parabolas

19. Intersection points of parabolas (2 topics)

1. Intersection points of a parabola with a line
2. Intersection points of parabolas

20. Quadratic inequalities (2 topics)

1. Quadratic inequalities
2. Solving a quadratic inequality

Chapter 6: Functions (40 topics)

21. Domain and range (6 topics)

1. Function and formula
2. Function rule
3. Intervals
4. Domain
5. Limited domain
6. Range

22. Power functions (4 topics)

1. Power functions
2. Transformations of power functions
3. Multiple transformations of power functions
4. Equations with power functions

23. Higher degree polynomials (7 topics)

1. Polynomials
2. The leading coefficient
3. Equations with polynomials
4. Solving higher degree polynomials by factoring out
5. Solving higher degree polynomials with factorization
6. Solving higher degree polynomials with the quadratic equation
7. Higher degree inequalities

24. Root functions (7 topics)

1. Root functions
2. Transformations of root functions (upwards, to the right, and relative to the x-axis)
3. Multiple transformations of root functions
4. Root equations
5. Solving root equations with substitution
6. Inverse functions
7. Determining the inverse function

25. Fractional functions (11 topics)

1. Asymptotes and hyperbolas
2. Power functions with negative exponents
3. Transformations of power functions with negative exponents (upwards, to the right, and relative to the x-axis)
4. Linear fractional functions
5. Determining asymptotes of a linear fractional function
6. Linear fractional equations
7. Solving linear fractional equations by cross multiplication
8. Inverse of linear fractional functions
9. Isolating a variable when determining the inverse
10. Quotient functions
11. Perforation

25. Limits and asymptotes (5 topics)

1. The concept of a limit
2. Limits in connection with asymptotes
3. Horizontal asymptotes
4. Vertical asymptotes
5. Diagonal asymptotes

Chapter 7: Exponential functions and logarithms (14 topics)

26. Exponential functions (3 topics)

1. The exponential function
2. Exponential equations
3. Transformations of the exponential function (upwards, to the right, and relative to the y-axis)

27. Logarithmic functions (11 topics)

1. The logarithmic function
2. Logarithmic equations
3. Exponential equations
4. Isolating variables
5. Rules for solving logarithms
6. Solving logarithmic equations with calculation rules and the quadratic formula
7. Change of base
8. Solving logarithmic equations using substitution
9. Solving exponential equations using substitution
10. Graph of logarithmic functions
11. Transformations of the logarithmic function (upwards, to the right, and relative to the x-axis)

Chapter 8: Differentiation (34 topics)

28. The derivative (6 topics)

1. The difference quotient
2. The difference quotient at a point
3. The difference quotient on an interval of length h
4. The tangent line
5. Finding the slope using the difference quotient
6. Calculating the derivative of a function

29. The derivative of power functions (4 topics)

1. The derivative of power functions
2. The power rule
3. The derivative of the root
4. The power rule with a constant

30. Sum and product rule (4 topics)

1. The constant rule
2. The sum rule
3. The product rule
4. The derivative of a product

31. Chain rule (2 topics)

1. Composite functions
2. The chain rule

32. The derivative of standard functions (8 topics)

1. The derivative of sine
2. The derivative of cosine
3. The derivative of tangent
4. The base e
5. The natural logarithm
6. The derivative of exponential functions
7. The derivative of logarithms
8. The derivative of the natural logarithm

33. The quotient rule (1 topic)

1. The quotient rule

34. Applications of derivatives (9 topics)

1. Determining the interval at which a function increases/decreases
2. Extreme values: local maxima and minima
3. Extreme values on a restricted domain
4. Global maxima and minima
5. Calculating extreme values
6. The second derivative
7. Types of increasing and decreasing
8. Inflection points
9. Higher order derivatives

Chapter 9: Trigonometry (27 topics)

35. Angles with sine, cosine, and tangent (16 topics)

1. Angles
2. Right-angled triangles
3. The sum of the three angles of a triangle
4. Pythagorean theorem
5. Sine in a right-angled triangle
6. Cosine in a right-angled triangle
7. Tangent in a right-angled triangle
8. The unit circle
9. Angles in radians
10. Large and negative angles
11. Symmetry in the unit circle
12. Special values of trigonometric functions
13. Addition formulas for trigonometric functions
14. Double-angle formulas
15. Law of sines
16. Law of cosines

36. Trigonometric functions (11 topics)

1. The sine function
2. The cosine function
3. The tangent function
4. Equilibrium, period and amplitude
5. Transformations of trigonometric functions (upwards, to the right, and relative to the x-axis)
6. Inverse function of sine
7. Inverse function of cosine
8. Inverse function of tangent
9. Properties of the arcsin
10. Properties of the arccos
11. Properties of the arctan

Chapter 10: Geometry (20 topics)

35. Lines (4 topics)

1. Different descriptions of a line
2. Angles between lines
3. Perpendicular lines
4. Distance point and line

35. Circles (5 topics)

1. Different descriptions of a circle
2. Intersections of a line and a circle
3. Tangent line to a circle
4. Intersection of circles
5. Distance to a circle

35. Parametric Curves (6 topics)

1. Parametric curves
2. Lissajous figures
3. Vectors
4. Length of vectors
5. Vectors and parametric equations
6. Derivatives of parametric curves

35. More geometry (5 topics)

1. Some notions on lines
2. Different descriptions of a circle
3. Calculations with circles
4. Some notions on triangles
5. Calculations with triangles

Chapter 11: Integration (23 topics)

37. Antiderivatives (7 topics)

1. The antiderivative of a function/the indefinite integral
2. The constant of integration
3. The antiderivative of a power function
4. Sum rule for finding antiderivatives
5. Constant rule for finding antiderivatives
6. Antiderivatives of known functions
7. Antiderivatives and the chain rule

38. The definite integral (8 topics)

1. The definite integral
2. Area
3. Calculating the area of an enclosed domain
4. Calculating the area of a surface between curves
5. Solid of revolution
6. The volume of a solid of revolution
7. The volume of a solid of revolution between two graphs
8. The volume of a solid of revolution around the y-axis

39. Integration techniques (8 topics)

1. Substitution method
2. Trigonometric integrals
3. Integration by parts
4. Repeated integration by parts
5. Repeated integration by parts using reduction
6. Known antiderivatives of some quotient functions
7. Long division with polynomials
8. Finding the antiderivatives of quotient functions