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Advanced Algebra - Introduction to Calculus




  • Real numbers system Review
  • Functions:                                                             i. Domain and range; evaluations; increasing and decreasing; interval notation;  even and odd; vertical line test; transformations (including reflections,  translations and dilations); basic graphs; combinations (sum, difference, product,  quotient and composition); inverse; horizontal line test;                                                                   ii. Linear review - equations; properties of slopes iii. Quadratic review - equations; factors; formula; discriminant; completing the  square; standard form for transformations.                    iv. Polynomials of degree greater than 2 - graphs; equations and solutions (zeros or intercepts); synthetic division; remainder theorem; factor theorem; rational zero theorem; Descartes rule of signs; bounds for roots.                                       v. Rational functions - domain and range; graphs; asymptote criteria (vertical, horizontal and slant); symmetry;                                                         vi. Exponential and Logarithmic - domain and range; graphs; solving equations; properties; different bases; simplifying; natural exponential and natural logarithmic functions; applications involving growth and decay.
  • Polar coordinate system:                        trigonometric functions and the unit circle; radian measure; periodicity; conversions from rectangular to polar and polar to rectangular in real numbers; graphs.
  • Complex numbers:                                      operations in a + bi form; conjugates; modulus; geometric representations; polar form; conversions from rectangular to polar and polar to rectangular form;multiplication  and division in polar form; powers and roots (De Moivre's theorem); roots of unity.
  • Additional topics;                              mathematical induction; conic sections (standard equations); finite sums and series ( including arithmetic and geometric sums and sums to infinity); review of binomial  expansion.


  • The Derivative:                                               Slope of curved lines; general limits; tangents to a curve at a point; limit (as h approaches 0) of    [f(x + h) - f(x)]/h; rules of differentiation (power,  product,quotient and chain); simple trigonometric derivatives.
  • Applications of the Derivative:                      including curve sketching; maxima and minima; first and  second derivative test; second derivative; concavity, points of inflection;  tangent and normal; distance and velocity problems; word problems  (including simultaneous equations involving derivatives); related rates.
  • The Integral:                                             Riemann sums and the definite integral; The  Fundamental Theorems of Calculus; rules for integration; simple  trigonometric integrals
  • Applications of the Integral:                       including simple differential equations (general and particular  solutions); areas of plane regions - enclosed by curve and x axis and  enclosed between two curves; volumes of solids of revolution (rotations  about the x axis); accelerated motion, velocity and distance problems.



bookmarkTextbook                 Advanced Mathematical Concepts - Precalculus with Applications: Gordon, Yunker, Crosswhite and Vanatta


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As the time available is limited and the material to be studied extensive, it is possible that not all your questions will be answered in the class time available. It is the student's responsibility to ensure that he understands the fundamentals of each topic covered. This will mean that the student must have the conscientiousness and perseverance to work on his own and with others (students and teacher) to find solutions to the problems he has been assigned. Please ASK me for help if you are having difficulties and cannot proceed. We will arrange a time to discuss the situation.


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