Algebra and Precalc Review

Algebra and Precalc Review - Algebra and Precalculus Review...

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Page 1 of 9 Algebra and Precalculus Review Aug 21, 2011 Things you Should know for Differential Calculus 8/12/2011 Formulas and Rules you are expected to know. Zero Product rule: or 0 = AB 0 = A 0 = B . Note: The symbol means if and only if. Trichotomy: Given an arbitrary real number, say x , then x is either positive ( ), negative ( ), or zero ( ). 0 > x 0 < x 0 = x The definition of absolute value _____________________ The Quadratic formula: _________________________ The Pythagorean Theorem: ________________________ Items from geometry geometric description of line in a plane. geometric description of a circle Formulas for the area and perimeter of the following planar regions Rectangle, Parallelogram, Trapezoid, Triangle, Circle Formulas for the surface area and volume of spheres, rectangular prisms, circular cylinders Trigonometry The definition of Sine, Cosine, and the other 4 trigonometric functions. Right triangle trigonometry Selected identities equivalent notation 1 cos sin 2 2 = + A A ( ) ( ) 1 cos sin 2 2 = + A A A A 2 2 sec 1 tan = + ( ) ( ) 2 2 sec 1 tan A A = + A A 2 2 csc cot 1 = + ( ) ( ) 2 2 csc cot 1 A A = + Double Angle formulas ) )(cos (sin 2 2 sin A A A = A A A 2 2 sin cos 2 cos = 1 ) (cos 2 2 = A A 2 sin 2 1 = and their offspring, the power reduction formulas 2 2 cos 1 sin 2 A A = 2 2 cos 1 cos 2 A A + = Law of Cosines: (draw the triangle and label the appropriate angles and legs per the formula) B ac c a b cos 2 2 2 2 + = Law of Sines: c C b B a A sin sin sin = = (draw the triangle and label the appropriate angles and legs per the formula) A “simplified” expression is one:
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Page 2 of 9 (1) in which all like terms are collected, (2) which does not contain negative exponents, (3) in which common factors are reduced, (4) where the result is not a complex fraction. i.e. Quotients do not contain fractions in the numerator or in the denominator. (5) in which radical expressions are in reduced form. Make sure that you can derive the simplified version in each of these 5 examples. Example 1. violates (1) Simplified expression: ) 4 )( 2 3 ( ) 4 5 )( 4 3 ( 2 x x x x + + 15 4 24 2 + + x x Example 2. 2 2 2 ) 5 ( ) 5 2 )( 1 3 ( ) 3 )( 5 ( x x x x x x + + + the numerator violates (1) Simplified expression: 2 2 2 ) 5 ( 5 2 3 x x x x + + + Example 3. 2 3 2 1 2 16 2 1 x x violates (2) and (3) Simplified expression: 2 3 8 2 1 x x Example 4. 2 3 1 3 2 ) 4 3 ( ) 4 ( ) 4 3 ( 3 1 x x x x violates (1) and (4) Simplified expression: 2 3 2 ) 4 3 ( 3 8 3 x x x + Example 5. 4 3 72 y x Simplified expression 2 2 6 y x x Each simplified expression above is consistent with the simplification requirements, however, further algebraic manipulation may be required depending on what one wishes to do with the expression. For example if one were interested in finding the values of
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Algebra and Precalc Review - Algebra and Precalculus Review...

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