Unit_1 - UNIT 1 ALGEBRA TRIGONOMETRY PARAMETRIC EQUATIONS...

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UNIT 1 ALGEBRA, TRIGONOMETRY, PARAMETRIC EQUATIONS INTRO: This Unit will briefly touch on a smorgasbord of topics from elementary algebra and trigonometry. The notion of parametric equations will be introduced. 1. Review of Some Basic Algebra Solving equations Difficulty with algebra skills is a frequent contributor to poor performance in university math- ematics courses. Although it is clearly impossible to give a meaningful review of problem areas in algebra and trigonometry in just the next few pages, we have chosen to touch on a few topics which arise in this course and have come to our attention as sometimes forgotten. The most fundamental elementary algebra skill is solving an algebraic equation. You should be able to solve for x an equation of the sort a + 1 b = c - a d + e x + c and arrive at the solution x = e ( ba + 1) b ( c - a ) - d ( ba + 1) - c The student is encouraged to try this problem. Obviously, the strategy must be to isolate the variable x on one side of the equation. Perhaps the most common mistake we encounter in simple algebra is mistreatment of compound denominators. That is to say, a fraction such as a b + c can not be simplified, while a + b c = a c + b c . Unfortunately, we see students writing a b + c = a b + a c ( wrong ) This is the same problem which often arises in inverting an expression. For example, the equation 1 y = a + 1 a + b can be solved for y by inverting both sides of the equation. y = 1 a + 1 a + b = a + b a ( a + b ) + 1 = a + b a 2 + ab + 1
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where the compound denominator was simplified by multiplying both numerator and denom- inator by ( a + b ). Needless to say, if we had begun to simplify the right hand side by writing 1 a + 1 a + b = 1 a + a + b ( wrong ) this would have been completely wrong. Note that combining different algebraic fractions can usually only be done by multiplying numerator and denominator of each fraction in such a way as to create a common denominator. 1 a + 1 a + b + 1 c = 1 a · ( a + b ) c ( a + b ) c + 1 a + b · ac ac + 1 c · a ( a + b ) a ( a + b ) = ( a + b ) c + ac + a ( a + b ) a ( a + b ) c = = 2 ac + bc + a 2 + ab a 2 c + abc Similarly, a fraction in the denominator can be simplified by multiplying the entire numerator and the entire denominator by the common denominator of the fractions in the denominator. a + c 1 a + 1 a + b = a + c 1 a + 1 a + b · a ( a + b ) a ( a + b ) = ( a + c ) a ( a + b ) 1 a · a ( a + b ) + 1 a + b · a ( a + b ) = ( a + c )( a 2 + ab ) ( a + b ) + a = = a 3 + a 2 b + a 2 c + abc 2 a + b Quadratic equations The standard equations studied in high school algebra in two dimensions involving quadratic powers of the variables are the following. In these a constant may be written as a 2 or b 2 to indicate it is positive, since squares of real numbers are always positive. Constants written as a or b may be either positive or negative. a 2 x 2 + a 2 y 2 = c 2 (circle) a 2 x 2 + b 2 y 2 = c 2 (ellipse) ( a 2 x 2 - b 2 y 2 = c 2 b 2 y 2 - a 2 x 2 = c 2 (hyperbola) ( y = a x 2 + b x + c x = a y 2 + b y + c (parabola) xy = c (hyperbola)
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The scientist or engineer should be able to name and sketch these equations on sight. This is most efficiently done by knowing the possible shapes and then checking the easy cases x = 0 and y = 0. For example, let us sketch the graph of the ellipse 9 x 2 + y 2 = 9. An ellipse is always
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