Fundamentals of Engineering Review RICHARD L. JONES FE MATH REVIEW ALGEBRA AND TRIG 8//00 Introduction - Algebra Cartesian Coordinates Lines and Linear Equations Quadratics Logs and exponents Inequalities and absolute values Simultaneous Equations Conic Sections Complex Numbers 8//00 y b y d Cartesian Coordinates b d x y y y a a x x x x b a y y y x a x b x Pythagorean Theorem b b a b a x y d x x y y a 8//00 Fundamentals of Engineering Review
Slopes of Lines slope m rise y run x y y m x x move from P (x,y ) y y x x to P(x, y) 8//00 Equation of a Line Using the equation for the slope of a line between points, we can develop the general equation for a line. m y y x x y y m x x y mx b m slope b y intercept 8//00 Parallel lines: Perpendicular lines: m - m Linear equation: 0 Ax By C m m As long as the order of both x and y is the equation is that of a straight line. If the order changes then it begins to take on different shapes. If y is of order and x is of order then the equation is Quadratic and will generate a parabola. 6 8//00 Fundamentals of Engineering Review
Example The equation y a a x where x and y are variables, and a is a constant. Which of the following are represented by the equation? (A) a rd order polynomial (B) a quadratic equation (C) a straight line (D) acceleration Answer: The terms with a in them are constants so they don t determine the degree of the function. The implied Degree of both variables is a. The answer is (C). 7 8//00 Example What is the slope of the line: y x Solution Converting to the general form we get 0 y - x - y x y x slope y intercept 8 8//00 y 6x - Example: What is the slope of the line perpendicular to Answer If line A is perpendicular to the line B, then m= - m m 6 m = - - m 6 9 8//00 Fundamentals of Engineering Review
Example What is the distance between the point -,0 and the y intercept of the line y x - 0? Answer : y x b 0, d y y x x 0 0 6 9 0 8//00 Solving Quadratic Equations Using the Quadratic Formula Factoring Complete the squares Matrix methods (later) 8//00 Quadratic Equations: Formula ax bx c 0 (y has been set to 0) Method : Quadratic Formula -b b ac x a Two imaginary solutions (no real solutions): ac > b One solution: ac = b Two real solution s: ac < b 8//00 Fundamentals of Engineering Review
For Example: 0 x x - x - 9 8 - - x, or, 8//00 Example: Will the roots of the equation below be real, complex, or imaginary? 0x x - where x is a real valued variable. 0 0x x - x 0 x -0x 0 0 0 00,60 Roots 0 answer will be complex 8//00 Quadratics: Factoring x x x x 0 x x 0 x x 8//00 Fundamentals of Engineering Review
Quadratics: Completing the square 6 0 x x st, make sure that the coefficient of is. x nd, take the coefficient of x and divide it by x x rd, add the resulting number to both sides of the equation. 8//00 Completing the square (cont) x x x x x or - 7 8//00 Logarithms and Exponentials 8 y y log x a x a a base log x ln x e.788... e irrational number 8//00 Fundamentals of Engineering Review 6
Logarithms and Exponentials (a few properties) 9 a b a b a b a b e e e ln ln ln a a e b b a ab b e e ln a bln a ln b loga b ln a a e ln ln ln a b 8//00 Log/ln example 0 log 7 ln 7 ln.96.099.77 8//00 x Absolute values always positive suppose a b c a c b if c is positive c a c b if c is negative Note that when you multiply both sides by a negative number the inequality changes. 8//00 Fundamentals of Engineering Review 7
Example solve x- x- x if x- 0 so, x x 7 8//00 Example solve x- x- x if x- 0 so, x x 7 x- x if x- 0 so, -x x x 8//00 Example solve x- x- x if x- 0 so, x x 7 x- x if x- 0 so, -x x x solution : x - and x 7 - x 7 8//00 Fundamentals of Engineering Review 8
Common Log Example What is the common log of 000? Identit y : n 0 log 0 n 000 0 0 0 0 log 0 log 0 8//00 x-y Natural Log Example 6 x-y What is the natual log of e? c identity : ln e c ln e x y Note that the natural log is sometimes called the Naperian logarithm 8//00 Conic Sections 7 A B C D x y x y E 0 8//00 Fundamentals of Engineering Review 9
Conic Sections (continued) 0 x xy y A B C Dx Ey 8 F B AC 0 8//00 Why are these called Conic Sections? 9 8//00 Parabola 0 If opens left or right the equation will fit y k p x h p p p 0 means opens right, p 0 means opens left p center at h,k,focus at h+,k p direction at x h - 8//00 Fundamentals of Engineering Review 0
Parabola (cont) If opens up or down the equation will fit p y k x h p 0 means opens up, p 0 means opens down p center at h,k, focus at h,k+ p direction at y k - 8//00 Ellipse B AC 0 x h y k a b center at h, k, width a height x -axis (h,k) h b y -axis b c a 8//00 Example What shape does this equation represent? y y x 0 x xy y x y A B C D E F 0 y x in parabola, either A or C = 0, not both AND B=0 0 0x 0xy Cy 7x 0 y F 0 y x 8 B AC 0 parabola 8//00 Fundamentals of Engineering Review
Complex Numbers ' j' and ' i' are used to represent complex numbers. ' i' is normally used in math and physics while ' j' is normally used in engineering (specifically electrical) i - i - i i j - j - j i 8//00 Complex Numbers Example j7 + 6 j9? 6 j7 j9 0 j6 8//00 Complex Numbers Example 6 Complete the following math equation. Your final answer should be in Polar Notation. i i e 80 tan rad.8.90 8//00 Fundamentals of Engineering Review
Complex Conjugates 7 z* x jy if z x jy z z * x y 8//00 Complex Conjugate example j solve : j j 6 j j j 6 j 9 6 j 8 j j j j 8 8//00 Complex Number Conversions 9 Imaginary j axis A A R ji R Real axis A R I R A cos I tan I A sin R 8//00 Fundamentals of Engineering Review
R ji j Example 0 A 9 6 I tan R tan. A A. 8//00 Simultaneous Equations A set of equations can be solved simultaneously if the number of unknowns is equal to the number of equations. There are several ways to solve them including via Matrix methods which will be discussed later. 8//00 Simultaneous Equations Method. Solve one equation for one variable and then substitute it into the next equation. 7 x y solve : x y x y x y 8//00 Fundamentals of Engineering Review
Simultaneous Equations Method. Solve one equation for one variable and then substitute it into the next equation. 7 x y solve : x y x y x y substituting : 7 y y 7 y y 9 y y 8//00 Simultaneous Equations Method. Solve one equation for one variable and then substitute it into the next equation. 7 x y solve : x y substituting : x y x y 7 y y 7 y y 9 y y and x y x x 8//00 Simultaneous Equations nd method: Multiply the equations by numbers such that when added together, only one variable will be left. 7 = x + y Solve : - = x - y 8//00 Fundamentals of Engineering Review
Simultaneous Equations 6 nd method: Multiply the equations by numbers such that when added together, only one variable will be left. 7 = x + y Solve : - = x - y Multiply both sides of second eqn by - 7 x y x y 8//00 Simultaneous Equations 7 nd method: Multiply the equations by numbers such that when added together, only one variable will be left. 7 = x + y Solve : - = x - y Multiply both sides of second eqn by - 7 x y x y 7 x y x y 8//00 Simultaneous Equations 8 nd method: Multiply the equations by numbers such that when added together, only one variable will be left. 7 = x + y Solve : - = x - y Multiply both sides of second eqn by - 7 x y x y 7 x y x y 9 0x y 8//00 Fundamentals of Engineering Review 6
Simultaneous Equations 9 nd method: Multiply the equations by numbers such that when added together, only one variable will be left. 7 x y x y 9 0x y y 9 x and y x 8//00 Previous Example Modified 0 A second look at the equations with a clearer mind: The equations already are set up such that if they were added y would fall out without any help. 7 x y x y 8//00 Previous Example Modified A second look at the equations with a clearer mind: The equations already are set up such that if they were added y would fall out without any help. 7 x y x y 6 x 8//00 Fundamentals of Engineering Review 7
Previous Example Modified A second look at the equations with a clearer mind: The equations already are set up such that if they were added y would fall out without any help. 7 x y x y 6 x x x y y y 8//00 Unit Circle Triangles A few identities Trigonometry basics 8//00 0,0 The Unit Circle 6,0 r C r Units radians 8//00 Fundamentals of Engineering Review 8
Y x P Distance d from point,0 is point P on the circle. y d sin d y coordinate X cos d x coordinate tan cot d d d d sin y sec d cos x cos x d d x c sec d tan y sin d y 8//00 Triangle point of view 6 sin cos opp hyp adj hyp tan sin cos cot tan opp adj adj opp Hyp adj sec cos csc sin hyp adj hyp opp opp 8//00 c x y A few identities hyp adj opp 7 adj op p opp hyp hyp adj cos sin cos sin sin sin cos cos sin Hyp 8//00 Fundamentals of Engineering Review 9