Assignment 4: Real World Quadratic Functions MAT222: Intermediate Algebra Professor Andrea Grych Assignment 4: Real World Quadratic Functions Managers and business people use quadratic equations on a daily basis in order to find out how much of a profit can be made. The following problem is an example of that. On page number 666 of the textbook‚ problem number 56 (Dugopolski‚ 2012) states that in order to get maximum profits‚ a chain store manager has been told by the main office
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pangit you CHAPTER 6: INVERSE CIRCULAR FUNCTIONS AND TRIGONOMETRIC EQUATIONS 1. INVERSE CIRCULAR FUNCTIONS • Horizontal Line Test o Any horizontal line will intersect the graph of a one-to-one function in at most one point • Inverse Function o The inverse function of the one-to-one function [pic] is defined as [pic] • Summary of Inverse Functions 1. In a one-to-one function‚ each x-value corresponds to only one y-value and
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MATH INVESTIGATION 4.2 FACTORIZATIONS on the Math Investigator determines if a number is prime or composite. If a number is composite‚ it prints all its factors‚ the number of factors‚ and its prime factorization. The numbers 1‚ 2‚ 4‚ and 6 have 1‚ 2‚ 3‚ and 4 factors‚ respectively: 1 has only 1 as a factor; 2 has 1 and 2 as factors; 4 has 1‚ 2‚ and 4 as factors; and 6 has 1‚ 2‚ 3‚ and 6 as factors. These factors are illustrated by the rectangles shown here. Starting Points for Investigations
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Objective Linear Bounds y Equality l Inequalities l Problem size m m l l y Nonlinear s Gradient needed y n Solver linpro quapro qld qpsolve optim neldermead optim_ga fminsearch optim_sa lsqrsolve leastsq optim/"nd" optim_moga semidef lmisolve Quadratic y l l s Nonlinear Least Squares Min-Max Multi-Obj. Semi-Def. y y l* n l m s l* l* l l optional y n n For the constraint columns‚ the letter "l" means linear‚ the letter "n" means nonlinear and "l*" means linear constraints
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the first three terms and last two or three terms. Last term will be the one where you substitute in r=n. [pic] See what cancels [pic] • Properties of the roots of polynomial equations If [pic] and [pic] are the roots of quadratic equation [pic]‚ then [pic] [pic] If [pic]‚ [pic] and [pic] are the roots of cubic equation [pic]‚ then [pic] [pic] [pic] If [pic]‚ [pic]‚ [pic] and [pic] are the roots of quartic equation [pic]‚ then [pic] [pic]
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volume. The concept used to solve the problem To calculate the volume of the cube‚ length x breadth x height was utilized. This gave a cubic equation. This equation was then differentiated which gave a quadratic formula. dydx was then equated = 0. The quadratic was then solved using the quadratic formula ( x=-b±b2-4ac2a) to obtain two values of (x). These values were then substituted into the second differential (d2ydx2). If the value substituted produced a negative value‚ then that will be the
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b b 16. a0 = 1 where a ∈ R‚ a = 0 1 1 17. a−n = n ‚ an = −n a a √ 18. ap/q = q ap 19. If am = an and a = ±1‚ a = 0 then m = n 20. If an = bn where n = 0‚ then a = ±b √ √ √ √ 21. If x‚ y are quadratic surds and if a + x = y‚ then a = 0 and x = y √ √ √ √ 22. If x‚ y are quadratic surds and if a + x = b + y then a = b and x = y 23. If a‚ m‚ n are positive real numbers and a = 1‚ then loga mn = loga m+loga n m 24. If a‚ m‚ n are positive real numbers‚ a = 1‚ then loga = loga m −
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1.1 SOLUTIONS Notes: The key exercises are 7 (or 11 or 12)‚ 19–22‚ and 25. For brevity‚ the symbols R1‚ R2‚…‚ stand for row 1 (or equation 1)‚ row 2 (or equation 2)‚ and so on. Additional notes are at the end of the section. 1. x1 + 5 x2 = 7 −2 x1 − 7 x2 = −5 1 −2 5 −7 7 −5 x1 + 5 x2 = 7 Replace R2 by R2 + (2)R1 and obtain: 3x2 = 9 x1 + 5 x2 = 7 x2 = 3 x1 1 0 1 0 1 0 5 3 5 1 0 1 7 9 7 3 −8 3 Scale R2 by 1/3: Replace R1 by R1
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questions. Time allowed : 60 minutes. There are 3 sections: 15 questions in section I‚ 15 in section II and 20 in section III. Section – I (Mental Ability) : Sets‚ Relations and Functions‚ Mathematical Induction‚ Logarithms‚ Complex Numbers & Quadratic Equations‚ Linear Inequations‚ Differentiation‚ Sequences and Series (A.P. & G.P. Misc.)‚ Trigonometry‚ Cartesian System of Rectangular Coordinates‚ Straight Lines and Family of Straight Lines‚ Circles‚ Conic Section‚ Trigonometry‚ Permutations and
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the drawing. It is a simple and direct way of getting the resultant force but is limited in precision. Component method can obtain the resultant force by getting two directions at right angles to each other and getting their summations using the Pythagorean Theorem. In getting the equilibrant of the given forces‚ a force table can be used (see Fig. 1.1). The equilibrant of a set of forces is the single force that must be obtained with the set of forces to maintain in the system in equilibrium.
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