Math1090 Final Review Exercises: Solving Matrices, Sequences, Inequalities, and Functions, Exams of Mathematics

Download Math1090 Final Review Exercises: Solving Matrices, Sequences, Inequalities, and Functions and more Exams Mathematics in PDF only on Docsity! 1 Math1090 Final Review Exercises (from old Final Exams) 1A) Find the inverse of the following matrix, if possible. If it’s not possible, then explain why.         50 21 A 1A _______________________ 1B) For            304 123 201 A ,        21 35 B and        53 24 C , perform the indicated matrix operations, if possible. If not possible, explain why. (a) TAA  TAA  =_________________________ (b) BC BC = __________________________ 1C) Use Gauss-Jordan Elimination to solve the following system. 10342 2794 4242    zyx zyx zyx x = ____________ y = ____________ z = ____________ 2 2A) Given the arithmetic sequence -2, 1, 4, 7, 10, … (a) Find the 100th term. 100th term = _________________ (b) Find the sum of the first 100 terms. Sum of first 100 terms = _________________ 2B) How much would have to be invested at the end of each year at 6% interest compounded annually to pay off a debt of $80,000 in 10 years? $_________________ 2C) A lottery prize worth $1,000,000 is awarded in payments of $10,000 five times a year for 20 years. Suppose the money is worth 20% compounded 5 times per year. (a) What is the interest rate, i? i = _________ (b) What is the number of compoundings, n? n = _________ (c) What is the formula used to find the present value of this prize? _________________________________________________ (d) What is the present value of this prize? $____________________ 5 4C) Find the maximum of the objective function f x , y=2x y subjected to the following constraints. 2432 10 0 0     yx yx y x Maximum value: ______________ at point ___________________ 6 5A) If the cost of production for a product is given by 8411)( 2  xxxC and the revenue is given by xxR 30)(  , (a) Find the profit function P(x). P(x) = ____________________________ (b) Find the break-even point(s). Break-even point(s): ________________________ 5B) If 100 feet of fence is used to fence in a rectangular yard, then the resulting area is given by Ax =x 50−x where x feet is the width of the rectangle and (50 – x) feet is the length. Determine the width and length that give the maximum area. Width for max area = _______________ Length for max area = ______________ 5C) Let f x=−x−124 . (a) Solve f(x) = 0 to find the x-intercepts. x-intercepts: _______________________________ (b) Find the vertex of the parabola. Vertex _______________________ (c) Sketch the graph, showing the vertex and x-intercepts. 7 6A) Suppose that the population of Smalltown, USA grows according to the formula P t =3200e0.025t where time t is measured in years. (a) What is the initial population of the town (at t = 0)? Initial population = ________________ (b) How long will it take the population to double? ________________ years (c) What is the population after 1 year? 6B) Use the properties of logarithms and the fact that 3.02log10  7.05log10  log107≈0.85 to find the values below. (a) 8log10 8log10 _________________ (b) 35log10 35log10 _________________ (c) 2log5 2log5 _________________ 6C) Rewrite x32log 2 in exponential form and solve for x. Exponential form: ____________________________ x = ______________________ 10 15. The total costs for a company to produce and sell x units of a product are given by C x =50050xx2 (in dollars) . The sale price for one item is $250. (a) Find the revenue function, R x . (b) Find the profit function, P x  . (c) Find the break-even point(s). (d) Find the number of items sold to get the maximum profit. 16. The population of Mathville was 12,000 in 1960 and 21,000 in 1980. The population growth of the city follows the formula P t =P0 e ht where t is the number of years after 1960. (a) Determine P0 and h . (b) Estimate the population of Mathville in the year 2000. (c) How many years after 1960 will the population grow to be 34,000? 17. Let y=x 24x3 . (a) Find the vertex of the parabola. (b) Tell if it's a minimum or maximum point. (c) Solve y = 0 to find the x-intercepts, if there are any. (d) Sketch the graph, showing the vertex and x-intercepts. 11 18. Solve for the exact value of x . log3x−2log3 5=3 19. Solve for x. ∣3−4x∣=13 . 20. The Utah Company manufactures a certain product that has a selling price of $40 per unit. Fixed costs are $1,600 and variable costs are $20 per unit. Determine the least number of units that must be sold for the company to have a profit of no less than $5,000. [All work must be shown; the guess-and-test method is not acceptable.] 21. A rectangular plot of land has an area of 18,000 square feet. If its length is five times its width, how much fencing would be required to surround the property? 22. For the following functions, answer the specified questions. f x = x1 3x220x25 g  x=−x (a) What is the domain of f x  ? __________________________________ (b) What is the domain of g  x ? __________________________________ (c) f −1 = _________________________ (d) f 0 = __________________________ (e)  f °g 2 = ________________________ 12 23. Graph the function y=∣x−2∣−1 and determine the x- and y-intercepts. x-intercept: ____________________________ y-intercept: ____________________________ 24. The students at a university buy 3,000 graphing calculators per year when they cost $50 each, and they buy 2,000 calculators per year when they cost $100 each. Let P be the price per calculator and Q be the quantity of calculators sold. Assuming the relationship between P and Q is linear, give an equation expressing P in terms of Q. 25. Find the value for x which maximizes the quadratic function f x =−x211x−24 . 26. Solve the following equations. (a) ln 2x7=0 (b) e2x=9 (c) log xlog 3=2 27. Jeremy wants to make one savings deposit today so that in 7 years, he will have $16,000. Given an interest rate of 4% compounded semiannually (twice a year), how much money should Jeremy deposit?