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wewew - May 16 MATH 113 ‘ Total Points 200 2005 FINAL...

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Unformatted text preview: May 16 MATH 113 ‘ Total Points: 200 2005 FINAL EXAM Follow directions carefully: Write your name section number and instructor’s name on this test and on each of the 7 answer sheets. Number the answer sheets 1 through 7 Do the problems on the answer sheets as directed on the test. Mark your answers clearly. You must show all appropriate work in order to receive credit for an answer. You may use graphing calculators, but show work algebraically and give exact answers where indicated. Work each of the following problems on the answer sheets as indicated. Use both front and back of a sheet if needed. Do i not use another answer sheet. Only the problems assigned to a particular sheet should appear on that sheet as indicated on the ¢ exam. Show all appropriate work. Clearly indicate your answer. If more than one answer is given, or if the answer cannot be found, you may lose points. POINTS MAY BE DEDUCTED IF DIRECTIONS ARE NOT FOLLOWED. Answer problem 1 on answer sheet #1 l. (a) Determine the slope and the y-intercept of the graph of the equation 2y + 6x = 14 l [5 6a.] (b) Is the line 2y + 6x = 14 fiom the problem above parallel, perpendicular, or neither parallel nor perpendicular to the line that passes through the points (2,3) and (-l,2)? Show algebraically how you determine your answer. (c) The diameter of a circle has endpoints (5, -4) and (1, »8). What are the coordinates cf the center of the circle? (d) Find the length of the radius of the circle where the endpoints of a diameter are the points given above. Write your answer as an exact answer. (6) Write the equation of the horizontal line that passes through the point (7, -10) (i) Write an equation for the function whose graph is a result of shifting the graph of the fimction y = 2" 5 units to the right and 6 units down. = x2 — 4x — 21 Determine a1 ebraicall the x- and - interce ts s of the a hs of the followin function: Answer problem 2 on answer sheet #2 1 - - - 1 —3 2. [6] (a) Simplify, if possible, writing with only positive exponents: [$) 2 + V4t2 +16 + aza 4 7 _ l [5] (b) Simplify: iii—”‘- 1 — + 2 x [6] (c) Solve for x: (x -- 4)(x +1) = 6 l 1 1 [6] (d) Solve for r: - + — = 2 r I7 f(X+h)-f(X) [5] (e) Given the function f (x) = 3x + 4 , find and simplify the difference quotient h 3 [4] (0 Evaluate: 201 + 2)! n=1 Answer problem 3 on answer sheet # 3 3. (a) Determine the zeros of the function f (x) = x3 - 3x2 — 4x [5] (b) If x = 5 is a zero of multiplicity 3 of a polynomial function, )2 then what do you know about the graph of f at the point where x is 5? [4] (c) What is the vertex of the function f (x) = 3(x + 6)2 — 7 ? [4] (d) Is the graph of the function f (x) = x3 — x symmetric with respect to the origin? Justifl your answer algebraically ‘ [4] ax + b _ r (e) f (x) = + F 1nd values for a, b, and e such that : The lines x = 3 and y = -2 are asymptotes of the x C graph of f, and the y-intercept is the point (0, - 2). [5] (f) The profit a company makes is given by the function P = 1000 + 300x —- 0.03362 where x is the amount the company spends on advertising. [6] (i) Determine the amount the company should spend on advertising in order have the maximum possible profit. (ii) What is the maximum possible profit ? Answer problem 4 on answer sheet #4 4. (a) For the function, f, whose graph is on the right below: (i) State the domain of f. [2] (ii) State the range of f. [2] (iii) Determine the value of f (—6) [2] (iv) Determine the interval(s) over which f is increasing. [4] (v) State whether the function, f, appears to be even, odd, or neither. Specify how you arrive at your answer. [3] (vi) Does this function have an inverse? Explain how you can tell. [3] (b) Determine algebraically the x- and y- intercepts(s) of the graphs of the following functions: . ,, 7 + x (1) f (x) = 11106 + 1) - 3 (n) g(X) = 2 _ x [6] [6] Answer problem 5 on answer sheet #5 5. (a) Write the exponential form of the equation log y t = r . [4] 1 (b) Write as a single logarithm: 3 In x2 + E ln(x2 — 9) [4] (c) Solve for x, showing your work algebraically: In em—3 = 43 [6] (d) Solve for x, showing your work algebraically: log 4 (x + 2) — 10g4 (x — 1) = 1 [5] (e) In a given situation, the amount of radium at time t is given by the model y = 40 6]”. The 1/2 life of radium is 1620 years. [6] (i) What is the initial amount of radium that is present ? (ii) How many grams remain after 1620 years ? Answer problem 6 on answer sheet #6 6. (a) A gutter is made from sheets of aluminum that are 34 inches wide. The edges are turned up (along the dotted lines on the accompanying figure) to form right angles. Write a quadratic function that represents the cross-sectional area of the gutter. [5] 4—— 34 ————> X X (b) A fence is to be constructed around a rectangular lot whose perimeter is 500 feet . The fencing to be used along the two lengths costs $30 per foot, while the fencing to be used for the two side widths costs just $15 per foot. The total cost of the fence is $12,000. SET UP a system of equations that you would use to determine the dimensions of the lot. SET UP the system only. You do not have to solve the system. [5] (c) Determine whether or not the point (-4, 0, 1) a solution of the following system of equations. Justify your answer: [5] x - 3y + z = -3 2x + y - 32 = -11 -3x - y - z = 11 __1 0 5 — 2 4 0 (d) Let A= 3 2 B= 13 C= 1 1 — 4 3 [10] (i) Find AB if possible. (ii) Find the matrix X, if 2A + X = C Show your work. Answer problem 7 on answer sheet #7 7. (a) $8,000 is invested in an account at a rate of 5.5%, compounded continuously. The amount in the account afier t years is given by: A = Pen. Determine how long it will take for the investment to double. [6] (b) The portion of the graph below that is outlined and marked “S” represents the solution region for the following set of inequalities. Determine the corner points (i.e. vertices) of the solution region. [6] f(x) 2x+3y_<_18 2x+y.<_10 x20 yZO X (0) Determine the maximum value of the objective function Z = 2x + 12y subject to the system of constraints above. [4] (c) A diet supplement is to be created using two different foods. Every ounce of food X contains 30 units of calcium and 15 units of vitamin C. Each ounce of food Y contains 10 units of calcium and 35 units of vitamin C. The minimum daily requirements in the diet are 100 units of calcium and 200 units of vitamin C. (i) Let x represent the number of ounces of food X, and let y represent the number of ounces of food Y. Write a system of inequalities you would use to determine the number of ounces of food X and Food Y that can be used in the diet supplement. SET UP THE SYSTEM ONLY. You do not need to graph it. [5] (ii) Determine whether or not the point (2,3) would be in the solution region of the system of inequalities, and interpret the meaning of this point in this situation. [4] THE END Have a good summer! ...
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