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MAT284-2005SpringA - Final Examination Monday May 9 2005...

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Unformatted text preview: Final Examination Monday, May 9, 2005. Print name: Sign name: Student Identification Number: Instructor (Barth or Lewis): MAT 284 10:15 am Version A Barth and Lewis — 12:15 pm TO RECEIVE CREDIT, work and reasoning should be shown for every problem except those indicated by (no work needed). There are 32 questions (each worth 4 points) on 10 pages. CHECK THAT YOU HAVE THE COMPLETE TEST! Do not write below this line ————-———-—_———_———_—_—_—_—_—_ Questions 1 to 9 Question 10 Questions 11 to 21 Question 22 Questions 23 to 32 Total Examination # 1 (Questions 1 to 9 and 10 and 22) Examination # 2 (Questions 11 to 21 and 22) Examination # 3 (Questions 23 to 32 and 10) 3. A company sells a product at $1 1 per unit. Its fixed cost is $15,000 and its variable cost per unit is $7. a) Find a function giving the company's profit P = P(q) if it produces and sells q units. PM) = b) At what level of production will there be a loss of $3,000? (Show your work) Iff(s) = 31 and g(s) = \/s + l,then (f0 g)(s) : s —2 (You need not simplify your answers.) The domain ofg(z) = is \/4—22 (Show your reasoning) 7. If 10Wx - 3) = 7, then x = (Show your work) The supply and demand equations (in some order), with prices in dollars, are: p = -2q + 48, p = 6q + 8. Ifa tax of 40¢ per unit is imposed on the supplier, a) write the supply equation b) write the demand equation (No work needed) The cost to produce 20 units is $110 and the cost to produce 10 units is $70. The cost 0 = C(q) in dollars is a linear function of output q, i.e., c and q are linearly related. Find 0 = C(q). (Show your work ) Express 2 logsx — 3log5(x+1) as a single logarithm: (Show your work) 8. A company has 1,000 units in stock, now selling at $3 per unit. Next month the price will increase by $1. The company wants total revenues from the sale ofthe units to be no less than $3,600. Let x be the maximum number of units that can be sold this month. Then x must satisfy the following equation or inequality (you need not simplify or solve). 9. Ifthe demand function is p = 200 - 2q and the total cost c = 300 + 60q, where q is the number of units, then the profit function is. (Show your reasoning) 10. A company holds a workshop for at least 30 people. If 30 people attend, the charge is $200 each, and the company will reduce the charge for eveflone by $4 for each person above 30 who attends. Let x be the number of people who attend. Write the company's total revenue in dollars as a function of x. (Here 305 x < 80. Do not simplify your answer or maximize R. (Show your work) R : [If you wish to let x be the number of decreases, you may do so, but check here and write R below]. R: 11. 12. 13. 14. 1 dx (No work needed) (5x3 +6): Differentiate y = x2 e(2X + 1) and do not simplify your answer. (Show your work) 4t+2 t3 Differentiate y = Differentiate y = — and do not simplify your answer. V 2x3 + 1 (Show your work) and do not simplify your answer. (Show your work) 2 . — 2 — 15. Find lim——-—X X 8 X94x2—5x+4 (Show your reasoning) 16. Find the slope ofthe curve y = — 8x + x4 at x = 1 (Show your work) 17. Let q be the number of units, let p be the price per unit in dollars, and let the demand function be p = 100 — 2q. Find the marginal revenue function. (Show your work) . d . 18. Let y = w3 and let w = 5 — x2. Use the cham rule to find a; at x = 1 (first state the cham rule): (Show your work) 19. 20. 21. 22. 2 , 4+x~x Evaluate 11m —- X“’°°2x3 — x +1 (No work needed) Differentiate y : ln[x5\/ l + x2] (You do not need to simplify your answer but show your work) Use the DEFINITION of derivative to find the derivative of f(x) = x2 — 1. First state the definition. (Show your work) _d_ In(4x2+6) dx [6 ] 2 (Show your work.) 23. Let y = x3 + 9x2 + 24x - 7. Find all values of x for which the curve is concave up. Show our work and, ifthere are none, state “none” y 24. Let f(x) = - x3 + 3x2 + 24x —8. Find the critical points of f(x). (Show your work and, ifthere are none, state “none”.) 25. The curve y = -x3 + 3x2 + 9x + 8 has a critical point at x = - 1. Determine ifthis point is a relative maximum, relative minimum, or neither. (Show your work) 26. 27. 28. 29. Let y = -x3 + 15x2 -48x + 2. Determine whether or not the point x = 5 is an inflection point. Justify your answer. (Show your work) The demand function is p = - 4q + 400, where q is the number of units and p is the price per unit in dollars. Find the output at which total revenue is the maximum. (Prove your conclusion) lfp = 800 -2q2, where q is the number of units and p is the price per unit, then find the (point) elasticity of demand at q = 10. (Show your work) 2 30. Suppose that the marginal revenue function is 500 - 12q , where q is the number of units. Find the demand function. (Show your work) 31. Presently, at $50 per sweater, a company is selling 400 sweaters, while at $40 per sweater it estimates that it will sell 500 sweaters. Find the (approximate) elasticity of demand. (Show your work) 32. Let the total cost function be c = q2 + 3q + 400,, where q is the number ofunits. At what level of output q will the average cost per unit be at the minimum. [You must show that you are at the minimum] There are 32 questions. Be sure that you have done each one ofthem. 10 ...
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