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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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