THE USE OF A CALCULATOR, CELL PHONE, OR ANY OTHER ELECTRONIC DEVICE IS NOT PERMITTED DURING THIS EXAMINATION.

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1 MATH 110 FINAL EXAM **Test** December 14, 2009 TEST VERSION A NAME STUDENT NUMBER INSTRUCTOR SECTION NUMBER This examination will be machine processed by the University Testing Service. Use only a number 2 pencil on your scantron. On your scantron identify your name, this course (Math 110) and the date. Code and blacken the corresponding circles on your scantron for your student I.D. number and class section number. Code in your test version. There are 25 multiple choice questions each worth six points. For each problem four possible answers are given, only one of which is correct. You should solve the problem, note the letter of the answer that you wish to give and blacken the corresponding space on the answer sheet. Mark only one choice; darken the circle completely (you should not be able to see the letter after you have darkened the circle). Check frequently to be sure the problem number on the test sheet is the same as the problem number of the answer sheet. THE USE OF A CALCULATOR, CELL PHONE, OR ANY OTHER ELECTRONIC DEVICE IS NOT PERMITTED DURING THIS EXAMINATION. THE USE OF NOTES OF ANY KIND IS NOT PERMITTED DURING THIS EXAMINATION. THERE ARE 25 PROBLEMS ON 10 PAGES, INCLUDING THIS ONE. CHECK YOUR BOOKLET WHEN THE EXAM BEGINS.

2 MATH 110 FINAL EXAM **Test**, TEST VERSION A PAGE 2 1. Find all vertical and horizontal asymptotes of the graph of f(x) = x2 5x + 6 x 3 8x x. a) Vertical asymptotes at x = 0, and x = 5; horizontal asymptote at y = 0. b) Vertical asymptotes at x = 0, x = 3, and x = 5; no horizontal asymptote. c) Vertical asymptotes at x = 0, x = 3, and x = 5; horizontal asymptote at y = 0. d) Vertical asymptotes at x = 2, and x = 3; horizontal asymptote at y = 0. { x if x 2 2. Let f(x) = x 2. Which of the following is true? 3x + 4 if x > 2 a) f(x) is continuous at x = 2 but not differentiable at x = 2. b) f(x) is differentiable at x = 2 but not continuous at x = 2. c) f(x) is neither continuous nor differentiable at x = 2. d) f(x) is both continuous and differentiable at x = If f(s) = s 1 s + 1, find f (1). a) 3 4 b) 1 2 c) 1 4 d) 2 3

3 MATH 110 FINAL EXAM **Test**, TEST VERSION A PAGE 3 4. The total revenue and total cost functions for producing and selling x items is given by R(x) = x x 2 20x C(x) = 10x x 300 where x is the number of items, and R(x) and C(x) are in dollars. Using the marginal profit function P (x), determine the marginal profit when 10 items are produced and sold. a) $340 b) $260 c) $240 d) $ Find the equation of the line tangent to the graph of y = x 2 ln(x) at the point (1, 0). a) y = 2x 1 b) y = x + 1 c) y = x 1 d) y = 2x Find the relative extrema (if any) of the function 2x 2 e x. a) relative maximum at x = 2; relative minimum at x = 0. b) relative maximum at x = 0; relative minimum at x = 2. c) relative maximum at x = 4; relative minimum at x = 0. d) relative maximum at x = 0; relative minimum at x = 4.

4 MATH 110 FINAL EXAM **Test**, TEST VERSION A PAGE 4 7. Find lim x 1 x 1 x 3 + x 2 2x. a) 1 3 b) 1 c) 0 d) A company s costs in dollars for producing x units of their product is C(x) = x2 x x + 1. What is the limiting value of the cost when the number of units produced grows infinitely large? a) $7 b) $317 c) $17 d) $ If f(t) = 2t, find the intervals where f is increasing and where f is decreasing. t a) Increasing on (, 1) and (1, ); decreasing elsewhere. b) Increasing on (1, ); decreasing elsewhere. c) Increasing on (, 1); decreasing elsewhere. d) Increasing on ( 1, 1); decreasing elsewhere.

5 MATH 110 FINAL EXAM **Test**, TEST VERSION A PAGE If y 2 xy 8x + 6 = 0 find dy dx a) 2 b) -2 c) 10 3 d) 0 at the point (1, 2). 11. A game box manufacturer determines that in order to sell x units, the price per unit in dollars must be p(x) = 250 x. The manufacturer also determines that the total cost of producing x units is given by C(x) = x. How many units must the company produce and sell in order to maximize profit? a) 200 b) 120 c) 175 d) For the demand x + 2p = 60, where p represents the price in dollars and x the number of units, determine the elasticity of demand when the price p is equal to $15. a) 1 b) 1 2 c) 2 d) 3 2

6 MATH 110 FINAL EXAM **Test**, TEST VERSION A PAGE How long will it take $5,000 to grow to $10,000 if the investment earns an interest rate of 8% per year compounded quarterly? a) ln(2) 4 ln(1.02) b) ln(2) 4 c) ln(2) ln(1.02) d) ln(2) ln(1.08) 14. A company is increasing production of protein bars at a rate of 10 cases per day. All cases produced can be sold. The daily demand function is given by p(x) = 100 x2, where x is 300 the number of cases produced and sold, and p is in dollars. Find the rate of change of the revenue with respect to the time in days when the daily production is 20 cases. (Hint: This is a related rates application) a) $700 per day b) $990 per day c) $960 per day d) $996 per day 15. If y = (x + 3) x, find y (1). (Hint: Use logarithmic differentiation.) a) 4 ln 4 b) 1 c) 1 + ln 4 d) ln 4

7 MATH 110 FINAL EXAM **Test**, TEST VERSION A PAGE 7 (x 3 + x 2 x + 1) 16. Find x 2 dx. a) x2 2 + x 1 x 2 1 x + C b) x2 2 + x ln x 1 x + C c) x ln x 1 x + C d) x2 2 + x ln x + 1 x + C 17. If the rate of change of the unit price p of sunglasses is given by p 250x (x) = (16 + x 2 ) 3/2 where x is the number of sunglasses that the supplier will make available to the market daily in hundreds and p is in dollars, find the supply equation p(x) for the sunglasses if the quantity the supplier is willing to make available is 300 pairs (x = 3) of sunglasses daily when the unit price is $50? a) b) x x c) (16 + x 2 ) 3/2 125 d) + 50 (16 + x 2 ) 3/2 18. Sales of the Penn State Learning Calculus tutorial software packages in the first t years of its operation are approximated by f(t) = 2t. What are the average yearly sales over the t the time interval 1 t 3? a) ln 2 2 b) ln 5 2 c) ln 5 d) 2 ln 2

8 MATH 110 FINAL EXAM **Test**, TEST VERSION A PAGE 8 e 3x 19. Find dx. 1 + e3x a) ln (1 + e 3x ) + C b) 3e 3x 1 + e 3x + C c) 1 3 ln (1 + e3x ) + C d) 1 3(1 + e 3x ) + C 20. Find 1 1 a) 64 5 b) 32 5 c) 16 3 d) x 2 (x 3 + 1) 4 dx. 21. Find the area of the region in the first quadrant that is enclosed by the graphs of y = x and y = x. a) 1 2 b) 3 2 c) 1 6 d) 2 3

9 MATH 110 FINAL EXAM **Test**, TEST VERSION A PAGE If the demand function for tasers used to control noise levels in math classrooms is p = D(x) = 6x + 14 and the corresponding supply function is p = S(x) = 4x + 4, determine the consumer surplus at the market equilibrium point. a) 2 b) 8 c) 4 d) A cable supplier is expected to generate $100,000 in revenue per year for the next 7 years. If the income is reinvested in the business at a rate of 8% per year compounded continuously, which expression below is the total accumulated value of this income stream at the end of 7 years? 7 a) e , 000e 0.08t dt 0 7 b) e , 000e 0.56t dt c) , 000e 0.56t dt d) 100, 000e 0.56

10 MATH 110 FINAL EXAM **Test**, TEST VERSION A PAGE Your math professor has decided that for reasons of mental health, he wishes to retire and return to his planet of origin. To continue his very modest lifestyle, he wishes to establish a fund from which he can withdraw $5000 per month for the next 20 years. If the fund earns 6% per year compounded continuously, how much money does he need now to establish the fund? (Hint: this is a present value problem) a) (12)(20) 0.06 (1 e 1.2 ) b) 60, (1 e 1.2 ) c) (12)(20) 0.06 (e1.2 1) d) 60, (e1.2 1) 25. A recently educated Math 110 student decides to make regular payments of $2000 annually into a retirement account paying 5% interest per year compounded continuously. How much will the student have in their retirement account after 30 years? 30 a) e (0.05)(30) 2, 000e 0.05t dx b) c) d) 0 2, (e0.05 1) 2, (e1.5 1) (0.05)(2, 000) (e 1.5 1) 30