FACULTY OF ARTS AND SCIENCE University of Toronto FINAL EXAMINATIONS, APRIL 2016 MAT 133Y1Y Calculus and Linear Algebra for Commerce

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1 FACULTY OF ARTS AND SCIENCE University of Toronto FINAL EXAMINATIONS, APRIL 206 MAT YY Calculus and Linear Algebra for Commerce Duration: Examiners: hours N. Hoell A. Igelfeld D. Reiss L. Shorser J. Tate LEAVE BLANK FAMILY NAME: GIVEN NAME: STUDENT NO: SIGNATURE: Question Mark MC /45 B /0 B2 /4 B /0 B4 /0 B5 / BONUS /5 TOTAL NOTE:. Aids Allowed: A non-graphing calculator, with empty memory, to be supplied by student. No calculator may be used that has a button with d dx and/or on it. 2. Instructions: Fill in the information on this page, and make sure your test booklet contains 4 pages.. This exam consists of 5 multiple choice questions, and 5 written-answer questions. For the multiple choice questions you can do your rough work in the test booklet, but you must record your answer by circling the appropriate letter on the front page with your pencil. Each correct answer is worth marks; a question left blank, or an incorrect answer or two answers for the same question is worth 0. For the writtenanswer questions, present your solutions in the space provided. The value of each written-answer question is indicated beside it. 4. Put your name and student number on each page of this examination. ANSWER BOX FOR PART A Circle the correct answer. A. B. C. D. E. 2. A. B. C. D. E.. A. B. C. D. E. 4. A. B. C. D. E. 5. A. B. C. D. E. 6. A. B. C. D. E. 7. A. B. C. D. E. 8. A. B. C. D. E. 9. A. B. C. D. E. 0. A. B. C. D. E.. A. B. C. D. E. 2. A. B. C. D. E.. A. B. C. D. E. 4. A. B. C. D. E. 5. A. B. C. D. E. Page of 28

2 Name: Student #: Record your answers on the front page PART A. MULTIPLE CHOICE. [ marks] If the demand function is given by p = 500 q and the total cost function by c = 5q then profit is maximized when q = A. 50 B. 00 C. 750 D. 25 E [ marks] Find the equation(s) of the tangent line(s) to the curve f(x) = x x that are parallel to the x-axis. A. x = 0, x =, x = B. y = 2, y = 2 C. y = 2 only D. x =, x = E. y =, y = Page 2 of 28

3 Name: Student #: Record your answers on the front page. [ marks] Given y2 x = y 2 x + 4x 4 then at x = and y = 2, dy dx = A. ln B. ln C. 5 D. ln E. 2 2 ln 2 4. [ marks] The curve f(x) = x is concave up when ex A. x > only B. x < 2 only C. x < only D. x > 2 only E. x > 2 only Page of 28

4 Name: Student #: Record your answers on the front page 5. [ marks] If a > 0 is a constant, lim A. does not exist B. C. D. 5 2 E. 2 x a 2 x a x = 6. [ marks] Let G(x) = x ln t dt for < x < 0. Then G(x) has: A. no relative extrema nor absolute extrema B. an absolute maximum, but no absolute minimum nor relative extrema C. an absolute minimum, but no absolute maximum nor relative extrema D. a relative maximum, but no absolute extrema E. a relative minimum, but no absolute extrema Page 4 of 28

5 Name: Student #: Record your answers on the front page 7. [ marks] The integral A. B. C. D. E e w dw e w dw e w dw e w dw (6x + ) e x2 +x 4 dx is equal to e w dw 8. [ marks] Find the average value of g(s) = s on the interval [0, 9]. A. 2 9 B. 2 C. 6 D. 2 E. Page 5 of 28

6 Name: Student #: Record your answers on the front page 9. [ marks] Find the present value of a continuous annuity at an annual rate of % compounded continuously for four years if the payment at time t is at the rate of $400 per year. A. $84.07 B. $679.4 C. $ D. $507.7 E. $ [ marks] dx = 4 x 2 A. B. 8 C. 8 D. 0 E. diverges Page 6 of 28

7 Name: Student #: Record your answers on the front page. [ marks] If f(x, y, z, w) = 7xyzw + 7yzw 7yw, then f x = A. 7yzw B. 7 C. 7yzw + 7x D. 0 E. 7x 2. [ marks] Assume that x, y, z > 0 and x, y, and z are independent variables. Then ( x 2 y r z 6) = r A. 0 B. rx 2 y r z 6 C. x 2 y r z 6 ln y D. x 2 y r z 6 ln r E. ln y Page 7 of 28

8 Name: Student #: Record your answers on the front page. [ marks] The joint demand functions for products A and B are q A = 0 p B p A q B = 20 + p A 2p B respectively. A. A and B are neither competitive nor complementary when 20 + p A 2p B = 0 B. A and B are competitive as long as 20 + p A 2p B > 0 and complementary as long as 20 + p A 2p B < 0 C. A and B are neither competitive nor complementary at any positive prices. D. A and B are competitive at all positive prices. E. A and B are complementary at all positive prices. 4. [ marks] If x, y, and z are independent variables A. 0 B. 6(xz) 6 y 5 C. 0x 5 y 5 z 6 D. 6(xy) 5 z 6 E. 6x 5 y 5 z 6 2 x y (xyz)6 = Page 8 of 28

9 Name: Student #: Record your answers on the front page 5. [ marks] Let z = f(x, y), x = g(r, s), and y = h(r, s). It is given that g(, 0) = g r (, 0) = 2 g(, ) = 2 g r (, ) = 5 g s (, 0) = g s (, ) = h(, 0) = h r (, 0) = 4 h s (, 0) = 5 h(, ) = 0 h r (, ) = h s (, ) = 2 f(, 0) = 6 f x (, 0) = 6 f y (, 0) = 7 f(, ) = f x (, ) = 2 f y (, ) = 4 f(, ) = 5 f x (, ) = f y (, ) = 2 At r = and s = 0, z r = A. 4 B. 0 5 C. 2 6 D. 5 E. Page 9 of 28

10 PART B. WRITTEN-ANSWER QUESTIONS B. [0 marks] An entrepreneur has an account that guarantees him % per year compounded annually on any money he places in it for the next 5 years. 0 years from now, he will need to withdraw $00, 000, and 5 years from new, another $00, 000. He is due to receive $20, 000 at the end of each year for the first 0 years and will place the proceeds into the account. To meet the remaining part of the required payments, he will sell used equipment after the first years and also place the proceeds into the account. How much money must he realize from selling the equipment to meet his obligations? Page 0 of 28

11 B2. [4 marks] (a) [7 marks] The production function for a company is given by P (l, k) = 20l k 2 where l denotes units of labour and k denotes units of capital. Labour costs $225/unit and capital costs $50/unit and the total cost of labour and capital is $270, 000. Use Lagrange multipliers to find the number of units of labour and capital that maximize production. You don t have to show that your answer is indeed a maximum. (b) [7 marks] Find and classify the critical points of f(x, y) = xy + 8 y + 8 x (x 0, y 0) Page of 28

12 B. [0 marks] Find the area of the finite region bounded by the graphs of x = y 2 + y 4 and y = x. Page 2 of 28

13 B4. [0 marks] (a) [7 marks] If y = f(x) is the most general solution to the differential equation dy dx = y x for x, y > 0 find f(x). (b) [ marks] What is the answer if f() = 2? Page of 28

14 B5. [ marks] Evaluate (a) [5 marks] x 2 ln xdx (b) [6 marks] x + x(x + ) 2 dx Bonus: [extra 5 marks possible] x + x(x + ) dx 2 Page 4 of 28

15 FACULTY OF ARTS AND SCIENCE University of Toronto FINAL EXAMINATIONS, APRIL 206 MAT YY Calculus and Linear Algebra for Commerce Duration: Examiners: hours N. Hoell A. Igelfeld D. Reiss L. Shorser J. Tate LEAVE BLANK FAMILY NAME: GIVEN NAME: STUDENT NO: SIGNATURE: Question Mark MC /45 B /0 B2 /4 B /0 B4 /0 B5 / BONUS /5 TOTAL NOTE:. Aids Allowed: A non-graphing calculator, with empty memory, to be supplied by student. No calculator may be used that has a button with d dx and/or on it. 2. Instructions: Fill in the information on this page, and make sure your test booklet contains 4 pages.. This exam consists of 5 multiple choice questions, and 5 written-answer questions. For the multiple choice questions you can do your rough work in the test booklet, but you must record your answer by circling the appropriate letter on the front page with your pencil. Each correct answer is worth marks; a question left blank, or an incorrect answer or two answers for the same question is worth 0. For the writtenanswer questions, present your solutions in the space provided. The value of each written-answer question is indicated beside it. 4. Put your name and student number on each page of this examination. ANSWER BOX FOR PART A Circle the correct answer. A. B. C. D. E. 2. A. B. C. D. E.. A. B. C. D. E. 4. A. B. C. D. E. 5. A. B. C. D. E. 6. A. B. C. D. E. 7. A. B. C. D. E. 8. A. B. C. D. E. 9. A. B. C. D. E. 0. A. B. C. D. E.. A. B. C. D. E. 2. A. B. C. D. E.. A. B. C. D. E. 4. A. B. C. D. E. 5. A. B. C. D. E. Page 5 of 28

16 PART A. MULTIPLE CHOICE B. [ marks] If the demand function is given by p = 500 q and the total cost function by c = 5q then profit is maximized when q = A. 50 B. 00 C. 750 D. 25 E π = R C = pq c = 500 q q (5q ) = 500 q 5q 2000 dπ dq = = 0 when q = 50 or q = 2500 q Note that on (0, 2500) dπ dπ > 0 and on (2500, ) dq dq forever afterwards. Maximum is at q = 2500, E < 0. So π increases from 0 to 2500 and decreases B2. [ marks] Find the equation(s) of the tangent line(s) to the curve f(x) = x x that are parallel to the x-axis. A. x = 0, x =, x = B. y = 2, y = 2 C. y = 2 only D. x =, x = E. y =, y = Parallel to x-axis means slope = 0, and the equation(s) are y = c constant. When is slope = 0? When f (x) = 0, x 2 = 0, x 2 = 0, x =, or x =. At x =, y = 2, line is y = 2. At x =, y = 2, line is y = 2. B Page 6 of 28

17 B. [ marks] Given y2 x = y 2 x + 4x 4 then at x = and y = 2, dy dx = A. ln B. ln C. 5 D. ln E. 2 2 ln 2 x 2yy y 2 x 2 = y 2 x + y 2 x ln(2) 2 x + 4 x =, y = 2 gives 4y 4 = y ln(2) y = 2 ln(2) + 8 y = ln(2) + 4 A B4. [ marks] The curve f(x) = x is concave up when ex A. x > only B. x < 2 only C. x < only D. x > 2 only E. x > 2 only f (x) > 0 f(x) = xe x f (x) = e x xe x f (x) = e x e x + xe x = e x (x 2) > 0 when x > 2 only E Page 7 of 28

18 B5. [ marks] If a > 0 is a constant, lim A. does not exist B. C. D. 5 2 E. 2 x a 2 x a x = So 0, use L Hôpital. 0 lim a 2 x = a 0 = x lim x a x = a 0 = a 2 x ln(a) ( 2 = lim x a x ln(a) ( ) x 2 2 a 2 x x a x = lim x 2 ) a 0 = 2 a = 2 0 C B6. [ marks] Let G(x) = x ln t dt for < x < 0. Then G(x) has: A. no relative extrema nor absolute extrema B. an absolute maximum, but no absolute minimum nor relative extrema C. an absolute minimum, but no absolute maximum nor relative extrema D. a relative maximum, but no absolute extrema E. a relative minimum, but no absolute extrema By the Fundamental Theorem of Calculus G (x) = ln x > 0 on (, 0) G is an increasing function and has no extrema of any kind on the open interval A Page 8 of 28

19 B7. [ marks] The integral A. B. C. D. E e w dw e w dw e w dw e w dw (6x + ) e x2 +x 4 dx is equal to e w dw Let w = x 2 + x 4, dw = (6x + )dx x = 0 = w = 4, x = = w = 0 (6x + )e x2 +x 4 dx = e w dw B B8. [ marks] Find the average value of g(s) = s on the interval [0, 9]. A. 2 9 B. 2 C. 6 D. 2 E. 9 Average = sds 9 0 = s/2 0 = /2 = 2 D Page 9 of 28

20 B9. [ marks] Find the present value of a continuous annuity at an annual rate of % compounded continuously for four years if the payment at time t is at the rate of $400 per year. A. $84.07 B. $679.4 C. $ D. $507.7 E. $ P.V. = = T f(t)e rt dt 400e 0.0t dt = e 0.0t 0 = 400 ( e 0.2 ) 0.0 = $507.7 D B0. [ marks] dx = 4 x 2 A. B. 8 C. 8 D. 0 E. diverges R lim x /2 dx = lim R 4 R 2x /2 R = lim R 4 ( 2 R ) = A Page 20 of 28

21 B. [ marks] If f(x, y, z, w) = 7xyzw + 7yzw 7yw, then f x = A. 7yzw B. 7 C. 7yzw + 7x D. 0 E. 7x Only the first term depends on x: f x = 7yzw A B2. [ marks] Assume that x, y, z > 0 and x, y, and z are independent variables. Then ( x 2 y r z 6) = r A. 0 B. rx 2 y r z 6 C. x 2 y r z 6 ln y D. x 2 y r z 6 ln r E. ln y ( x 2 y r z 6) = x 2 z 6 r r (yr ) = x 2 z 6 y r ln y C Page 2 of 28

22 B. [ marks] The joint demand functions for products A and B are q A = 0 p B p A q B = 20 + p A 2p B respectively. A. A and B are neither competitive nor complementary when 20 + p A 2p B = 0 B. A and B are competitive as long as 20 + p A 2p B > 0 and complementary as long as 20 + p A 2p B < 0 C. A and B are neither competitive nor complementary at any positive prices. D. A and B are competitive at all positive prices. E. A and B are complementary at all positive prices. q A 5 = > 0 at all positive prices p B p A pb q B = > 0 at all positive prices p A The goods are competitive at all positive prices D B4. [ marks] If x, y, and z are independent variables A. 0 B. 6(xz) 6 y 5 C. 0x 5 y 5 z 6 D. 6(xy) 5 z 6 E. 6x 5 y 5 z 6 Alternatively 2 x y (xyz)6 = 2 x y (xyz)6 = 2 ( x 6 y 6 z 6) x y = ( 6x 6 y 5 z 6) x = 6x 5 y 5 z 6 E 2 x y (xyz)6 = [ 6(xyz) 5 xz ] x = 0(xyz) 4 yz xz + 6(xyz) 5 z = 0x 5 y 5 z 6 + 6x 5 y 5 z 6 = 6x 5 y 5 z 6 as before Page 22 of 28

23 B5. [ marks] Let z = f(x, y), x = g(r, s), and y = h(r, s). It is given that g(, 0) = g r (, 0) = 2 g(, ) = 2 g r (, ) = 5 g s (, 0) = g s (, ) = h(, 0) = h r (, 0) = 4 h s (, 0) = 5 h(, ) = 0 h r (, ) = h s (, ) = 2 f(, 0) = 6 f x (, 0) = 6 f y (, 0) = 7 f(, ) = f x (, ) = 2 f y (, ) = 4 f(, ) = 5 f x (, ) = f y (, ) = 2 At r = and s = 0, z r = A. 4 B. 0 5 C. 2 6 D. 5 E. z r = z x x r + z y y r = f x g r + f y h r We want g r (, 0) = 2 and h r(, 0) = 4 Also, when r = and s = 0, x = g(, 0) =, y = h(, 0) = So we want f x (, ) = and f y (, ) = 2 z r = ( 2 ) 2 ( 4 ) = E Page 2 of 28

24 PART B. WRITTEN-ANSWER QUESTIONS B. [0 marks] An entrepreneur has an account that guarantees him % per year compounded annually on any money he places in it for the next 5 years. 0 years from now, he will need to withdraw $00, 000, and 5 years from new, another $00, 000. He is due to receive $20, 000 at the end of each year for the first 0 years and will place the proceeds into the account. To meet the remaining part of the required payments, he will sell used equipment after the first years and also place the proceeds into the account. How much money must he realize from selling the equipment to meet his obligations? Let x be the proceeds he will need from the sale of equipment , , , , 000 x (00, 000) (00, 000) Set Income = Expenses at any moment: At 5, for example, Solving for x, 20, 000s (.0) 5 + x(.0) 2 = 00, 000(.0) , 000 The equation at : 20, 000s (.0) 7 + x = 00, 000(.0) , 000(.0) 2 Alternatively, x = $27, [ x(.0) , 000s ] is how much money he will have after 0 years. If he then pays 00, 000, x(.0) , 000s , 000 remains in the account. After [ a further 5 years, x(.0) , 000s , 000 ] (.0) 5 must equal 00, 000. Multiplying this out and solving for x gives the same answer (and is in fact just setting the equation of value at 5 instead of at. Page 24 of 28

25 B2. [4 marks] (a) [7 marks] The production function for a company is given by P (l, k) = 20l k 2 where l denotes units of labour and k denotes units of capital. Labour costs $225/unit and capital costs $50/unit and the total cost of labour and capital is $270, 000. Use Lagrange multipliers to find the number of units of labour and capital that maximize production. You don t have to show that your answer is indeed a maximum. Budget constraint: 225l + 50k = 270, 000 Subbing into budget: L = 20l k 2 λ(225l 50k 270, 000) L l = 20 l 2 2 k 225λ = 0 and Lλ = 0 is the budget constraint L k = 40 l k 50λ = 0 20 l 2 2 k = 225λ 40 l k = 50λ dividing the second equation by the first 2 l k = l k = so k = l (b) [7 marks] Find and classify the critical points of 225l + 450l = 270, l = 270, 000 l = 400 k = 200 f(x, y) = xy + 8 y + 8 x (x 0, y 0) f x = y 8 x 2 f y = x 8 y 2 y = 8 x 2, x = 8 y 2 = x2 = 64 y 4 = x 2 = y2 64 y =8 y4 64 8y = y 4, y 0 so 8 = y and y = 2 x = 8 y 2 = 8 4 = 2 Crit pits: x = 2, y = 2 only f xx = 6 f x yy = 6 y f xy = = f yx D = f xx f yy (f xy ) 2 = 256 x y D(2, 2) = 256 = > 0 so local extremem and 8 8 f xx (2, 2) = 6 = 2 > 0 so local min at (2, 2) 8 Page 25 of 28

26 B. [0 marks] Find the area of the finite region bounded by the graphs of x = y 2 + y 4 and y = x. x = y y = y 2 + y 4 0 = y 2 + 4y 5 0 = (y + 5)(y ) y = 5 y = x = 6 x = 0 (0,) (6,-5) A = = = 5 5 [ ( y) (y 2 + y 4) ] dy (5 4y y 2 ))dy [5y 2y 2 y ] 5 = ( ) ( ) = = = = Page 26 of 28

27 B4. [0 marks] (a) [7 marks] If y = f(x) is the most general solution to the differential equation dy dx = y x for x, y > 0 find f(x). dy y = dx x ln y = ln x + c (no x, y necessary since x > 0, y > 0) y = e ln x+c = e ln x e c y = Ax (b) [ marks] What is the answer if f() = 2? 2 = A So A = 2 and y = 2x Page 27 of 28

28 B5. [ marks] Evaluate (a) [5 marks] x 2 ln xdx Let u = ln x, dv = x 2 dx du = dx x, v = x x 2 ln xdx = x ln x x 2 dx = x x ln x 9 x + C (b) [6 marks] x + x(x + ) 2 dx x + x(x + ) = A 2 x + B x + + C (x + ) 2 A(x + ) 2 + Bx(x + ) + Cx = x + x = 0 = A = x = = C = 2 = C = 2 x = say = 4A + 2B + C = B + 2 = 4 so B = [ x + x(x + ) dx = 2 x ] x dx (x + ) 2 = ln x ln x + 2 x + + C Bonus: [extra 5 marks possible] x + x(x + ) dx 2 R x + lim dx = lim R x(x + ) 2 R = lim R [ ln x ln x + 2 ] R from (b) x + ( R ln R + 2 ) (ln ln 2 ) R + R R But so ln R + R + 0 and 2 R + 0 The answer is ln 2 +. Page 28 of 28