QuesSol_2002 - FACULTY OF ARTS AND SCIENCE University of...

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FACULTY OF ARTS AND SCIENCE University of Toronto FINAL EXAMINATIONS, APRIL/MAY 2002 MAT 133Y1Y Calculus and Linear Algebra for Commerce PART A. MULTIPLE CHOICE 1. [3 marks] lim x →- 8 - x + 8 | 64 - x 2 | A equals 1 B equals 0 C equals 1 16 D equals - 1 16 E does not exist 2. [3 marks] The system of linear equations 2 x 1 - 4 x 2 - 6 x 3 = 2 x 1 - 2 x 2 - x 3 = 1 - x 1 + 2 x 2 + 5 x 3 = - 1 has A no solution B unique solution with x 3 = 0 C infinitely many solutions with x 2 = 0 D infinitely many solutions with x 3 = 0 E infinitely many solutions with x 1 = 0 1
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3. [3 marks] The equation ( xy ) 3 = 4 y - 3 defines y implicitly as a function of x near the point x = 1 , y = 1 . At this point y 0 is A undefined B 3 C 0 D - 1 E - 3 4. [3 marks] If f ( x ) = x x 2 + 1 , then A f has an absolute maximum and an absolute minimum B f has an absolute maximum but only a local minimum C f has an absolute minimum but only a local maximum D f has no absolute maximum and no absolute minimum, but only a local maximum and a local minimum E f has no absolute or local extrema at all 5. [3 marks] If the total revenue function is r ( q ) = 2500 q ln(10 q + 10) , where q is the number of units of some commodity, then the marginal revenue when q = 100 is closest to A 310 B 36 , 139 C 356 D 248 E 2 , 475 2
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6. [3 marks] lim x 1 2 x - 2 ln x A = 2 ln 2 B = 2 C = 0 D = 1 2 E does not exist 7. [3 marks] y=ln x a x y b 1 x The area of the shaded region is given by: A Z a 0 ln y dy B Z b 0 ( a - ln x ) dx C Z a 0 e y dy D Z a 0 ( b - e y ) dy E Z b 0 ln x dx - Z b 1 ln x dx 3
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8. [3 marks] If A and B are real numbers such that x + 9 x 2 - 9 = A x + 3 + B x - 3 (wherever both sides of the equality are defined) then A = A - 2 B 0 C 1 D - 1 E 2 9. [3 marks] Z 1 0 xe 3 x dx = A 1 3 B 1 + 2 e 3 9 C - 1 + 4 e 3 9 D 1 + 2 e 3 3 E - 1 - 4 e 3 9 10. [3 marks] Z 2 dx x A equals 2 B diverges C equals 1 D equals 2 2 E equals 2 4
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11. [3 marks] From time t = 0 until time t = 8 years, cash flows into an account which earns interest continuously at a rate of 5% per year. If, at time t years, the rate of the cash flow is 1000 e . 3 t dollars per year, then the present value of the total cash flow (to the nearest dollar) is A $25 , 556 B $27 , 038 C $21 , 297 D $18 , 254 E $33 , 411 12. [3 marks] If y
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