QuesSol_2007 - FACULTY OF ARTS AND SCIENCE University of...

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FACULTY OF ARTS AND SCIENCE University of Toronto FINAL EXAMINATIONS, APRIL/MAY 2007 MAT 133Y1Y Calculus and Linear Algebra for Commerce PART A. MULTIPLE CHOICE 1. [3 marks] The system of equations x + y + z + u + v = 1 y + z + 2 u = 2 z + u + 2 v = 3 has A no solutions B a unique solution C infinitely many solutions with one parameter D infinitely many solutions with two parameters E infinitely many solutions with three parameters 2. [3 marks] Let h ( x ) = x 2 - 1 x - 1 for x 6 = 1 a - 2 for x = 1 Then h ( x ) is continuous everywhere for a equal to A 4 B 3 C 1 D 2 E - 2 1
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3. [3 marks] If f ( x ) = ( x + 1) 1 x + ln x , f 0 ( x ) = A ( x + 1 ) 1 x + ln x h ( - 1 x 2 + 1 x ) ln ( x + 1 ) + 1 x + ln x · 1 2( x + 1) i B e ( 1 x + ln x ) ln ( x + 1 ) h - ( 1 x 2 + 1 x ) ln ( x + 1 ) + 1 x + ln x · 1 2 x ( x + 1) i C ( x + 1 ) 1 x + ln x h ( - 1 x 2 + 1 x ) ln ( x + 1 ) + 1 x + ln x · 1 ( x + 1) i D e ( 1 x + ln x ) ln ( x + 1 ) h ( - 1 x 2 + 1 x ) ln ( x + 1 ) 2 x + 1 x + ln x · 1 2 x ( x + 1) i E ( x + 1 ) 1 x + ln x h ( - 1 x 2 + 1 x ) ln ( x + 1 ) + 1 x + ln x · 1 2 x ( x + 1) i 4. [3 marks] The slope of the tangent line to the curve x 3 + 3 y 2 = 4 at (1 , 1) is A undefined B - 1 3 C 4 3 D 3 4 E - 1 2 2
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5. [3 marks] On the interval ( 1 2 , 2 ) , the function f ( x ) = e - ( x 2 + x - 2 ) has A a local minimum but no local maximum B a local maximum but no local minimum C a local maximum and a local minimum D neither a local maximum nor a local minimum E two local minima and no local maximum 6. [3 marks] lim x →∞ ( x 2 + 1) 1 x 2 +2 A does not exist B = e 1 2 C = e D = 0 E = 1 7. [3 marks] When using the Trapezoidal Rule with the interval [ - 2 , 6] divided into n = 4 subintervals the approximate value of Z 6 - 2 p 1 + x 2 dx is closest to A 23 . 04 B 21 . 55 C 22 . 19 D 15 . 36 E 14 . 37 3
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8. [3 marks] Z 1 - 1 x 2 3 dx (2 + x 5 3 ) 3 = A - 1 3 B 3 5 ln 27 C - 64 27 D 1 5 E 4 15 9. [3 marks] Z e 1 x ln x dx = A e 2 - e + 1 2 B 1 2 C e 2 2 D e 2 + 1 4 E 0 4
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10. [3 marks] Z 3 x - 4 ( x - 1)( x - 2) dx = A 3 ln | ( x - 1)( x - 2) | + C B 3 ln x - 2 x - 1 + C C 3 ln x - 1 x - 2 + C D ln ( x - 1)( x - 2) 2 + C E ln x - 1 ( x - 2) 2 + C 11. [3 marks] Z 8 1 3 x dx A diverges B = 6 C = - 6 D = 4 E = - 4 5
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12. [3 marks] If f ( x, y, z ) = x 2 y 3 z 4 , f xyz (2 , - 1 , - 2) = A - 3 8 B - 3 2 C - 1 4 D 3 2 E 3 8 13. [3 marks] If two products called A and B have the joint demand functions q A ( p A , p B ) = 1000 - 24 p A + 2 p 2 A - 40 p B + p 2 B and q B ( p A , p B ) = 2000 + 60 p A
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