CALCULUS_III_(MATH_200)_1ST_TRIMESTER_2017.docx - END OF 1st TRIMESTER 2017(SB EXAMINATION SCHOOL DEPARTMENT UNIT CODE UNIT TITLE TIME SCIENCE AND

CALCULUS_III_(MATH_200)_1ST_TRIMESTER_2017.docx - END OF...

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END OF 1 st TRIMESTER 2017 (SB) EXAMINATION SCHOOL : SCIENCE AND TECHNOLOGY DEPARTMENT : PURE AND APPLIED SCIENCES UNIT CODE : MATH 200 UNIT TITLE : CALCULUS III TIME : 2 HOURS INSTRUCTIONS Answer question one and any other two questions Question One (30 marks) a(i) Express the point ( 2, π 3 ) in Cartesian coordinates (3mks) ii) Express ( 5, 12 ) in polar coordinates (3mks) b) Replace the following polar equation by the equivalent Cartesian equation r 2 + 2 r 2 Cosθ Sin θ = 1 . (3mks) c) Find (i) Lim n →∞ 4 n 2 + 6 n + 3 n 2 5 (4mks) d) If f ( x , y ) = 2 xSiny + 3 y 2 Cosx; find: f x + f y (4mks) e) Evaluate 0 3 0 1 ( 2 xy 2 + 2 yCosx ) dx dy . (4mks) f) Find the Taylor series and Taylor polynomials generated by f ( x ) = Sinx at x = 0 , if the degree of polynomial n = 5 (5mks) g) Determine the area enclosed between the curves y = x 2 + 1 and y = 7 x (4mks) Question Two (20marks) Page 1 of 3
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a) By finding the sum of the series n = 1 1 n ( n + 1 ) , show that the series converges (5mks) b) Find all the second partial derivatives of f ( x , y ) = x 3 + x 2 y 3
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