CALCULUS_III_(MATH_200)_3RD_TRIMESTER_2018 (1).docx - KENYA METHODIST UNIVERSITY END OF 3RD TRIMESTER 2018(FT EXAMINATION SCHOOL SCIENCE AND TECHNOLOGY

CALCULUS_III_(MATH_200)_3RD_TRIMESTER_2018 (1).docx - KENYA...

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KENYA METHODIST UNIVERSITY END OF 3 RD TRIMESTER 2018 (FT) EXAMINATION SCHOOL : SCIENCE AND TECHNOLOGY DEPARTMENT : PURE AND APPLIED SCIENCES UNIT CODE : MATH 200 UNIT TITLE : CALCULUS III TIME : 2 HOURS INSTRUCTION TO CANDIDATES: You should have the following for this examination. - Mathematical tables. - Scientific Calculator. Answer question ONE (COMPULSORY) and any other TWO questions. Maximum marks for each part of a question are as shown. Do not write anything on the question paper. Question 1 (One) (30 Marks) a) Find the cylindrical co-ordinate of the point whose Cartesian co-ordinates are ( 2 2 , 2 2 , 3 ¿ (2 Marks) b) Find the equation of the line y = 3 x + 2 in polar coordinates. (2 Marks) c) Evaluate lim n→∞ n 2 + n + 1 2 n 2 + 3 (4 Marks) Page 1 of 3/MATH 200
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d) Replace the Cartesian equation by the equivalent spherical coordinate equation x 2 9 + y 2 4 = 1 (4 Marks) e) Determine whether the series n = 1 cos n n 2 is convergent or divergent. (6 Marks) f) Find a power series representation for the function f ( x ) = 1 ( 1 x ) 2 (6 Marks) g) given that f ( x , y ) = x 3 y x y 3 x 2 + y 2 find f x and f y (6 Marks) Question 2 (Two) (20 Marks) a) Evaluate
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Unformatted text preview: lim ¿ x → − ∞ 5 ( 2 x ) + 4 ( 3 x ) 3 ( 2 x ) − 3 x ¿ (6 Marks) b) By first changing the Cartesian integral to the equivalent polar integral, evaluate ∫ a ∫ √ a 2 − x 2 ( x 2 + y 2 ) dydx (7 Marks) c) Compute the double integral over the indicated rectangle. Page 2 of 3/MATH 200 ∬ R ❑ 6 x y 2 dydx Where R=[2,4] × [ 1,2 ] (7 Marks) Question 3 (Three) (20 Marks) a) Apply Taylor series to evaluate lim x→ e x − 1 − x x 2 (6 Marks) b) Find a power series representation for ln ( 1 + x ) (6 Marks) c) Show that the third Taylor polynomial of f ( x ) = sinx at x = a is given by T 3 ( x ) = x − x 3 6 .use your polynomial to estimate sin 0.1 (8 Marks) Question 4 (Four) (20 Marks) a) A homogeneous triangle has vertices (0,0), (1,0), (1,3).Find the coordinates of its center of mass. (10 Marks) b) Find the volume under the surface z = √ 1 − x 2 and above the triangle formed by y = x ,x = 1 and the x axis (10 Marks) Page 3 of 3/MATH 200...
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