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Sma 2270: Calculus Iii Question Paper

Sma 2270: Calculus Iii

Course:Bachelor Of Science In Engineering

Institution: Dedan Kimathi University Of Technology question papers

Exam Year:2011

SMA 2270 CALCULUS III (Engineering) Page 1 of 3
Page 1 of 3
DEDAN KIMATHI UNIVERSITY OF TECHNOLOGY UNIVERSITY EXAMINATIONS 2011/2012 SECOND YEAR FIRST SEMESTER EXAMINATION FOR BACHELOR OF SCIENCE IN ENGINEERING (CIVIL, EEE, TIE, MECHATRONICS) SMA 2270: CALCULUS III DATE: 11TH AUGUST 2011 TIME: 2.00 PM – 4.00 PM QUESTION ONE (30 Marks) – (Compulsory)
(a) State the Mean Value Theorem. [2 Marks]
(b) (i) Let be a function. State what is meant by a function being continuous at a point . [3 Marks]
(ii) Determine if the function below is continuous at . [5 Marks]
(c) Evaluate [5 Marks]
(d) Determine the center of mass of a uniform triangular laminar bounded by the lines , and . [6 Marks]
(e) Find the volume of the solid under over the rectangle . [5 Marks]
(f) If , find and . [4 marks]
SMA 2270 CALCULUS III (Engineering) Page 2 of 3
Page 2 of 3
QUESTION TWO (20 Marks) – Optional
(a) i) State the ratio test for convergence of infinite series. [2 Marks]
ii) Hence determine whether converges or diverges. [5 Marks]
(b) Establish the reduction formula for hence evaluate . [7 Marks]
(c) Determine whether the integral is convergent or divergent. [6 Marks]
QUESTION THREE (20 Marks) – Optional
(a) Find a value of such that the conclusion of the MVT is satisfied for from . [3 Marks]
(b) Show that [4 Marks]
Hence using the ratio test determine whether the series below converges or diverges. [7 Marks]
(c) Find , and the centre of mass of the region bounded by , , with density . [6 Marks]
QUESTION FOUR (20 Marks) – Optional
(a) Find the eigenvalues and the eigenvectors corresponding to the smallest eigenvalue of the matrix [10 Marks]
(b) Use the integral test to test whether converges or diverges. [4 Marks]
SMA 2270 CALCULUS III (Engineering) Page 3 of 3
Page 3 of 3
(c) Find the Taylor series of the function about and determine its radius and interval of convergence. [6 Marks]
QUESTION FIVE (20 Marks) – Optional
(a) Find the inverse of the matrix using cofactors. [5 Marks]
(b) Use Cramer’s rule to solve the system [4 Marks]
(c) Using the ratio test, examine the absolute convergence [6 Marks]
(d) Find the Maclaurins series for upto the term containing . Hence evaluate to 3 decimal places. [5 Marks]

Sma 2270: Calculus Iii Question Paper

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