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Sma 2277: Calculus Iii Question Paper

Sma 2277: Calculus Iii

Course:Bachelor Of Science In Mechanical Engineering

Institution: Dedan Kimathi University Of Technology question papers

Exam Year:2013

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DEDAN KIMATHI UNIVERSITY OF TECHNOLOGY
2012/2013 UNIVERSITY EXAMINATIONS
SECOND YEAR FIRST SEMESTER EXAMINATION FOR BACHELOR OF SCIENCE IN MECHANICAL ENGINEERING
SMA 2277: Calculus III
DATE: 19TH AUGUST 2013 TIME: 2.00 PM – 4.00 PM
===============================================================
Instruction: Answer Question One And Any Other Two Questions
QUESTION ONE (30 Marks) – Compulsory
(a) State the mean value theorem and give its interpretation. (3 Marks)
(b) Use differentials to approximate the change in ( ,) =-2 +5 -3+7 if ( , ) changes from (-2,3) to (-2.02,3.01) . (4 marks)
(c) Use L’Hospital’s rule to evaluate (5 marks)
(d) Determine if the function below is continuous at =-2. (5 Marks)
( ) =
(e) Evaluate the following integral (4 Marks)
(f) Find a maclaurin series for ( ) =upto the term containing (5 Marks)
(g) Find the length of a curve = between x = 0 and x = 4 (4 Marks)
QUESTION TWO (20 Marks) – Optional
(a) (i) State Rolle’s theorem. (2 Marks)
(ii) Verify Rolle’s theorem for (x) = on . (6 Marks)
(b) Expand ln about = up to the term containing ( and use it to evaluate ln1.01. (6 Marks)
(c) Express the complex exponential as a complex trigonometric expression. (3 marks)
(d) Evaluate (3 marks)
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QUESTION THREE (20 Marks) – Optional
(a) Evaluate the following integral by changing the order of integration (5 Marks)
(b) Determine whether the integral is convergent or divergent. (6 Marks)
(c) Show that the length of the section of the curve parametrically by from is equal to 8a (5 marks)
(d) The kinetic energy of a body with mass m and velocity v is . Show that: = (4 marks)
QUESTION FOUR (20 Marks) – Optional
(a) Evaluate the double integral ?R where = [-1,3] ×[ 0,]. (4 Marks)
(b) Show that if =ln then =0 (6 Marks)
(c) Using ratio test determine the convergence or divergence of =1. (5 Marks)
(d) Find the volume generated when the area enclosed by the -axis and the curve =3- is rotated about the -axis. (5 Mark)
QUESTION FIVE (20 Marks) – Optional
(a) Find the total derivative given =+3 +5, = cos and =-sin leaving your answer in terms of . (4 Marks)
(b) Find , and the centre of mass of the region bounded by =4 -,= 0, =0 with density .(6 Marks)
(c) Find the reduction formula of n= hence find 4= . (7 Marks)
(d) Test the convergence of. (3 Marks)

Sma 2277: Calculus Iii Question Paper

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