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Calculus 1 Exam

This document is an examination paper for various Bachelor of Science degrees at Dedan Kimathi University of Technology, specifically for the course SMA 2172 Calculus I. It includes instructions for answering questions, a variety of calculus problems, and topics such as continuity, differentiation, and parametric equations. The exam was conducted on July 8, 2014, and consists of multiple questions with varying marks.

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81 views3 pages

Calculus 1 Exam

This document is an examination paper for various Bachelor of Science degrees at Dedan Kimathi University of Technology, specifically for the course SMA 2172 Calculus I. It includes instructions for answering questions, a variety of calculus problems, and topics such as continuity, differentiation, and parametric equations. The exam was conducted on July 8, 2014, and consists of multiple questions with varying marks.

Uploaded by

brightonodhiambo645
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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DEDAN KIMATHI UNIVERSITY OF TECHNOLOGY

UNIVERSITY EXAMINATIONS 2013/2014

FIRST YEAR SUPPLEMENTARY /SPECIAL EXAMINATION


FOR THE DEGREE IN BACHELOR OF SCIENCE IN INDUSTRIAL CHEMISTRY,
LEATHER TECHNOLOGY,MECHATRONICS ENGINEERING, BACHELOR OF
SCIENCE IN CIVIL ENGINEERING, BACHELOR OF SCIENCE IN
MECHANICAL ENGINEERING, BACHELOR OF SCIENCE IN ELECTRICAL &
ELECTRONIC ENGINEERING, BACHELOR OF SCIENCE IN
TELECOMMUNICATION AND INFORMATION ENGINEERING, BACHELOR OF
SCIENCE IN GEOMATIC ENGINEERING & GEOSPATIAL INFORMATION
SYSTEMS AND BACHELOR OF SCIENCE IN GEOSPATIAL INFORMATION
SCIENCE

SMA 2172 CALCULUS I

DATE: 8TH JULY 2014 TIME: 8.30AM – 10.30AM

INSTRUCTIONS: ANSWER QUESTION ONE (COMPULSORY) AND ANY


OTHER TWO QUESTIONS

QUESTION 1 (30 MARKS)


a) What is continuity of a function f(x)? (3 marks)

b) Find the points at which the functions f x if x 1


1 if x 1
is continuous. (3 marks)
c) Find for the function y (4 marks)
d) Differentiate f x 3 cos 5x 2 . (3 marks)
e) Find if y t 2tand x 4 t . (3 marks)
f) Show that if, f x sec ax, then f x a sec ax tan ax (3 marks)
g) Given that 2y 5x 2 7y 0. Find (3 marks)
h) Determine the differential coefficient of the function: y √3x 4x 1 (4 marks)
i) Given the function 3 2, find:
(i) f(x + a) (2 marks)
(ii) f(2x + 3) (2 marks)

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QUESTION 2 (20 MARKS)
√
a) Estimate the value of lim → (3 marks)
b) If = 2 + 8 − , then by first principle method, evaluate, f x . (3 marks)
c) If = sin 2 , evaluate, f x . (2 marks)
d) If = − 4 + 5 + 2, find, (2 marks)
e) Determine the differential coefficient of: = 2 + 5 4 − + 1 (3 marks)
f) Find if = 3 (4 marks)
g) Show that, ℎ (sin ) = sec . (3 marks)

QUESTION 3 (30 MARKS)


a) If = ( ), find (4 marks)
b) Differentiate:
i) = 2 (4 marks)
ii) = sin tan 2 (4 marks)
c) Differentiate = 2 − 3 + 3 cos 3 with respect to y. (4 marks)
d) Using logarithmic differentiation, evaluate the derivative of:
/ √
= (4 marks)

QUESTION 4 (20 MARKS)


a) The parametric equations of a function are given by: = 3 2 and = 2 .
Determine expressions for:
(i) (3 marks)

(ii) (3 marks)
b) Given the function = − 4 ,
i) Find the coordinates of the stationary points on the curve. (3 marks)
ii) Use the function in (i) above to state the nature of these stationary points.
(3 marks)
iii) Sketch the graph the function. (2 marks)

QUESTION 5 (20 MARKS)


a) A stone is dropped into a lake, creating a circular ripple that travels outward at a
speed of 60 cm/s. Find the rate at which the area within the circle is increasing after
(a) 1 s, (b) 3 s, and (c) 5 s. What can you conclude? (7 marks)
b) An open cylindrical tank of height h m and radius r m has a capacity of 1m3.
i) Show that h = 1/( ) (3 marks)
ii) Show that its internal surface area = + . (4 marks)
iii) Determine the maximum value of S. (3 marks)

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iv) Evaluate: 2 + − (3 marks)

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