UNIVERSITY OF TECHNOLOGY, JAMAICA

FACULTY OF SCIENCE AND SPORT

SCHOOL OF MATHEMATICS AND STATISTICS

Final Examination, Semester 2

Module Name: Introduction to Calculus / Calculus 1

Module Code: MAT 1004 / MAT 2003

Date: April / May 2010

Theory / Practical: Theory

Groups: BPHARM, BSC1, PBCMS, MT, Community Colleges (CMCS), SGIS1,

ASTAT1, BSc.AS1, BSc.SE

Duration: Two (2) hours

Instructions:

1. This question paper consists of four (4) printed pages, which includes a cover page,

six (6) questions and a formulae sheet.

2. You are required to ANSWER ANY FOUR (4) questions in the answer booklet

provided.

3. Full marks will be awarded for full workings / explanations.

4. You are allowed to use silent electronic calculators.

5. Begin the answer to each question on a fresh page and number your solutions

carefully.

1

DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.

QUESTION 1

a) Evaluate the following limits, if they exist:

(i) [2marks]

(ii) [2marks]

(iii) [3marks]

b) A function, f(x), is defined by

.

Determine, if any, the points of discontinuity. [4marks]

c) A colony of insects were under investigation and it was reported that the population size

can be modelled by

The time, t, is measured in days and P(t) represents the size of the population at the t th

day. Ignoring effects such as migration and death, calculate the average growth rate of the

insects from day one to three. [4marks]

QUESTION 2

a) Find the derivative of the function using FIRST PRINCIPLES.

[5marks]

b) Find for the following and simplify as far as possible:

(i) [4marks]

(ii) [3marks]

(iii) [3marks]

QUESTION 3

a) Find:

(i) [2marks]

(ii) [2marks]

(iii) [6marks]

2

b) Find for . [5marks]

QUESTION 4

a) Find the first derivative, , for . [5marks]

b) Find the equation of the tangent to the curve at the point where x = 2.

[4marks]

c) Find:

i. [2marks]

ii. [2marks]

iii. [2marks]

QUESTION 5

a) Find , by using the method of partial fractions.

[6marks]

b) Given ,

(i) Find the coordinates of the stationary points.

(ii) Determine the nature of the stationary points.

(iii) Sketch the curve and clearly label the stationary points. [4+3+2marks]

QUESTION 6

a) Find the general solution to the differential equation given by

[4marks]

b) Find the area enclosed by the curve , the lines x = 0, x = 3 and the x-axis.

Give your answer correct to two decimal places. [4marks]

c) Find given that . [4marks]

d) A function, f(x), is defined by

Given that f(x) is continuous at x = 3, find the value of p. [3marks]

****END OF PAPER****

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FORMULAE SHEET

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