UNIVERSITY OF TECHNOLOGY, JAMAICA

FACULTY OF SCIENCE AND SPORT

FINAL EXAMINATION ( SEMESTER 1 )

SUBJECT: INTRODUCTION TO CALCULUS (MAT 1004)

GROUP: COMMUNITY COLLEGES/ SCIT DATE: DECEMBER 2009

DURATION: TWO (2) HOURS

INSTRUCTIONS:

1. Candidates are required to ANSWER ANY FOUR (4) questions.

2. This exam paper consists of five(5) questions, each worth 15 marks.

3. All workings must be clearly shown.

4. The use of silent electronic calculators is allowed.

5. Begin the answer to each question on a new page.

6. A list of useful formulae is attached.

_____________________________________________________________________

Question 1

a) Evaluate the following limits, if they exist:

i)

ii)

iii) [ 3 + 4 + 3 marks ]

b) Given the function, f(x), defined by

f(x) =

i) Determine the value of b if the function is continuous at x = 1. [ 3 marks ]

ii) Evaluate 2f(-1)-3f(2) [2

marks]

c) Find the derivative of using FIRST PRINCIPLES. [5 marks]

1

Question 2

a) By applying an appropriate rule, find for the following functions of x:

i)

ii)

iii)

iv) [4 + 4 + 4 + 4 marks ]

b) Find the equation of the tangent line to the curve at the point (2, 4).

[4 marks]

Question 3

a) Find an expression for for . [ 5 marks ]

b) Given the function

,

i) Find the coordinates of all the critical points of the function.

ii) Determine the nature of each critical point

iii) Sketch the graph of the function, clearly labelling the critical points.

[ 4 + 3 + 3 marks ]

c) Find [5 marks]

Question 4

a) Perform the following integrations:

i)

ii)

iii)

iv)

[ 3 + 5 + 4 + 4 marks ]

b) Find the general solution of the differential equation:

[ 4 marks ]

Question 5

2

a) Using logarithmic differentiation, find for:

i) [5 marks]

ii) [5 marks]

b) Resolve into partial fractions, and hence find

[10 marks]

END OF TEST

SOME USEFUL FORMULAE

1.

2.

3.

4.

5.

6.

7.

3