University of Guyana
Faculty of Technology
Division of Engineering Mathematics
EMT
111 ENGINEERING MATHS
Course Description:
This
course covers foundation topics in Algebra, Analysis & Calculus, Discrete Mathematics,
Geometry and Trigonometry.
Course Objectives:
This
course is designed for students to develop:
Form of instruction: lectures, tutorials.
Lecturer: Laurel Benn
Course
Assessment:
Exams50% 
Coursework 50% 

Tests
25% Quizzes
10% Assignments
15% 
Main
Course Texts:
Course
Content:
Week 
Topics 
Core Learning
Outcomes 
1 
Arithmetic
of real numbers 
Students should be able to: ·
carry
out the operations add, subtract, multiply and divide on both positive and
negative numbers ·
express
an integer as a product of prime factors ·
calculate
the highest common factor and lowest common multiple of a set of integers ·
obtain
the modulus of a number ·
understand
the rules governing the existence of powers of a number ·
combine
powers of a number ·
evaluate
negative powers of a number ·
express
a fraction in its lowest form ·
carry
out arithmetic operations on fractions ·
represent
roots as fractional powers ·
express
a fraction in decimal form and viceversa ·
carry
out arithmetic operations on numbers in decimal form ·
round
numerical values to a specified number of decimal places or significant
figures ·
understand
the concept of ratio and solve problems requiring the use of ratios ·
understand
the scientific notation form of a number ·
manipulate
logarithms 
12 
Algebraic expressions and
formulae 
Students should be able to: ·
add
and subtract algebraic expressions and simplify the result ·
multiply
two algebraic expressions, removing brackets ·
evaluate
algebraic expressions using the rules of precedence ·
change
the subject of a formula ·
distinguish
between an identity and an equation ·
obtain
the solution of a linear equation ·
recognise
the kinds of solution for two simultaneous equations ·
understand
the terms direct proportion, inverse proportion and joint proportion ·
solve
simple problems involving proportion ·
factorise
a quadratic expression ·
carry
out the operations add, subtract, multiply and divide on algebraic fractions ·
interpret
simple inequalities in terms of intervals on the real line ·
solve
simple inequalities, both geometrically and algebraically ·
interpret
inequalities which involve the absolute value of a quantity. 
23 
Linear laws 
Students should be able to: ·
understand
the Cartesian coordinate system ·
plot
points on a graph using Cartesian coordinates ·
understand
the terms 'gradient' and 'intercept'
with reference to straight lines ·
obtain
and use the equation y = mx + c ·
obtain
and use the equation of a line with known gradient through a given point ·
obtain
and use the equation of a line through two given points ·
use
the intercept form of the equation of a straight line ·
use
the general equation ax + by + c = 0 ·
determine
algebraically whether two points lie on the same side of a straight line ·
recognize
when two lines are parallel ·
recognize
when two lines are perpendicular ·
obtain
the solution of two simultaneous equations in two unknowns using graphical
and algebraic methods ·
interpret
simultaneous linear inequalities in terms of regions in the plane ·
reduce
a relationship to linear form. 
3 
Quadratics, cubics, polynomials

Students should be able to: ·
recognize
the graphs of y = x 2 and
y
= x 2 ·
understand
the effects of translation and scaling on the graph of y =
x 2 ·
rewrite
a quadratic expression by completing the square ·
use
the rewritten form to sketch the graph of the general expression ax 2 + bx + c ·
determine
the intercepts on the axes of the graph of
y
= ax 2 + bx + c ·
determine
the highest or lowest point on the graph of y =
ax 2 +
bx + c ·
sketch
the graph of a quadratic expression ·
state
the criterion that determines the number of roots of a quadratic equation ·
solve
the equation ax 2 + bx + c = 0 via factorization, by completing the square
and by the formula ·
recognize
the graphs of y = x 3 and
y
= x 3 ·
recognize
the main features of the graph of y = ax 3 + bx 2 + cx + d ·
recognize
the main features of the graphs of quartic polynomials ·
state
and use the remainder theorem ·
derive
the factor theorem ·
factorize
simple polynomials as a product of linear and quadratic factors 
4 
Functions and
their inverses 
Students should be able to: ·
define
a function, its domain and its range ·
use
the notation f (x) ·
determine
the domain and range of simple functions ·
relate
a pictorial representation of a function to its graph and to its algebraic
definition ·
determine
whether a function is injective, surjective, bijective ·
understand
how a graphical translation can alter a functional description ·
understand
how a reflection in either axis can alter a functional description ·
understand
how a scaling transformation can alter a functional description ·
determine
the domain and range of simple composite functions ·
use
appropriate software to plot the graph of a function ·
obtain
the inverse of a function by a pictorial representation, graphically or
algebraically ·
determine
the domain and range of the inverse of a function ·
determine
any restrictions on f (x)
for the inverse to be a function ·
obtain
the inverse of a composite function ·
recognize
the properties of the function 1/x ·
understand
the concept of the limit of a function. 
5 
Sequences, series, binomial
expansions 
Students should be able to: ·
define
a sequence and a series and distinguish between them ·
recognize
an arithmetic progression and its component parts ·
find
the general term of an arithmetic progression ·
find
the sum of an arithmetic series ·
recognize
a geometric progression and its component parts ·
find
the general term of a geometric progression ·
find
the sum of a finite geometric series ·
interpret
the term 'sum' in relation to an infinite geometric series ·
find
the sum of an infinite geometric series when it exists ·
find
the arithmetic mean of two numbers ·
find
the geometric mean of two numbers ·
obtain
the binomial expansions of (a +
b)^{s}, (1 + x )^{s }for s
a rational number ·
use
the binomial expansion to obtain approximations to simple rational functions 
5 
Logarithmic and exponential
functions 
Students should be able to: ·
recognize
the graphs of the power law function ·
define
the exponential function and sketch its graph ·
define
the logarithmic function as the inverse of the exponential function ·
use
the laws of logarithms to simplify expressions ·
solve
equations involving exponential and logarithmic functions ·
solve
problems using growth and decay models. 
6 
Rates of change and
differentiation 
Students should be able to: ·
define
average and instantaneous rates of change of a function ·
understand
how the derivative of a function at a point is defined ·
recognize
the derivative of a function as the instantaneous rate of change ·
interpret
the derivative as the gradient at a point on a graph ·
distinguish
between 'derivative' and 'derived function' ·
use
the notations, f ′(x) , y′ etc. ·
use
a table of the derived functions of simple functions ·
recall
the derived function of each of the standard functions ·
use
the multiple, sum, product and quotient rules ·
use
the chain rule ·
relate
the derivative of a function to the gradient of a tangent to its graph ·
obtain
the equation of the tangent and normal to the graph of a function. 
7 
Stationary points, maximum and
minimum values 
Students should be able to: ·
use
the derived function to find where a function is increasing or decreasing ·
define
a stationary point of a function ·
distinguish
between a turning point and a stationary point ·
locate
a turning point using the first derivative of a function ·
classify
turning points using first derivatives ·
obtain
the second derived function of simple functions ·
classify
stationary points using second derivatives. 
8 
Indefinite integration 
Students should be able to: ·
reverse
the process of differentiation to obtain an indefinite integral for simple
functions ·
understand
the role of the arbitrary constant ·
use
a table of indefinite integrals of simple functions ·
understand
and use the notation for indefinite integrals ·
use
the constant multiple rule and the sum rule ·
use
indefinite integration to solve practical problems such as obtaining velocity
from a formula for acceleration or displacement from a formula for velocity.. 
9 
Definite integration,
applications to areas and volumes 
Students should be able to: ·
understand
the idea of a definite integral as the limit of a sum ·
realize
the importance of the Fundamental Theorem of the Calculus ·
obtain
definite integrals of simple functions ·
use
the main properties of definite integrals ·
calculate
the area under a graph and recognize the meaning of a negative value ·
calculate
the area between two curves ·
calculate
the volume of a solid of revolution ·
use
trapezium and Simpson's rules to approximate the value of a definite
integral. 
10 
Sets

Students should be able to : ·
understand
the concepts of a set, a subset and the empty set ·
determine
whether an item belongs to a given set or not ·
use
and interpret Venn diagrams ·
find
the union and intersection of two given sets ·
apply
the laws of set algebra 
1011 
Geometry 
Students should be able to : ·
recognise
the different types of angle ·
identify
the equal angles produced by a transversal cutting parallel lines ·
identify
the different types of triangle ·
state
and use the formula for the sum of the interior angles of a polygon ·
calculate
the area of a triangle ·
use
the rules for identifying congruent triangles ·
know
when two triangles are similar ·
state
and use Pythagoras' theorem ·
understand
radian measure ·
convert
from degrees to radians and viceversa ·
state
and use the formulae for the circumference of a circle and the area of a disc
·
calculate
the length of a circular arc ·
calculate
the areas of a sector and of a segment of a circle ·
quote
formulae for the area of simple plane figures ·
quote
formulae for the volume of elementary solids: a cylinder, a pyramid, a tetrahedron,
a cone and a sphere ·
quote
formulae for the surface area of elementary solids: a cylinder, a cone and a sphere
·
sketch
simple orthographic views of elementary solids ·
understand
the basic concept of a geometric transformation in the plane ·
recognise
examples of a metric transformation (isometry) and affine transformation (similitude)
·
obtain
the image of a plane figure in a defined geometric transformation: a translation
in a given direction, a rotation about a given centre, a symmetry with
respect to the centre or to the axis, scaling to a centre by a given ratio. 
1112 
Trigonometry 
Students should be able to: ·
define
the sine, cosine and tangent of an acute angle ·
define
the reciprocal ratios cosecant, secant and cotangent ·
state
and use the fundamental identities arising from Pythagoras' theorem ·
relate
the trigonometric ratios of an angle to those of its complement ·
relate
the trigonometric ratios of an angle to those of its supplement ·
state
in which quadrants each trigonometric ratio is positive (the CAST rule) ·
state
and apply the sine rule ·
state
and apply the cosine rule ·
calculate
the area of a triangle from the lengths of two sides and the included angle ·
solve
a triangle given sufficient information about its sides and angles ·
recognise
when there is no triangle possible and when two triangles can be found. 
1213 
Coordinate geometry 
Students should be able to: ·
calculate
the distance between two points ·
find
the position of a point which divides a line segment in a given ratio ·
find
the angle between two straight lines ·
calculate
the distance of a given point from a given line ·
calculate
the area of a triangle knowing the coordinates of its vertices ·
give
simple examples of a locus ·
recognise
and interpret the equation of a circle in standard form and state its radius
and centre ·
convert
the general equation of a circle to standard form ·
recognise
the parametric equations of a circle ·
derive
the main properties of a circle, including the equation of the tangent at a point
·
define
a parabola as a locus ·
recognise
and interpret the equation of a parabola in standard form and state its vertex,
focus, axis, parameter and directrix ·
recognise
the parametric equation of a parabola ·
derive
the main properties of a parabola, including the equation of the tangent at a
point ·
understand
the concept of parametric representation of a curve ·
use
polar coordinates and convert to and from Cartesian coordinates 
13 
Trigonometric functions and
applications 
Students should be able to: ·
define
the term periodic function ·
sketch
the graphs of sin x, cos x and
tan x and describe their main features ·
deduce
the graphs of the reciprocal functions cosec, sec and cot ·
deduce
the nature of the graphs of asin
x , a cos x , a tan x ·
deduce
the nature of the graphs of sin ax
, cosax , tan ax ·
deduce
the nature of the graphs of sin(x + a), a +
sin x, etc ·
solve
the equations sin x = c, cos x = c, tan x = c ·
use
the expression asin(wt + f) to represent an oscillation
and relate the parameters to the motion ·
rewrite
the expression a coswt + bsinwt as a single cosine or sine formula. 
14 
Trigonometric identities 
Students should be able to: ·
obtain
and use the compound angle formulae and double angle formulae ·
obtain
and use the product formulae ·
solve
simple problems using these identities 