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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:    

Exams-50%

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 vice-versa

·         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

 

1-2

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.

 

2-3

Linear laws

Students should be able to:

 

·         understand the Cartesian co-ordinate system

·         plot points on a graph using Cartesian co-ordinates

·         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

 

10-11

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 vice-versa

·         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.

 

11-12

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.

 

12-13

Co-ordinate 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 co-ordinates 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 co-ordinates and convert to and from Cartesian co-ordinates

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