 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:

• a fluency and confidence with numbers.
• a fluency and confidence with algebra.
• a knowledge of trigonometric functions.
• an understanding of basic calculus and its application to ‘real-world ‘ situations.

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