Mathematics

1. National Diploma for Industrial Technician
Civil Engineering
Module : Engineering Mathematics
Module No : CEM 1-2
Module Value : 01
60 Hours ( Hours per Week)
Rational :
Assessment Method : AE
Aim of the module to enable the students to:-
01.
A ARITHMATIC
INDICES
01. Evaluate expression involving positive, negative, and fractional indices.
1.1 Define the term basic, indices reciprocal, in terms of an
1.2 Apply the following rules
Axn
an
=am+n
, am
+a n
=am-n
, (am
)n
= amn
1.3 Deduce a0
= 1 for all a, a-n
= a and that a n = na
1.4 Evaluate expression, which combine positive, negative and fractional indices
STARNDERD FORM
02. Use standard form
2.1 Express a deanery number standard form
2.2 Convert to normal decimal form a number given in standard form
2.3 Add, subtract, Multiply and divide to numbers given in standard form
ENGINEERING MATHEMATICS Unit No: C 1-1
C

2. COMM0N LOGRITHMS
03. Evaluate expressions, powers and roots using logarithms
3.1 Define the invers of 10x
= y as x= log10y
3.2 Use log tables to determine the logarithms of given numbers
3.3 Use log tables to determine the number given its logarithms
3.4 Understand and use the first low if logarithms (log a/b = log a – log b)
3.5 Understand and use the third low of logarithms (log Np
= p log N)
3.6 Use logarithms to evaluate powers and roots
3.7 Apply lows of logarithms to evaluate expression
3.8 Apply lows of logarithms to solve equations
B ALGEBRA
04. Evaluation and transformation of formulae
4.1 Evaluate by substitution of given formulae
4.2. Transpose formulae which the subject contains roots, powers and more than one
variables
DIRECT AND INVERSE VARIATION
05. Illustrate direct and invers variations
5.1 Identify depended and independent variables
5.2 State the relationship between to variables which are
a) Directly proportional
b) Inversely proportional
5.3 Calculate the Co-efficient of proportionality from given data
5.4 State that for invers proportionality the product of variables is constant
5.5 Solve problems involving
a) Hook’s law

3. SIMPLE AND LINEAR SIMULATANEOUS EQUAVATIONS
06. Solve simple and linear simultaneous equations
. 6.1 Solve linear equations with in one unknown
6.2 Form and solve simple equations from particle applications
6.3 Solve of simultaneous linear equations with two and three unknowns
QUADRATIC FUNCTIONS AND QUADRATIC EQUATIONS
07. Understand quadratic functions and solve quadratic equations
7.1 Factorise quadratic expressions
7.2 Define the roots of and equations
7.3 Solve quadratic equations, with real roots be factorizations
7.4 Solve quadratic equations, which provide real roots by the formula
7.5 Form and solve quadratic equations, which are mathematical model of particle
problems
7.6 Solve algebraically, simultaneous quadratic and linear equations
C GRAPHS
STRAIGHT LINE
08. Understand parameters of straight - line low and its applications
8.1 Select and label axes
8.2 Plot coordinator and the graphs
8.3 Plot three points from co-ordinates determine from an equations of the form
y=mx+c where m and c are given numerical values and draw straight line through
the points
8.4 Determine the intercept with the y-axis and relates it to the m value
8.5 Determine the gradient of the straight line graph and relate it to the m value
8.6 Distinguish between positive and negative gradients
8.7 Plot co-ordinates from a set of experimental data
8.8 Draw the best straight line
8.9 Determine the gradient and intercept with the y-axis and deduce the equation
8.10 Determine the law of the straight line graph from the co-ordinates of two points
on the line

4. D MENSURATION
ARES OF PLANE FIGURES
09. Calculate areas of plane figures using formulae
9.1 Using given formulae to calculate areas of triangles, square, rectangle,
parallelogram, circle symmetrical, trapezium
AREAS AND VOLUMES OF COMPOSITE AND IRREGULER FIGURES
10. Calculate areas and volume and composite and irregular figure
10.1 Calculate the volume and surface areas of pyramids
10.2 Calculate the surface areas and volume of frustum
10.3 Calculate the total area and volume of composite figures
IRREGULAR AREAS AND VOLUMES BY NUMERICAL METHODS
11. Irregular areas and volumes by numerical methods
11.1 Calculate areas of irregular shapes using the mid-ordinate and trapezoidal rules
11.2 Apply Simpson’s rule to calculate area of irregular shape
11.3 Use the prismoidal rule to calculate volumes where appropriate
E GEOMETRY AND TRIGNOMETRY
TYPES AND PROPERTIES OF TRIANGLES
12. Understand the type and properties of triangle
12.1 State the angle sum of a triangle
12.2 Identify the types of triangles as acute angled, right angle obtuse angled,
equilateral, isosceles
12.3 Compare two triangles for similarity and determined and determine and unknown
side or angle of second triangle

5. PROPERTIES OF ACIRCLE AND RADIEN MEASURE
13. Understand the geometric properties of a circle
13.1 Identify radius, diameter, circumference, chord tangent, secant, sector, segment
and are of a circle and solve problems
13.2 Define the radian
13.3 Convert degree measure to radians and vice versa
13.4 Express angular rotation in multiples of radians
13.5 Use formulae s=r0
and A=1/2r2
0 for arc length and of a sector respectively and
solve problems
BASIC TRIGNOMETRIC RATIOS AND SOLUTIONS OF TRIANGLES
14. Solve triangles of angles and lengths of sides, using trig. ratios and trig. rules
14.1 Define, sin, cosine and tangent for acute angles
14.2 Obtain the three trigonometric functions of a given acute angles from a table or
calculator and vice versa
14.3 Calculate the surd forms the three trigonometric functions of a given acute angle
form a table or calculator and vice versa
14.4 Calculate the surd forms the three trigonometric functions for angles o0
, 300
, 450
,
600
, 900
, 1200
, 1350
, 1500
, 1800
14.5 State the relationship
Sin (90-θ) = Cos θ
Cos (90-θ) = Sin θ
Sin (A±B) = Sin A Cos B ± Cos A Sin B
Cos (A±B) = Cos A Cos B ± Sin A Sin B
Sin (180-θ) = Sin θ
Cos (180-θ) = - Cos θ
14.6 Solve problem by using Sin 2A, Cos 2A, Sin 3A Cos 3A, Tan 3A for A
14.7 State and use the Sin rule
A/Sin A = B/Sin B = C/Sin C
14.8 State and use the Cosine rule
A2
= b2
+ c2
– 2bc Cos A
14.9 Apply Sin and Cosine rules to solve practical problems
14.10 Calculate the area of any triangle using the formulae ½ ab sin c and s(s-a)(s-b)
(s-c)

6. F CALCULUS
DIFFERENTATION
15. Differentiate simple algebraic, trigonometric and exponential functions and
determined maximum, minimum of function
15.1 Differentiate function functions of the form y= axn
, n=0,1,2,3….
15.2 Differentiate simple algebraic function of the form
Y=axn
+bxn-1
+………………….
15.3 Differentiate functions of the form y=a cos θ, y=b sin θ, y= c tan θ
15.4 Define the differential property of the exponential function
15.5 Find the derivatives of the functions of log ex, ex, ax
15.6 Evaluate the derivatives at given points pf the functions in 5
15.7 State the basic rules for the derivatives of sum, products, quotient and function of a
function
15.8 Determine the derivatives of a various combination of any two function
15.9 Evaluate the derivatives in 8 at given points
15.10 Determine and evaluate second derivatives of algebraic, trigonometric, exponential and
logarithmic functions
15.11 Determine maximum and minimum of functions
15.12 Solve problems involving maximum relevant to the technology
INTERGRATION
16. Integrate algebraic, trigonometric and exponential functions, and determine areas
and volumes
16.1 Determine indefinite integrals of simple algebraic, and trigonometric functions
16.2 Recognize the need to include and arbitrary constant of trigonometric functions
16.3 Define by dx by as the area under the curve between ordinates at x=a, dx=b
16.4 Valuate by de by (0(x) b=0(b)-0(a) for simple functions where 0 (x) is the definite integral
of y (x)
16.5 Determine areas by applying the definite integral for simple algebraic and trigonometric
functions
16.6 Determine indefinite integrates of functions involving sin ax, Cos as, eax
16.7 Evaluate definite integrals involving sin az, Cos ax eax

7. 16.8 Determine the volume of revolution of area iven in 3 as yd2x and calculate the volumes
of a cone, sphere, frustum of a cone frustum of a sphere etc. using above formulae
16.9 use methods of substitution to determine integrals
E.g. x dx/1+x2
; Substitute U=1+x2
16.10 Use integration by parts to determine integrals of the form
xex
dx x sin ex dx, x2
loge x dx, loge dx etc.
G. Statistics;
 Probability
 Measures of central tendency – mean, median and mode
 Measures of variation – range, fractiles, variance, standard deviation
 Normal distribution curve and its applications
 Applications of descriptive statistics in construction
H. Graphical Techniques;
 Basic concepts in drawing graphs (scales, axes, gradients, intercepts)
 Determination of gradient & intercept of a straight line, identifying the trends indicated by the
slope or gradient of a graph
 Upper and lower limits of acceptability applicable to data entered on a graph or chart
 Drawing lines of best fit
 Uses of charts and tables and their interpretation
Drawing of graphs in log and semi-log scales