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Technical English - I 2nd week

MATHEMATICS

AND RELATED SUBJECTS

Numeric characters (Decimal system 0 to 9; Binary 0 and 1; Hexadecimal A-F & 0-9): 0-Zero 1-One 2-Two 3-Three 4-Four 5-Five 6-Six 7-Seven 8-Eight 9-Nine

integer & real numbers / rational number / fraction / numerator / denominator / sign

Numbers: Integer numbers; 7, 24, 177 Real numbers; 5.4, 8.333, 1.25

• • -2 -1 0 +1 +2 • •

Sign Sign (not necessarily)

± Plus & minus sign is mostly used to represent a range (For instance ±0.25m means from -0.25m to +0.25m or 40 ±4 °C denotes the temperature from 36 °C to 44 °C)

Even numbers; 2, 6, 8, 184 Odd numbers; 1, 7, 9, 83

Negative numbers; -24 -14.05 Rational numbers; 1/2 -2/7 5/4

5/4 numerator

denominator

fraction / decimal separator / quotient / numerator / denominator / infinite number

Real portion (fraction)

Integer part

Decimal separator . (Dot) US standard , (Comma) European standard

1.25 25.1

4

11

4

5

Real number:

Numerical forms: 1.25 125x10-2 125EE-2

Infinite numbers:

6666666.03

2

Infinite

Denominator

Numerator Fraction

Quotient

plus / minus / multiply / division / equal / reverse / power / index / infinite

X2 Power (Super script form) X1 Index (Sub script form)

Undefined Reverse Square root Absolute value

Plus Minus Multiply Division Equal Not equal

Greater than Less than Equal or Greater Equal or less Approximately equal

x0

1

x

1xx

Addition Subtraction Multiplication Division

a+b a -c axb a/b

sigma / pi / factorial / delta / absolute / infinity / logarithm / exponential power

Numerical forms:

alpha / beta / gamma / delta / epsilon / zeta / eta / theta / iota / kappa / lambda / mu nu / xi / omicron / pi / rho / sigma / tau / upsilon / phi / chi / psi / omega

Numerical forms:

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function / input / equation / relationship / commutative / associative / distributive

Functions and equations

Function Equation

f(x)=-0.4x2+2.4x+0.25 y=-0.4x2+2.4x+0.25 x=1.2 f(x) Relationship between x & y Input FUNCTION output variables

Algebra Algebra provides the rules which allow complex mathematical relationship to be

condensed or expanded. The general rules for changing the form of a mathematical relationship are:

Commutative law for addition: a + b = b + a

Commutative law for multiplication: a b = b a Associative law for addition: a + ( b + c ) = ( a + b ) + c Associative law for multiplication: a ( b c ) = ( a b ) c

Distributive law: a ( b + c ) = a b + a c

variable / unknown / simultaneous / constant / substitute / linear / quadratic

Linear equations 2x + 3y = 12 3x + 4y = 5 Solution of the linear equations Graphical method (intersection point of the lines) By substitution method (Solve one variable from the first equation, substitute it into the second one or others) Matrix methods A is the coefficient matrix

B is the constant values vector

A S B Solution vector = B * inverse A (Other matrix methods: Gauss elimination and Gauss-Jordan Elimination methods)

Quadratic Equations 2x2 – 3xy + y2 = 15

x2 – 2xy + y2 = 9

(One equation is not enough for the solution. n unknowns require n independent equations

to solve for the simultaneous value of x and y.)

Simultaneous (Linear or Quadratic) Equations

y

x

2 3 3 4

x y

12 5 ( ) ( ) = ( )

polynomial / exponent / power / concave up & down / turning point / inflection point

A Polynomial can be expressed in terms that only have positive integer exponents and the operations of addition, subtraction, and multiplication.

A polynomial function is a function similar to other functions. The difference is that it is involving only non-negative integer powers of x. A polynomial of degree n is a function of the form

f(x) = anxn + an-1xn-1 + . . . + a2x2 + a1x + a0

A turning point is a point where the graph of the function changes from sloping downwards to sloping upwards, or vice versa. m=n-1 (m is the number of the turning points)

(x7 – 2x4 + 5)3x = 0

is a polynomial equation since all of the variables have integer positive exponents.

turning point

inflection point

In many scientific and engineering applications we may need to find the slope of a curve at a specific point since it tells us the rate of change at a particular instant. The slope of a curve y = f(x) at the point P means the slope of the tangent at the point P. Here, x (independent) and y (dependent) are variables.

slope / curve / rate of change / tangent / independent / dependent / concept

By definition, the slope (m) is the rate of the change in y and x. m=Δy/Δx=(y2−y1)/(x2−x1) m can be found when the equation of a straight line which is a tangent to the curve,

is known. But what if it's not so?

This lead us to apply the concept of limit.

algebraic method / slope / curve / tangent / indeterminate / limit problem

An algebraic method to find the slope of y = f(x) at P. The idea of defining the slope in this way was discovered in the seventeenth century by Newton and Leibnitz.

If Q move closer and closer to P, the line PQ will get closer and closer to the tangent at P and so the slope of PQ gets closer to the slope that we want.

Now the problem has become a limit problem since which is indeterminate. 0

0

dx

dy

Newton quotient

In this limit problem, the ratio of two polynomials is in an indeterminate form, since x = 5 makes both the numerator and denominator equal to zero. Solution: Organizing the function(s) Finding the limit using derivative

(0/0, 0.∞ and ∞/∞ are an indeterminate form)

function / variable / infinity / indeterminate / limit / derivative

We may interested in what happens to the value of a function as the independent variable gets very close to a particular value.

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function / quotient / limit / differentiable / derivative

Limit

dx

dfxf )('

h

xfhxf

h

)()(lim

0

(f prime) Derivative (f double prime) Second derivative 'f ''f )''( ff

If the Newton quotient has a limit when h approaches 0, that is, the quotient approaches a limit as h approaches 0, it means f-function

is differentiable. Thus, this limit is called the derivative of f at x.

For example; The function f(x)=x2 is differentiable, its derivative is 2x.

Newton quotient

If the dependent variable depends on more than one variables such as y = f(x, h, t)

This time it is called as “partial derivation” and written as ∂y/∂x, ∂y/∂h, ∂y/∂t.

integration / anti-differentiation / inverse / definite / indefinite /

dxxf )(

Simply, integration (or anti-differentiation) is the inverse of differentiation

Derivative Integration

Indefinite integration Definite integration

)(')( xfxf )(')( xfxf

dx

dfxf )('

Cx

dxx 6

65 5.10

6

1

6

2

6

662

1

62

1

5

xdxx

b

adxxfA )(

b

adxxgxfA )]()([

dxxfVb

a2

)(

integration / definite integration / surface area / volume calculation

shape / mensuration area / perimeter circumference

boundary / length width / height

axis / radius /side square / circle

rectangle / triangle ellipse / parallelogram

annulus / trapezoid polygon

Mensuration is the branch of mathematics which deals with the study of Geometric shapes, their area, volume and related parameters.

cube / sphere / cylinder / cone / prism / pyramid / volume

V : Volume ℓ : Length h : Height w: Width r : Radius

clockwise / counterclockwise / anticlockwise / acute & right & obtuse & straight & reflex & full & adjacent & complementary & supplementary & congruent angles

Types of angles: Acute angle Right angle Obtuse angle Straight angle Reflex angle Full angle Adjacent angle Complementary angle Supplementary angle Congruent angles Corresponding angles Interior angles Exterior angles

Non-parallel Straight lines

d Parallel lines

Anticlockwise clockwise

Acute angle Right angle Obtuse angle

Straight angle Reflex angle Full angle

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trigonometry / angle / degree / radian /gradian / hypotenuse / opposite / adjacent

For any right triangle, there are six trig ratios: Sine (sin) cosine (cos) tangent (tan) cosecant (csc) secant (sec) cotangent (cot)

(Cos ϴ , Sin ϴ)

Warning!... degree or radian