Republic of Iraq
Ministry of Education
General Directorate of Curricula
MATHEMATICS
6
Biological Branch
REVISION
A Specialized committee in the Ministry of Education
Eleventh Translanted Edition
2021-1443
Scientific Supervisor: Marwah Flayylh Hasan
Art Supervisor: Salah Saad Mihsen
Book Designer: Salah Saad Mihsen
The Geometrical Drawings: Salah Saad Mihsen
‫أﺳﺘﻨﺎداً اﻟﻰ اﻟﻘﺎﻧﻮن ﻳﻮزع ﻣﺠﺎﻧ ًﺎ وﳝﻨﻊ ﺑﻴﻌﻪ وﺗﺪاوﻟﻪ ﻓﻲ اﻻﺳﻮاق‬
Introduction
The maths subject is considered one of the basic courses that helps students to acquaive
educational abilities to develop their thinking and solving problems and it helping to deal
with di�cult situations in their life .
As a starting point of attention by the Ministry of Education represented by the General
Directorate of Curricula to develop the curricula in general and specially of maths in
order to go along with the technological scientific .
This book is part of the of the project to diversify education for sixth – graders / (Biology
Branch).
The book consists of six chapters:
The first chapter is (Complex Numbers) which contains:
Find the roots and their properties ,Solving second –order equation .in complex number
, polar coordinates , Modulus and Argument of complex numbers .
The second chapter is (Conic Sections) which contains:
The standard equation .for Parabola , Ellipse and Hyperbola, and eccentricity of each
them.
The third chapter is (Application of Di�erentiation) which contains: Higher-order
derivatives, Related Rates.
Rolle›s and Mean value theorem ,the derivative test for increasing and decreasing for a
function, local Max and Min. Concavity and inflection opint, the second derivative test
for Local Max. and Min graphing functions and optimization (Max ,Min) problems.
The fourth chapter is ( Integration )which contains: Integration and it find integration of
Algebraic, Logarithmic , Exponential and Circular functions find the area between x-axis
and the curve , and the area between two curves ,find the volume of revolution .
The fi�h chapter is (Ordinary Di�erential Equation) with contains: find (Degree, Order and
Solution) and solving equation by separation of variable and by homogenous equations.
The sixth chapter is ( Space Geometry ) which contains: Dihderd angle and
perpendicular planes ,and Orthogonal Projection on plane .
We hope God help us to serve our country and our sons
Authors
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(16) : 6 - 42
(18) : 43 - 73
(48) : 74 - 133
(26) : 134 - 175
(13) : 176 - 197
(9) : 198 - 214
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w = { 0 , 1 , 2 , 3 , . . .}
Whole Numbers W = {0 , 1 , 2 , . . . }
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Note
Complex Number
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Equality
EqualityofofComplex
ComplexNumbers
Numbers
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Addition of Complex Numbers
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z : c + z = z+ c = 0
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z = -c
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1-4
Multiplying Complex Numbers
Find the result of the following
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Complex Number Conjugate
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2 - 1 Conic Sections
2-1
In this section we give geometric definitions of parabolas, ellipses, and hyperbol and derive their
standard equations. They are called conic sections. because its consist from intersecting a right
circular cone. with a plane as shown in Figure 2 - 1
parabola
Ellipse
Figure 2 - 1
Hyperbol
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In this example, the length of real axis is equal to the length of conjugate axis, like this kind of
hyperbola it is called hyperboliod right- angle, because the fore points are formed square shape.
The value of the eccentricity equal to constant value 2
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(n)
y(n) = d (n)y= f(n)(x)
d X
h f(a) , h = h - a
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+
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(Optional)
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The point (0 , f(0)) = (0,0) is inflection point
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Inflection point
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This test fails , we use the first derivative test instead
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dividing by 3
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18 - X
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For example
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Let
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