Coordinate Lines

Coordinate Lines

In 1637 Rene' Descarts published a philosophical work called Discourse on the Method of Rightly Conducting the Reason . In the back of that book was an appendix that the British philosopher John Stuart Mill described as, ``The greatest single step ever made in the progress of the exact sciences." In that appendix Rene' Descartes linked together algebra and geometry, thereby creating a new subject called analytic geometry ; it gave a way of describing algebraic formula by geometric curves and, conversely , geometric curves by algebraic formulas .

Figure 1.1.4

     In analytic geometry , the key step is to establish a correspondence between real numbers and points on a line . This is done by arbitrarily designative one of the two directions along the line as the positive direction and the other as the negative direction  . The positive direction is usually marked with an arrowhead as in Figure 1.1.4 ; for horizontal lines the positive direction is generally taken to the right. A unit of measurements is then chosen and an arbitrary point , called the origin , is selected anywhere along the line. The line, the origin, the positive direction , and the unit of measurement define what is called a coordinate line or sometimes a real line. With each real number we can associate a point on the line as follows :

$\bullet$ Associate the origin with the number $0$
$\bullet$ Associate with each positive number $r$ the point that is a distance of $r$ units in the positive direction from the origin.
$\bullet$ Associate with each negative number $-r$ the point that is a distance of $r$ units in the negative direction from the origin

     The real number corresponding to a point on the line is called the coordinate of the point .


Example 1 In Figure 1.1.5 we have marked the locations of the points with coordinates $-4,-3,-1.75,-\cfrac{1}{2},\sqrt{2} ,\pi$ and $4$. The location of $\pi$ and $\sqrt{2}$ which are approximate , were obtained from their decimal approximations, $\pi\approx 3.14$ and $\sqrt{2}\approx 1.41$

Figure 1.1.5

     It is evident from the way in which real numbers and points on a coordinate line are related that each real number corresponds to a single point and each point corresponds to a single real number . To describe this fact we say that the real numbers and the points on a coordinate line are in one-to-one correspondence 
 

Table Of Integrals calculus

Table Of Integrals 

Elementary Integrals

1. $\displaystyle\int \mathrm{d}u =u+C$ 

2. $\displaystyle\int a \mathrm{d}u = au+C$

3. $\displaystyle\int [f(u)+g(u)] = \displaystyle\int f(u) \mathrm{d}u +\displaystyle\int g(u) \mathrm{d}u$

4. $\displaystyle\int u^n \mathrm{d}u =\cfrac{u^{n+1}}{n+1}+C        (n \neq -1)$

5. $\displaystyle\int \cfrac{\mathrm{d}u}{u} = \ln |u|+C$

Integrals Containing $a+bu$

6. $\displaystyle\int \cfrac{u\mathrm{d}u}{a+bu} =\cfrac{1}{b^2}[bu-a\ln|a+bu|]+C$

7. $\displaystyle\int \cfrac{u^2\mathrm{d}u}{a+bu} =\cfrac{1}{b^3}\left[\cfrac{1}{2}(a+bu)^2-2a(a+bu)+a^2\ln|a+bu|  \right]+C$

8. $\displaystyle\int \cfrac{u\mathrm{d}u}{(a+bu)^2} = \cfrac{1}{b^2}\left[\cfrac{a}{a+bu}+\ln |a+bu|\right]+C$

9. $\displaystyle\int\cfrac{u^2\mathrm{d}u}{(a+bu)^2} =\cfrac{1}{b^3}\left[bu-\cfrac{a^2}{a+bu}-2a\ln|a+bu|\right]+C$

10. $\displaystyle\int \cfrac{u\mathrm{d}u}{(a+bu)^3}=\cfrac{1}{b^2}\left[\cfrac{a}{2(a+bu)^2}-\cfrac{1}{a+bu} \right]+C$

11. $\displaystyle\int \cfrac{mathrm{d}u}{u(a+bu)} =\cfrac{1}{a}\ln\left|\cfrac{u}{a+bu}\right|+C$

12. $\displaystyle\int \cfrac{\mathrm{d}u}{u^2(a+bu)}=-\cfrac{1}{au}+\cfrac{b}{a^2}\ln\left|\cfrac{a+bu}{u}\right| +C$

13. $\displaystyle\int \cfrac{\mathrm{d}u}{u(a+bu)^2} =\cfrac{1}{a(a+bu)}+\cfrac{1}{a^2}\ln \left|\cfrac{u}{a+bu}\right| +C$

Integrals Containing $\sqrt{a+bu}$ 

14. $\displaystyle\int u\sqrt{a+bu}\mathrm{d}u=\cfrac{2}{15b^2}(3bu-2a)(a+bu)^{\cfrac{3}{2}}+C$

15. $\displaystyle\int u^2\sqrt{a+bu}\mathrm{d}u =\cfrac{2}{105b^3}(15b^2u^2-12abu+8a^2)(a+bu)^{\cfrac{3}{2}}+C$

16. $\displaystyle\int u^n\sqrt{a+bu}\mathrm{d}u =\cfrac{2u^n(a+bu)^{\cfrac{3}{2}}}{b(2n+3)}-\cfrac{2an}{b(2n+3)}\displaystyle\int u^{n-1}\sqrt{a+bu}\mathrm{d}u$

17. $\displaystyle\int\cfrac{u\mathrm{d}u}{\sqrt{a+bu}}=\cfrac{2}{3b^2}(bu-2a)\sqrt{a+bu}+C$

18. $\displaystyle\int \cfrac{u^2\mathrm{d}u}{\sqrt{a+bu}}=\cfrac{2}{15b^3}(3b^2u^2-4abu+8a^2)\sqrt{a+bu}+C$

19. $\displaystyle\int\cfrac{u^n\mathrm{d}u}{\sqrt{a+bu}}=\cfrac{2u^n\sqrt{a+bu}}{b(2n+1)}-\cfrac{2an}{b(2n+1)}\displaystyle\int\cfrac{u^{n-1}\mathrm{d}u}{\sqrt{a+bu}}$

20. $\displaystyle\int\cfrac{\mathrm{d}u}{u\sqrt{a+bu}} = \begin{cases}  \cfrac{1}{\sqrt{a}}\ln \left|\cfrac{\sqrt{a+bu}-\sqrt{a}}{\sqrt{a+bu}+\sqrt{a}}\right|+C & (a>0) \\ \cfrac{2}{\sqrt{-a}}\tan^{-1}\sqrt{\cfrac{a+bu}{-a}}+C & (a<0) \end{cases}$

21. $\displaystyle\int\cfrac{\mathrm{d}u}{u^n\sqrt{a+bu}}=-\cfrac{\sqrt{a+bu}}{a(n-1)u^{n-1}}-\cfrac{b(2n-3)}{2a(n-1)}\displaystyle\int\cfrac{\mathrm{d}u}{u^{n-1}\sqrt{a+bu}}$

22. $\displaystyle\int\cfrac{\sqrt{a+bu}\mathrm{d}u}{u}=2\sqrt{a+bu}+a\displaystyle\int\cfrac{\mathrm{d}u}{u\sqrt{a+bu}}$

23. $\displaystyle\int\cfrac{\sqrt{a+bu}\mathrm{d}u}{u^n}=-\cfrac{(a+bu)^{\cfrac{3}{2}}}{a(n-1)u^{n-1}}-\cfrac{b(2n-5)}{2a(n-1)}\displaystyle\int\cfrac{\sqrt{a+bu}\mathrm{d}u}{u^{n-1}}$

Integrals Containing $a^2\pm u^2$ $(a>0)$

24. $\displaystyle\int\cfrac{\mathrm{d}u}{a^2+u^2}=\cfrac{1}{a}\tan ^{-1}\cfrac{u}{a}+C$

25. $\displaystyle\int\cfrac{\mathrm{d}u}{a^2-u^2}=\cfrac{1}{2a}\ln\left|\cfrac{u+a}{u-a}\right| +C$

26. $\displaystyle\int\cfrac{\mathrm{d}u}{u^2-a^2}=\cfrac{1}{2a}\ln\left|\cfrac{u-a}{u+a}\right|+C$

Geometry Formulas

Geometry Formulas 

$A$ = area , $S$ = lateral surface area , $V$ = volume , $h$ = height , $B$ =  area of base , $r$ = radius , $l$ = slant height ,$C$ = circumference , $s$ = arc length   

Parallelogram

$A=nh$

Triangle

 

 $A=\cfrac{1}{2}bh$

 

Trapezoid 

$A=\cfrac{1}{2}(a+b)h$

 

Circle

 $A=\pi r^2 ,C=2\pi r$

Sector

 

 $A=\cfrac{1}{2}r^2 \theta ,s=r\theta$

Right Circular Cylinder

 

  $V=\pi r^2 ,S =2\pi rh$

 

Right Circular Cone

 

 $V=\cfrac{1}{3}\pi r^2 h, S=\pi rl$ 

 

Sphere

 $V=\cfrac{4}{3}\pi r^3 , S =4\pi r^2$

Coordinates , Graphs , Lines

Real Numbers , Intervals, And Inequalities 

     Classification Of Real Numbers 

     The simplest numbers are the natural numbers : 

$1,2,3,4,5, \dots $

     The natural numbers form a subset of a larger class of numbers called the $integers$

$\dots ,-4,-3,-2,-1,0,1,2,3,4,\dots $

     The integers in turn are a subset of a still larger class of numbers called the $rational numbers$ . With the exception that division by $zero$ is ruled out, the rational numbers are formed by taking ratios of integers . Example 

$\cfrac{2}{3},\cfrac{7}{5},\cfrac{6}{1},\cfrac{0}{9},-\cfrac{5}{2}\left(=\cfrac{-5}{2}=\cfrac{5}{-2}\right)$

     Observe that every integer is also a rational number because an integer $p$ can be written as the ration $p=p/1$.

     The early Greeks believed that the size of every physical quantity could, in theory, be represented by a rational number . They reasoned that size of a physical quantity must consist of a certain whole number of units plus some fraction $m/n$  of an additional unit. This idea was shattered in the fifth century B.C. by Hippasus of Metapontum who demonstrated the existence of irrational numbers, that is,numbers that cannot be expressed as the ratio of integers.Using geometric methods , he showed that the hypotenuse of the right triangle in Figure 1.1.1 cannot be expressed as the ratio of integers, thereby proving that $\sqrt{2}$ is an irrational number. Other examples of irrational numbers are

Figure 1.1.1

$\sqrt{3},\sqrt{5}, 1+\sqrt{2},\sqrt[3]{7}, \pi ,\cos 19^\circ$ 

The rational and irrational numbers together comprise a  larger class of numbers,called real numbers or sometimes the real number system.

In computations with real numbers, division by zero is never allowed because a relation ship of the form $y=p/0$ would imply that .

$0\dot y=p$

If $p$ is different from zero, this equation is contradictory ; and if $p$ is equal to zero, this equation is satisfied by any number $y$, so the ratio $0/0$ does not have a unique value - a situation that is mathematically unsatisfactory . For these reasons such symbols as $p/0$ and $0/0$ are not assigned a value ; they are said to be undefined . 

Because the square of a real number cannot be negative, the equation

$x^2=-1$

has no solution on the real number system . In the eighteenth century mathematicians remedied this problem by inventing a new number, which they denoted by

$i=\sqrt{-1}$

and which they defined to have the property $i^2=-1$. This, in turn, led to the development of the complex numbers, which are numbers of the form 

$a+bi$

where $a$ and $b$ are real numbers. Some examples are 

$2+3i\;[a=2,b=3],\; 3-4i\;[a=3,b=-4],\;6i \;[a=0,b=6],\; \cfrac{2}{3} [a=\cfrac{2}{3},b=0]$

Observe that every real number $a$ is also a complex number because it can be written as 

$a=a+0i$

Thus, the real numbers are a subset of the complex numbers. Those complex numbers that are not real numbers are called imaginary numbers . We shall be concerned primarily with real numbers; however, complex numbers will arise in the course of solving equations . For example, the solutions of the quadratic equation 

$ax^2+bx+c=0$

which are given by the quadratic formula

$x=\cfrac{-b\pm \sqrt{b^2-4ac}}{2a}$

 are imaginary if the quantity $b^2-4ac$ [called the discriminant of (1)] is negative.

The hierarchy of numbers is summarized in Figure 1.1.2

Figure  1.1.2

Rational and irrational numbers can be distinguished by their decimal representations . Rational numbers have decimals that are repeating , by which we mean that there is some point in the decimal representation at which the digits that follow consist of a fixed block of integers repeated over and over. For example,

$\cfrac{4}{3} =1.333\dots ,\cfrac{3}{11}=.272727\dots ,\cfrac{5}{7}=.714285714285714285\dots $

Repeating decimal that consist of zeros from some point on are called terminating decimal. Some example are 

$\cfrac{1}{2}=.50000 \dots ,\cfrac{12}{4}=3.0000\dots ,\cfrac{8}{25} = .32000\dots $

It is usual to omit the repetitive zeros in terminating decimals . For example,

$\cfrac{1}{2}=.5 ,\cfrac{12}{4}=3 , \cfrac{8}{25}=.32$

Moreover, it has become common to denote repeating decimals by writing the repeating digits only once , but with a bar over them to indicate the repetition . For example,

$\cfrac{4}{3}=1.\overline{3}, \cfrac{3}{11} =.\overline{27}, \cfrac{5}{7}=.\overline{714285}$


Rational numbers are represented by repeating decimal, and conversely every repeating decimal represents a rational number. Thus , the irrational numbers can be viewed as those real numbers that are represented by nonrepeating decimals. For example , the decimal

$.101001000100001000001 \dots$

does not repeat because the number of zeros between the ones keeps growing . Thus , it represents an irrational number .

     Irrational numbers cannot be represented with perfect accuracy in decimal notation . For example $\pi$ is only approximated by the decimal $3.14$ . Moreover , on matter how many decimal places we use , even if we compute $\pi$ to $2000$ places, as in Figure 1.1.3, we still have only an approximation to $\pi$

Figure 1.1.3

  
REMARK . Beginning mathematics students are sometimes taught to approximate $pi$ by $\cfrac{22}{7}$ Note, however , that

$\cfrac{27}{7}=3.\overline{142857}$

is a rational number whose decimal representation begins to differ from $\pi$ in the third decimal place .