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Functions And Graphs

2 fungsi dan grafik

2-1 FUNCTIONS

Physical chemistry, like all the physical sciences, is concerned with the dependence of one or more variables of a system upon other variables of the system. For example, suppose we wished to know how the volume of a gas varies with temperature. With a little experimentation in the laboratory, we would find that the volume of a gas varies with temperature in a very specific way. Careful measurements would show that the volume of a gas V1 at any temperature t on the Celsius scale obeys the specific law

V1 = V0 (1 + αt) (2-1)

2 - 1 fungsi kimia fisik, sukai semua ilmu eksakta,mempunyai kaitan dengan ketergantungan

dari satu atau lebih variabel dari satu sistem pada saat variabel lain dari sistem. antara lain, seandainya kita mengingin untuk mengetahui bagaimana volume dari satu gas membedakan dengan suhu. dengan cecah percobaan pada Laboratoria, kita akan menemukan bahwa volume dari satu gas membedakan dengan suhu pada satu sangat spesifik cara. pengukuran saksama akan memperlihatkan bahwa volume dari satu gas V1 di apapun temperatur t pada celcius menskalakan mematuhi hukum spesifik

Where V0 is the volume of the gas at 0˚C and α is a constant known as the coefficient of expansion the gas. This equation predicts that there is a one to one correspondence between the volume of a gas and its temperature. That is, for every value of value of t substituted into equation (2.1), there is a corresponding value for V1

di mana V0 adalah volume dari gas di dalam memasuki ' c dan satu adalah satu telah tetap dikenal sebagai koefisien muai gas. penyamaan ini meramalkan yang terdapat sebuah sesuatu ke satu penyesuaian di antara volume dari satu gas dan ini temperatur. Itu adalah, bagi setiap nilai berharga t mengganti ke dalam penyamaan (2. 1 ), terdapat sebuah nilai sesuai untuk V1. .

Let us define a collection of temperatures as a mathematical set T = (t1,t2,t3,t4,...) andThe corresponding volumes as another set V = {V1,V2,V3,V4,...}. A mathematical set isdefined as a collection of numbers, each member of the set called an element, so we see that our collection of numbers, each member of the set called an element, so we see that our collection of temperatures and volumes satisfies this definition. If there is associated with each element of set T at least one element in the other set V, then this association is said to constitute a function from T to V, written f : t → V1. that is, the function takes every

element in set T into the corresponding element in set V. we see the equation (2-1) satisfies this condition . since Vt is a function of t- that is, the value of V depends on the value of t- the above expression can be written f : t → f(t), where Vt = f(t). remember that f(t), read "f of t," does not mean f is multiplied by t, but that f(t) is the value of V0(1+ α t)at t. hence, we can write

f(t) = V1 ( 1 + αt ) (2-2)

Mari kita mendefinisikan satu koleksi suhu sebagai satu setelan matematis t = (t1,t2,t3,t4,...) dan volume sesuai sebagai setelan lain v = {v1,v2,v3,v4,...}. Satu setelan matematis didefinisikan sebagai satu koleksi angka, masing-masing anggota dari dipanggil yang cocok satu elemen, sehingga kita melihat bahwa koleksi kita dari angka, masing-masing anggota dari dipanggil yang cocok satu elemen, sehingga kita melihat bahwa koleksi kita dari suhu dan volume memuaskan definisi ini. kalau di situ dihubungkan dengan masing-masing elemen setelan t paling tidak sesuatu unsur di yang lain menyetel v, kemudian asosiasi ini dikatakan untuk mendasari satu fungsi dari t ke v, ditulis f:t -> V1. bahwa adalah, fungsi mengambil tiap-tiap elemen di setelan t ke dalam elemen sesuai di setelan V. kita melihat penyamaan (2 - 1) puaskan kondisi ini. sejak v1 adalah satu fungsi dari t yang adalah, nilai dari v bergantung kepada nilai dari t ekspresi di atas dapat ditulis f:t -> f ( t), di mana V1=f (t ). ingat bahwa f (t ), bacaan "f t," tidak berarti f diperbanyak oleh t, tapi itu f (t ) adalah nilai dari V0 (1 + di)di t. maka, kita dapat tulis (2 - 2)

A function, then, is defined as a correspondence between elements of two mathematical sets.

Sebuah fungsi, kemudian, didefinisikan sebagai satu penyesuaian di antara unsur dua setelan matematis.

In the above example, V1 was considered to be a function of only a single variable t. such an equation , V1 = f(t), can be represented by a series of points on a two dimensional

Cartesian coordinate system. Physicochemical system, however, usually depend on more than one variable. Thus, it is necessary to extend the definition of function given above to include function of more than one variable. For example, we find experimentally that the volume of a gas will vary with temperature according to equation (2-2) only if the pressure of the gas is held constant. Thus, the volume of a gas not only a function of temperature, but also is a function of pressure. Careful measurements in the laboratory will show that for most gases at or around room temperature and one atmosphere pressure the law relating the volume of a gas simultaneously to the temperature and the pressure of a gas is the well-known ideal gas law

V = RTP = f (T,P) (2-3)

Where R is a constant. equation (2-3) implies that there is a one-to-one correspondence between three sets of numbers: a set of volumes, V = {V1,V2,V3, V4,...}; a set of temperatures on the absolute temperature scale, T = {T1,T2,T3,T4,...}; and a set of pressure, P = {P1,P2,P3,P4,...}. These three sets can be represented graphically on a three-dimensional coordinate system by plotting V along one axis, T along a second axis, P along the third axis. Such graphs of P, V, and T commonly are called Phase diagrams.

2-2 GRAPHICAL REPRESENTATION OF FUNCTION

As we saw above, one of the most convenient ways to represent the functional dependence of the variables of the system is by the use of coordinate system. This is because each set of numbers is easily represented by a coordinate axis, and the graphs that result give an immediate visual representation of the behaviour. In this section we shall explore several types of graphical representation of function. We begin with functions that describe a linear dependence between the variables.

Equations of the First Degree. Equation that define functions showing a linear dependence between variables are known as equations of the first degree, or first-degree equation . These function describe a dependence commonly called the direct proportion, and the equation are called first-degree equations because all the variables in these equations are raised only to the first power. First degree equations have the general form

f (x) = mx + b (2-4)

Persamaan tingkat satu. Persamaan yang mendefinisikan fungsi ketergantungan sebuah garis lurus diantara variabel yang dikenal dengan tingkat satu dari sebuah persamaan, atau persamaan tingkat satu. Fungsi ini mendeskripsikan ketergantungan yang biasanya disebut proporsi langsung, dan persamaan ini disebut persamaan tingkat satu

karena semua variabel di persamaan ini dinaikkan hanya untuk pangkat pertama. Persamaan tingkat pertama punya bentuk umum

f(x)=mx+b (2 - 4)

Where m and b are constants. As an example of a first-degree equation, consider the

equation ˚F = 95 ˚C + 32. Illustrated in fig. 2-1 this familiar equation describes the

temperature of a system on the Fahrenheit scale and its relationship to the temperature on the Celsius temperature scale. Note that, indeed, the equation is that of a straight line. From the graph we can determine the significance of the constant m and b in Equation (2-4). let us consider the latter, b = 32. We can see that the line crosses the ˚F-axis at the point where

the value of ˚C is zero . If we substitute zero for ˚C into the equation ˚F = 95 ˚C + 32, we

obtain ˚F = 32. But this is just the value of b in the equation. Hence, (0,b) are the coordinates of the point where the line crosses the ˚F-or y-axis, and thus b is called the y-intercept. It also may be of interest to consider the value of ˚C for which ˚F = f(˚C) = 0, this point represents the point where the line crosses the ˚C-or x-axis, and is known as the zero of the function. Rearranging equation (2-4) gives

0 = mx + b or x = - bm (2-5)

˚F240

(Figure 2-1) Graph of ˚F = 95 ˚C+32

Dimana m dan b konstan. Misalnya pada persamaan tingkat satu, pertimbangkan persamaan ˚F=9/5˚C+32, dijelaskan dalam gambar 2.1. Persamaan yang sudah lazim ini menjelaskan temperatur dari suatu sistem dalam skala Fahrenheit dan hubungannya dengan skala Celcius dalam skala temperatur. Melihat hal itu, bahwasannya, persamaan adalah sebagai suatu garis lurus. Dari grafik kita bisa menentukan arti dari nilai m dan b dalam persamaan (2-4). Mari kita perhatikan yang terakhir, b=32. Kita dapat melihat persilangan garis sumbu ˚F pada titik dimana harga ˚C adalah nol. Apabila kita membagi nol untuk ˚C dalam persamaan ˚F=9/5˚C+32, kita memperoleh ˚F=32. Tapi ini hanyalah nilai dari b dalam persamaan. Karenanya, (0,b) adalah kordinat dari titik dimana persilangan garis ˚F atau sumbu y dan demikian b disebut intercept y. Itu juga mungkin menarik untuk mempertimbangkan nilai dari ˚C yang mana ˚F = f(˚C) = 0. Titik ini menggambarkan titik dimana persilangan garis ˚C atau sumbu x, dan hal itu dikenal sebagai fungsi nol. Susunan ulang persamaan (2-4) yang diberikan

0 = mx + b atau x = - bm (2-5)

˚F

(Gambar 2-1) Grafik dari ˚F = 95 ˚C+32

240

For the value of x, dividing both sides of the equation by a and rearranging the equation gives

x2 + ba x = -

ca (2-11)

Next, adding (b2/4a2) to both sides of the equation to complete the square gives

x2 + ba

x + b 2

4 a2 = b2

4 a2 - ca (2-12)

Or (x+ b2a )

2

= b2 – 4 ac

4a 2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

(x,y)

(x,y)

(x,y)

y

Figure 2-4 Variation of slope as a function of x.

Taking the square root of both sides of the equation gives

(x+ b2a ) = ±√b2−4 ac

2a(2-13)

or

x = −b±√b2−4 ac2a

(2-14)

Which is well known quadratic formula.

Sometimes the zero’s of the equation, called the roots, can be determined by the factoring method. For example, consider the equation

x2 – 3x + 2 = 0 (2-15)

Which can be factored into the terms

(x – 1) ( x – 2) = 0 (2-16)

The roots of the equation now can be found by solving the equations

(x – 1) = 0 and (x – 2) = 0 (2-17)

Which gives x = 1 and x = 2. Substituting a =1, b =-3, and c = 2 into the quadratic formula, equation (2-14), yields the same results.

in cases where the equation defining a particular physical situation is a second -degree equation (or even one of higher order), there arises a problem that is not present when one simply considers the pure mathematics, as we have done above. Since quadratic equations necessarily have two roots, we must decide, in cases where both roots are not the same, which root correctly represents physical situation, even though both are mathematically correct, consider the equilibrium equation

A + B = C + D

Assume that initially the concentrations of A,B,C,and D are each 1 molar. Suppose we wish to determine the equilibrium concentration of A,B,C, and D given that the equilibrium constant in terms of molar concentration, Kc , equals 50. If we assume that at equilibrium the concentration at C is (1+x), molar, then the equilibrium concentration of A,B,C, and D must be (1 - x),(1 - x), and (1 + x) molar, respectively. Substituting these values into the equilibrium constant equation

We have Kc = (C)(D)( A)(B)

50 = (1+x )(1+x)(1−x )(1−x )

(2-18)

Rearranging Equation (2-18) gives the quadratic equation

49x2 – 102x + 49 = 0 (2-19)

Substituting the values a = 49, b=-102, and c =49 into equation (2-14) yields the two solutions x = 1.3 and x 0.75.

We know must decide which value of x is physically correct. If choose x = 1.3, the equilibrium concentration of A and B will be negative numbers, which physically does not make sense. Thus, the physically correct value for x must be 0.75, giving for the equilibrium concentration of A,B,C, and D : 0,25M, 0,25M,1,75M, and 1,75M, respectively. We see, then, that although both roots were mathematically correct, only one root made sense physically.

Exponential and Logarithmic Functions. Exponential functions are functions whose defining equation is written in the general form

f(x) = ax (2-20)

Where a > 0. An important exponential function that is used extensively in physical chemistry, and indeed in the physical sciences as a whole, is the function

f(x) = ex (2-21)

Where the constant e is a non-terminating, non-repeating decimal having the value, to five significant figures,

e = limx →0

(1+x )1x = 1 +

11! +

12! +

13! + ......

= 2, 7183

This function is illustrated in fig. 2-5. note that all exponentials have the point (0,1) in common, since a˚=1 for any a. also note that there are no zero’s to the function, since the

function approaches zero as x approaches -∞. The expression limx→0 means that (1+x)1/x

approaches a value of 2, 7183 as x approaches 0, and is read "in the limit that x approaches zero." the physical significance of the constant e will be discussed in chapter 3.

There is a direct relationship between exponential functions and logarithmic function. The power of exponent to which the constant a is raised in the equation y = axis called the logarithm of y to the base a and is written

loga y = x (2-22)

The logarithmic function log2 y = x is illustrated in fig. 2-6. Note, as in the case of exponential functions, that the point (0, 1) is common to all logarithmic functions, since loga 1 = 0 for any a. logarithms have many useful properties and are an important tool in the study of physical chemistry. For this reason the general properties of logarithms are reviewed in chapter 3.

Figure 2-5 graph of y = ex

y

x

Section 2-2 Graphical Representation of Functions

y

x

Figure 2-6 graph of log2 y = x

Circular Functions. A circle is defined as the locus off all points in a plane that are at a constant distance from a fixed point. Circles are described by the equation

(x - a)2 + (y - b)2 = r2 (2-23) Where a and b are the coordinates of the center of the circle (the fixed point) and r is the radius. A unit circle is one with its center at the origin and a radius equal to unity.

x2 + y2 = 1 (2-24)

Consider, now, the triangle inscribed in the unit circle shown in fig. 2-7. Let us define three functions: sine (abbreviated sin), which takes the angle (Ө) into the y-coordinate of a point (x,y), cosine (abbreviated cos), which takes the angle (Ө) into the x-coordinate of the point (x,y), and tangent (abbreviated tan), which is the ratio of y

Figure 2-7 Graph of the unit circle x2 + y2 = 1

TABLE 2-1 DEPENDENCE OF r ON Ө FOR THE FUNCTION r = A cos Ө

(x,y)

Ө Ө Ө(degree) r (degree) r (degree) r 0 1.000A 135 -0.707A 270 0 30 0.866A 150 -0.866A 300 0.500A 45 0.707A 180 -1.000A 315 0.707A 60 0.500A 210 -0.866A 330 0.866A 90 0 225 -0.707A 360 1.000A 120 -0.500A 240 -0.500A

To x. Thus,sin Ө = ycos Ө = x

tan Ө = yx

sin ӨcosӨ (2-25)

These functions are called circular or trigonometric functions. Note that equations (2-25) are just the transformation equations (1-4) with r =1. It is interesting to compare the graphs of functions, such as sin Ө and cos Ө, in linear coordinates (coordinates in which Ө is plotted along one axis) to those in plane polar coordinates. Consider, for example the equation r = A cos Ө, where A is a constant. Such an equation can be used to describe the wave properties of p-type atomic orbitals in two dimensions, the functional dependence of r upon Ө can be seen in table 2-1.

When r versus Ө is plotted in linear coordinates [shown in fig. 2-8(a)], the typical cosine curve results. On the other hand, when r versus 0 is plotted in polar coordinates [shown in fig. 2-8(b)], the more familiar shape of the p-orbital can be seen. It is important to note that both graphs are equivalent, the shapes of the curves depending merely on the choice of coordinate system.

TABEL 2-1 KETERGANTUNGAN DARI r PADA Ө UNTUK FUNGSI r = A cos Ө

Ө Ө Ө(degree) r (degree) r (degree) r 0 1.000A 135 -0.707A 270 0 30 0.866A 150 -0.866A 300 0.500A 45 0.707A 180 -1.000A 315 0.707A 60 0.500A 210 -0.866A 330 0.866A 90 0 225 -0.707A 360 1.000A 120 -0.500A 240 -0.500A

ke x. Dengan demikian,

sin Ө = ycos Ө = x

tan Ө = yx

sin ӨcosӨ (2-25)

Fungsi ini disebut sirkuler (bundar) atau fungsi trigonometri. Melihat persamaan itu (2-25) hanyalah persamaan transformasi (1-4) dengan r = 1. Sangat menarik untuk membandingkan grafik dari fungsi, seperti sin Ө dan cos Ө , dalam koordinat garis lurus (koordinat yang mana Ө direncanakan sepanjang satu sumbu) untuk itu dalam koordinat polar pesawat. Pertimbangkan, antara lain pesamaan r = A cos Ө , dimana A adalah tetap. Persamaan tersebut bisa digunakan untuk menjelaskan sifat gelombang dari orbital p atom sejenis pada dua dimensi. Ketergantungan fungsional dari r pada Ө dapat dilihat dalam Tabel 2-1.

Saat r banding Ө direncanakan dalam koordinat garis lurus (ditunjukkan dalam Gambar 2-8 (a)), hasil kurva kosinus membelok. Di sisi lain, ketika r banding Ө direncanakan dalam koordinat polar (ditunjukkan dalam Gambar 2-8 (b)), yang lebih dikenal dengan bentuk dari orbital p dapat dilihat. 1 Penting untuk dicatat bahwa kedua grafik yang ekuivalen, bentuk dari kurva hanya bergantung pada pemilihan sistem koordinat.

2-3 ROOTS TO POLYNOMIAL EQUATIONS

We saw in the previous sections that the zeroes of the function (the roots) can be found easily if the equations are first- or second-degree equations. But how do we find the roots to equations that are not linear or quadratic? Before the age of computers this was not a simple task. One standard way to find the roots of polynomial equation without using a computer is to graph the function. For example, consider the equation

y = x4 + x3 – 3x2 – x + 1

In polar coordinates, negative values of r have no meaning, so we are actually plotting |r|= A cos Ө

2-3 AKAR PERSAMAAN POLINOMIAL

Kita melihat pada bab sebelumnya bahwa fungsi dari nol (akar) dapat ditemukan dengan mudah apabila pada persamaan tingkat satu atau dua. Tapi bagaimana kita menemukan akar untuk persamaan yang tidak berupa garis lurus atau persaman kuadrat? Sebelum masa komputer, ini bukanlah tugas sederhana. Satu standar cara untuk menemukan akar dari

persamaan polinomial tanpa mengunakan komputer adalah membuat grafik fungsi. Antara lain,pertimbangkan persamaan

y = x4 + x3 – 3x2 – x + 1

Di koordinat polar, nilai negatif dari r tidak punya arti, sehingga kita sebenarnya merencanakan |r|= A cos Ө

Section 2-3 Roots to Polynomial Equations

(a) (b)

Figure 2.8 Graphs of r = A cos Ө plotted in (a) linear coordinates and (b) polar coordinates.

If we plot this function from x = -3 to x = +3, we obtain the graph shown in fig. 2-9. The roots to the equation are the values of x for which y = 0, or the points on graph where the graph crosses the x-axis. Careful examination of the graph will show that the roots are x = -2.09, x = -0.74, x = + 0.47, and x = + 1.36. In

Fig. 2-9 Graph of y = x4 + x3 – 3x2 – x + 1.

Chapter 11 we shall discuss numerical methods of finding roots to polynomial equations using a computer.

SUGGESTED READINGS

1. BRADLEY, GERALD L, And SMITH, KARI. J., Calculus, Prentice-Hall, Inc., Upper Saddle River, NJ, 1995.2. 3.4.

PROBLEMS

1. Determine the zero’s of the following functions:abcde

2. Plot the following functions in plane polar coordinates from 0 to 2 (phi) (remeber that in polar coordinates, negative Values of r have no meaning):abc

3. Plot the following functions in Cartesian coordinates:abcd

4. Plot the following functions choosing suitable coordinate axes:

abcdef

5. Plot the functions in problem 4 choosing coordinates so that a straight lineresults.

6. Evaluate ....... for the following:ab

7. Radioactive decay is a first - order processIn which the concentration of the radioactive material C is related to time r by the equation

C= C0e-kt Where Co and k are constants (e is the

Exponential). Given the following data, determine the values of Co and k by plotting the data in such a way that a straight line result.

8. Using the graphical method, determine the roots of the following equations:abcd