E x a m p l e : ( 12 ) : ( + 4 ) = 3.

Transcription

1 Algebraic Transformations Operations with negative and positive numbers Absolute value (modulus): for a negative number this is a positive number, received by changing the sign " " by " + "; for a positive number and zero this is the number itself. The designation of an absolute value (modulus) of a number is the two straight brackets insideof which the number is written. E x a m p l e s : 5 = 5, 7 = 7, 0 = 0. Addition: 1) at addition of two numbers of the same sign their absolute values are added and before the sum their common sign is written. E x a m p l e s : ( + 6 ) + ( + 5 ) = 11 ; ( 6 ) + ( 5 ) = 11 ; 2) at addition of two numbers with different signs their absolute values are subtracted(the smaller from the greater) and a sign of a number, having a greater absolute value is chosen. E x a m p l e s : ( 6 ) + ( + 9 ) = 3 ; ( 6 ) + ( + 3 ) = 3. Subtraction: it is possible to change subtraction of two numbers by addition, thereat a minuend saves its sign, and a subtrahend is taken with the back sign. E x a m p l e s : ( + 8 ) ( + 5 ) = ( + 8 ) + ( 5 ) = 3; ( + 8 ) ( 5 ) = ( + 8 ) + ( + 5 ) = 13; ( 8 ) ( 5 ) = ( 8 ) + ( + 5 ) = 3; ( 8 ) ( + 5 ) = ( 8 ) + ( 5 ) = 13. Multiplication: at multiplication of two numbers their absolute values are multiplied, and a product has the sign " + ", if signs of factors are the same, and " ", if the signs are different. The next scheme ( a rule of signs at multiplication) is useful: + + = + + = + = = + At multiplication of some factors (two and more ) a product has the sign " + ", if a number of negative factors is even, and the sign " ", if this number is odd. E x a m p l e : Division: at division of two numbers the first absolutevalue is divided by the second and a quotient has the sign " + ", if signs of dividend and divisor are the same, and " ", if they are different. The same rule of signs as at multiplication acts: + : + = + + : = : + = : = + E x a m p l e : ( 12 ) : ( + 4 ) = 3.

2 Monomials and polynomials Monomial is a product of two or some factors, each of them is either a number, or a letter, or a power of a letter. For example, 3 a 2 b 4, b d 3, 17 a b c are monomials. A single number or a single letter may be also considered as a monomial. Any factor of a monomial may be called a coefficient. Often only a numerical factor is called a coefficient. Monomials are called similar or like ones, if they are identical or differed only by coefficients. Therefore, if two or some monomials have identical letters or their powers, they are also similar (like) ones. Degree of monomial is a sum of exponents of the powers of all its letters. Addition of monomials. If among a sum of monomials there are similar ones, he sum can be reduced to the more simple form: a x 3 y 2 5 b 3 x 3 y 2 + c 5 x 3 y 2 = ( a 5 b 3 + c 5 ) x 3 y 2. This operation is called reducing of like terms. Operation, done here, is called also taking out of brackets. Multiplication of monomials. A product of some monomials can be simplified, only if it has powers of the same letters or numerical coefficients. In this case exponents of the powers are added and numerical coefficients are multiplied. E x a m p l e : 5 a x 3 z 8 ( 7 a 3 x 3 y 2 ) = 35 a 4 x 6 y 2 z 8. Division of monomials. A quotient of two monomials can be simplified, if a dividend and a divisor have some powers of the same letters or numerical coefficients. In this case an exponent of the power in a divisor is subtracted from an exponent of the power in a dividend; a numerical coefficient of a dividend is divided by a numerical coefficient of a divisor. E x a m p l e : 35 a 4 x 3 z 9 : 7 a x 2 z 6 = 5 a 3 x z 3. Polynomial is an algebraic sum of monomials. Degree of polynomial is the most of degrees of monomials, forming this polynomial. Multiplication of sums and polynomials: a product of the sum of two or some expressions by any expression is equal to the sum of the products of each of the addends by this expression: ( p+ q+ r ) a = pa+ qa+ ra opening of brackets. Instead of the letters p, q, r, a any expressions can be taken. E x a m p l e : ( x+ y+ z )( a+ b )= x( a+ b )+ y( a+ b ) + z( a+ b ) = = xa + xb + ya + yb + za + zb. A product of sums is equal to the sum of all possible products of each addend of one sum to each addend of the other sum. 1. Examples

3 2. Solving Problems Using Formulas of Abridged Multiplication

4 Formulas of abridged multiplication [1] ( a + b )² = a² + 2ab + b², [2] ( a b )² = a² 2ab + b², [3] ( a + b ) ( a b ) = a² b², [4] ( a + b )³ = a³ + 3a² b + 3ab² + b³, [5] ( a b )³ = a ³ 3a² b + 3ab² b³, [6] ( a + b )( a² ab + b² ) = a³ + b³, [7] ( a b )( a ² + ab + b² ) = a³ b³. E x a m p l e : Calculate 99³ using the formula [5]. S o l u t i o n : 99³ = (100 1)³ = = Examples 1. 2.

5 3.

6 Solving Problems With Fractions And Proportions Vulgar (simple) fractions A part of a unit or some equal parts of a unit is called a vulgar (simple) fraction. A number of equal parts into which a unit has been divided, is called a denominator; a number of these taken parts, is called a numerator. A fraction record: Here 3 a numerator, 7 a denominator. If a numerator is less than a denominator, then the fraction is less than 1 and called a proper fraction. If a numerator is equal to a denominator, the fraction is equal to 1. If a numerator is greater than a denominator, the fraction is greater than 1. In both last cases the fraction is called an improper fraction. If a numerator is divisible by a denominator, then this fraction is equal to a quotient: 63 / 7 = 9. If a division is executed with a remainder, then this improper fraction can be presented as a mixed number: Here 9 an incomplete quotient ( an integer part of the mixed number ), 2 a remainder ( a numerator of the fractional part ), 7 a denominator. It is often necessary to solve a reverse problem to convert a mixed number into a fraction. For this purpose, multiply an integer part of a mixed number by a denominator and add a numerator of a fractional part. It will be a numerator of a vulgar fraction, and its denominator is saved the same. Reciprocal fractions are two fractions whose product is 1. For example, 3 / 7 and 7 / 3 ; 15 / 1 and 1 / 15 and so on.

7 Operations with vulgar fractions Extension of a fraction. A fraction value isn t changed, if to multiply its numerator and denominator by the same non-zero number. This transformation of a fraction is called an extension of a fraction. For instance: Cancellation of a fraction. A fraction value isn t changed, if to divide its numerator and denominator by the same non-zero number. This transformation of a fraction is called a cancellation of a fraction. For instance: Comparison of fractions. From two fractions with the same numerators that one is more, a denominator of which is less: From two fractions with the same denominators that one is more, a numerator of which is more: To compare two fractions, which have different both numerators and denominators, it is necessary to extend them to reduce to the same denominators. E x a m p l e. Compare the fractions: S o l u t i o n. Multiply numerator and denominator of the first fraction - by denominator of the second fraction and numerator and denominator of the second fraction - by denominator of the first fraction: The used transformation of fractions is called a reducing of fractions to a common denominator. Addition and subtraction of fractions. If denominators of fractions are the same, then in order to add the fractions it is necessary to add their numerators; in order to subtract the fractions it is necessary to subtract their numerators (in the same order). The received sum or difference will

8 be a numerator of the result; a denominator is saved the same. If denominators of fractions are different, before these operations it is necessary to reduce fractions to a common denominator. At addition of mixed numbers a sum of integer parts and a sum of fractional parts are found separately. At subtracting mixed numbers we recommend at first to reduce the mixed numbers to improper fractions, then to subtract these fractions and after this to convert the result into a mixed number again (in case of need). E x a m p l e. Multiplication of fractions. To multiply some number by a fraction means to multiply it by a numerator and to divide a product by a denominator. Hence, we have the general rule for multiplication of fractions: to multiply one fraction by another it is necessary to multiply separately their numerators and denominators and to divide the first product by the second. E x a m p l e. Division of fractions. To divide some number by a fraction it is necessary to multiply this number by a reciprocal fraction. This rule follows from the definition of division. E x a m p l e. Ratio and proportion. Proportionality Ratio is a quotient of dividing one number by another. Proportion an equality of two ratios. For instance: 12 : 20 = 3 : 5; a : b = c : d. Border terms of the proportion: 12 and 5 in the first proportion; a and d in the second proportion. Middle terms of the proportion: 20 and 3 in the first proportion; b and c in the second proportion. The main property of a proportion: A product of border terms of a proportion is equal to a product of its middle terms. Two mutually dependent values are called proportional ones, if a ratio of their values is saved as invariable. This invariable ratio of proportional values is called a factor of a proportionality. E x a m p l e. A mass of any substance is proportional to its volume. For instance, 2 liters of mercury weigh 27.2 kg, 5 liters weigh 68 kg, 7 liters weigh 95.2 kg. A ratio of mercury mass to its volume ( factor of a proportionality ) will be equal to: Thus, a factor of a proportionality in this example is density.

9 Vector s Multiplication Vectors This is a vector: A vector has magnitude (how long it is) and direction: The length of the line shows its magnitude and the arrowhead points in the direction. We can add two vectors by simply joining them head-to-tail: And it doesn't matter which order we add them, we get the same result: Subtracting We can also subtract one vector from another: first we reverse the direction of the vector we want to subtract, then add them as usual: Notation A vector is often written in bold, like a or b.

10 A vector can also be written as the letters of its head and tail with an arrow above it, like this: Calculations Now... how do we do the calculations? The most common way is to break up a vector into x and y pieces, like this: The vector a is broken up into the two vectors a x and a y Adding Vectors And we can add the vectors by adding the x parts then adding the y parts: The vector (8.13) and the vector (26,7) add up to the vector (34.20) Example: add the vectors a = (8.13) and b = (26,7) c = a + b c = (8.13) + (26,7) = ( ) = (34.20) Subtracting Vectors Remember: to subtract, first reverse the vector we want to subtract, then add. Example: subtract k = (4,5) from v = (12,2) a = v + k a = (12,2) + (4,5) = (12,2) + ( 4, 5) = (12 4,2 5) = (8, 3) Magnitude of a Vector The magnitude of a vector is shown by two vertical bars on either side of the vector:

11 a OR it can be written with double vertical bars (so as not to confuse it with absolute value): a We use Pythagoras' theorem to calculate it: a = ( x 2 + y 2 ) Example: what is the magnitude of the vector b = (6,8)? b = ( ) = ( ) = 100 = 10 A vector with magnitude 1 is called a Unit Vector. Vector vs Scalar When using vectors we call an ordinary number a "scalar". Scalar: just a number (like 7 or 0,32)... definitely not a vector. A vector is often written in bold, so we know it is not a scalar: so c is a vector, it has magnitude and direction but c is just a value, like 3 or 12,4 Example: kb is actually the scalar k times the vector b. Multiplying a Vector by a Scalar When we multiply a vector by a scalar it is called "scaling" a vector, because we change how big or small the vector is. Example: multiply the vector m = (7,3) by the scalar 3 a = 3m = (3 7,3 3) = (21,9) It still points in the same direction, but is 3 times longer (And now you know why numbers are called "scalars", because they "scale" the vector up or down.)

12 Multiplying a Vector by a Vector (Dot Product and Cross Product) How do we multiply two vectors together? There is more than one way! The scalar or Dot Product (the result is a scalar). The vector or Cross Product (the result is a vector). (Read those pages for more details.) More Than 2 Dimensions The vectors we have been looking at have been 2 dimensional, but vectors work perfectly well in 3 or more dimensions: Example: add the vectors a = (3,7,4) and b = (2,9.11) c = a + b c = (3,7,4) + (2,9.11) = (3+2,7+9,4+11) = (5.16,15) Example: subtract (1,2,3,4) from (3,3,3,3) (3,3,3,3) + (1,2,3,4) = (3,3,3,3) + ( 1, 2, 3, 4) = (3 1,3 2,3 3,3 4) = (2,1,0, 1) Example: what is the magnitude of the vector w = (1, 2,3)?

13 w = ( ( 2) ) = ( ) = 14 Magnitude and Direction We may know a vector's magnitude and direction, but want its x and y lengths (or vice versa): <=> Vector a in Polar Coordinates Vector a in Cartesian Coordinates You can read how to convert them at Polar and Cartesian Coordinates, but here is a quick summary: From Polar Coordinates (r,θ) to Cartesian Coordinates (x,y) x = r cos( θ ) y = r sin( θ ) From Cartesian Coordinates (x,y) to Polar Coordinates (r,θ) r = ( x 2 + y 2 ) θ = tan -1 ( y / x )

14 Examples Solving Problems on Proving Identities 1. 2.

15 3. 4.

16 Linear Functions and Their Graphs. Systems of Liner Equations Constants and variables Applying mathematics in studying of laws of nature and using them in technique, we meet with constants and variables. A variable is a value, which can be changed at the conditions of the considered problem; a constant cannot be changed at these conditions. The same value can be a constant for one problem and a variable for the other. E x a m p l e. An acceleration of a gravity is a constant for the same width of Earth, but it changes depending on a width, i.e. in other words is a variable. Variables are marked usually by the last letters of the Latin alphabet: x, y, z, and constants by the first ones: a, b, c,. Functional dependence between two variables Two variables x and y are tied by a functional dependence, if for each value of one of them it is possible to receive by the certain rule one or some values of another. E x a m p l e. A temperature T of water boiling and atmosphere pressure p are tied by a functional dependence, because each value of pressure corresponds to a certain value of the temperature and inversely. So, if p = 1 bar, then T = 100 C; if p = 0.5 bar, then T = 81.6 C. A variable, values of which are given, is called an argument or an independent variable; the other variable, values of which are found by the certain rule is called a function. Usually an argument is marked as x, and a function is marked as y. If only one value of function corresponds to each value of argument, this function is called a single-valued function; otherwise, if there are many corresponding values, this function is called a multiple-valued function ( two-valued, three-valued and etc.). E x a m p l e. A body is thrown upwards; h is its height over a ground, t is the time, passed from a throwing moment. h is a single-valued function of t, but t is a two-valued function of h, because the body is on the same height twice: the first time at an assent, the second time at a fall. The formula binding variables h and t ( initial velocity v 0 and an acceleration of a gravity g are constants here ), shows that we have only one value of h at the given t, and two values of t at the given h ( they are determined by solving the quadratic equation ). Representation of function by formula and table Many of functions can be represented ( exactly or approximately ) by simple formulas. For example, the dependence between an area S of a circle and its radius r is given by the formula S = r 2 ; the previous example shows the dependence between a height h of a thrown body and a flying time t. But this formula is in fact an approximate one, because it does not consider neither a resistance of air nor a weakening of Earth gravity by a height. It is very often impossible to represent a functional dependence by a formula, or this formula is an uncomfortable for calculations. In these cases a function is represented by a table or a graph.

17 E x a m p l e. The functional dependence between a pressure p and a temperature of water boiling T cannot be presented by the one formula, so it is It is obvious, that any table cannot contain all values of argument, but an available for practice table must contain so many values, that they are enough to work or to receive additional values by interpolating the existing ones. Coordinates. Graphical representation of functions Coordinates. Two mutually perpendicular straight lines XX and YY ( Fig.1 ) form a coordinate system, called Cartesian coordinates. Straight lines XX and YY are called axes of coordinates. The axis XX is called an x-axis, the axis YY an y-axis. The point O of their intersection is called an origin of coordinates. An arbitrary scale is selected on each axis of coordinates. Find projections P and Q of a point M to the coordinate axes XX and YY. The segment OP on the axis XX and a number x, measuring its length according to the selected scale, is called an abscissa or x-coordinate of a point M ; the segment OQ on axis YY and a number y, measuring its length - an ordinate or y-coordinate of a point M. Values x = OP and y = OQ are called Cartesian coordinates ( or simply coordinates ) of a point M. They are considered as positive or negative according to the adopted positive and negative directions of coordinate axes. Usually positive abscissas are placed by right on an axis XX ; positive ordinates by upwards on an axis YY. On Fig.1 we see: a point M has an abscissa x = 2, an ordinate y = 3; a point K has an abscissa x = 4, an ordinate y = 2.5. This can be written as: M ( 2, 3 ), K ( 4, 2.5 ). So, for each point on a plane a pair of numbers (x, y) corresponds, and inversely, for each pair of real numbers (x, y) the one point on a plane corresponds.

18 Graphical representation of functions. To represent a functional dependence y = f ( x ) as a graph it is necessary: 1) to write a set of values of the function and its argument in a table: 2) To transfer the coordinates of the function points from the table to a coordinate system, marking according to the selected scale a set of x-coordinates on x-axis and a set of y-coordinates on y-axis ( Fig.2 ). As a result a set of points A, B, C,..., F will be plotted in our coordinate system. 3) Joining marked points A, B, C,..., F by a smooth curve, we receive a graph of the given functional dependence. Such graphical representation of a function permits to visualize a behavior of the function, but has an insufficient attainable accuracy. It s possible, that intermediate points, not plotted on a graph, lie far from the drawing smooth curve. Good results also depend essentially on a successful choice of scales. That is why, you should define a graph of a function as a locus, coordinates of points of which M (x, y) are tied by the given functional dependence. 1. Proportional values. If variables y and x are direct proportional, then the functionaldependence between them is represented by the equation: y = kx, where k is a constant a factor of proportionality. A graph of a direct proportionality is a straight line, going through an origin of coordinatesand forming with an x-axis an angle, a tangent of which is equal to k : tan = k ( Fig.8 ).

19 Therefore, a factor of proportionality is called also a slope. There are shown three graphswith k = 1/3, k = 1 and k = 3 on Fig Linear function. If variables y and x are tied by the 1-st degree equation: A x + B y = C, ( at least one of numbers A or B is non-zero ), then a graph of the functional dependence is a straight line. If C = 0, then it goes through an origin of coordinates, otherwise - not.graphs of linear functions for different combinations of A, B, C are represented on Fig.9. Systems of two simultaneous linear equations in two unknowns Systems of two simultaneous linear equations in two unknowns have the shape: where a, b, c, d, e, f numerical coefficients; x, y unknowns. Solution of these simultaneous equations can be found by two basic methods: Substitution. 1). From one equation we express one of unknowns, for example x, by coefficients and another unknown y : x = ( c by ) / a, (2)

20 2). Substitute in the second equation instead of x : d ( c by ) / a + ey = f. 3). Now, solving the last equation, find y : y = ( af cd ) / ( ae bd ). 4). Substitute this value for y in the expression (2) instead of y : x = ( ce bf ) / ( ae bd ). E x a m p l e. Solve the system of simultaneous equations: From the first equation express x by coefficients and y : x = ( 2y + 4 ) / 3. Substitute this expression into the second equation and find y : ( 2y + 4 ) / 3 + 3y = 5, hence y = 1. Now find x, substituting the found value instead of y into expression for x: x = ( ) / 3, from here x = 2. Addition or subtraction. This method consists in the following. 1). Multiply both sides of the first equation of the system (1) by ( d ) and both sides of the second equation by a and add them: From here we receive: y = ( af cd ) / ( ae bd ). 2). Substitute the found value of y into any equation of the original system (1) : ax + b( af cd ) / ( ae bd ) = c. 3). Find another unknown x : x = ( ce bf ) / ( ae bd ). E x a m p l e. Solve the system of simultaneous equations:

21 by the second way ( addition or subtraction ). Multiply the first equation by 1, the second by 3 and add them: 2. From here y = 1. Substitute this value into the second equation ( is it possible to substitute this into the first equation? ): 3x + 9 = 15, hence, x = The second order determinants. We saw, that formulas for solution of the system of two simultaneous linear equations in two unknowns have the shape: x = ( ce bf ) / ( ae bd ), y = ( af cd ) / ( ae bd ). (3) These formulas can be remembered very easily, if to introduce for their numerators and denominators the next symbol:, which will be used to mean an expression: ps qr. This expression is received by crosswise multiplication of numbers p, q, r, s : and the following subtraction of one product from another: ps qr. The sign + is taken for a product of numbers, located on the diagonal, going from the left upper number to the right lower number. The sign for another diagonal, going from the right upper number to the left lower number. For example, The expression is called the second order determinant. Cramer s rule. Using the determinants, the formulas (3) can be written as:

22 Formulas ( 4 ) are called Cramer s rule for solution of the system of two simultaneous linear equations in two unknowns. E x a m p l e. Solve the system of simultaneous equations using Cramer s rule. S o l u t i o n. Here a = 1, b = 1, c = 12, d = 2, e = 3, f = 14. Investigation of solutions of a system of two simultaneous linear equations in two unknowns shows, that depending on coefficients three different cases are possible: 1) coefficients at unknowns in equations are disproportionate: a : d b : e, in this case the system of simultaneous linear equations has a single solution, presented by formulas (4) ; 2) all coefficients of equations are proportional: a: d = b: e = c: f, in this case the system of simultaneous linear equations has an infinite set of solutions, because we have actually one equation instead of two. E x a m p l e. In the system

23 and this system has an infinite set of solutions. ( Why? ) Dividing the first equation by 2 and the second - by 3, we ll receive two identical equations: that is one equation in two unknowns, which has an infinite set of solutions. 3) coefficients at unknowns are proportional, but disproportionate to free terms: a : d = b : e c : f, in this case the system of simultaneous linear equations has no solutions, because we have here contradictory equations. Graphical solving of equations Graphical representation of functions permits to solve approximately any equation in one unknown and a system of two simultaneous equations in two unknowns. To solve a system of two simultaneous equations in two unknowns x and y graphically, we consider each of the equations as a functional dependence between variables x and y and build graphs for these two functions. Coordinates of intersection points of these graphs give us the found values for x and y ( i.e. a solution of this system of simultaneous equations ). According to these graphs the approximate coordinates of the intersection point K are: x = 1.25, y = 2.5. The exact solution of these simultaneous equations is:

24 Quadratic Trinomial. Solving of Quadratic Equations Factoring of a quadratic trinomial Each quadratic trinomial ax 2 + bx+ c can be resolved to factors of the first degree by the next way. Solve the quadratic equation ax 2 + bx+ c = 0. If x 1 and x 2 are the roots of this equation, then ax 2 + bx+ c = a ( x x 1 ) ( x x 2 ). This affirmation can be proved using either formulas for roots of a non-reduced quadratic equation or Viete s theorem. ( Check it, please! ). E x a m p l e. Resolve to the first degree factors the trinomial: 2x 2 4 x 6. S o l u t i o n. At first we solve the equation: 2x 2 4x 6 = 0. Its roots are: x 1 = 1 and x 2 = 3. Hence, 2x 2 4x 6 = 2 ( x + 1 ) ( x 3 ). ( Open the brackets and check the result, please ). Quadratic equation Quadratic equation. Reduced quadratic equation. Non-reduced quadratic equation. Pure quadratic equation. A quadratic equation is an algebraic equation of the second degree: ax 2 + bx + c = 0, (1) where a, b, c the given numerical or literal coefficients, x an unknown. If a = 0, then this equation becomes a linear one. Therefore, we ll consider here only a 0. So, it is possible to divide all terms of the equation by a and then we receive: x 2 + px + q = 0, (2) where p=b/a, q=c/a. This quadratic equation is called a reduced one. The equation (1) is called a non-reduced quadratic equation. If b or c (or both) is equal to zero, then this equation is called a pure one. The examples of pure quadratic equations are following: 4x 2 12 = 0, x 2 + 5x = 0, x 2 = 36. Solution of a quadratic equation Formulas for solution of non-reduced and reduced quadratic equation. In general case of a non-reduced quadratic equation: ax 2 + bx + c = 0, its roots are found by the formula:

25 If to divide all terms of a non-reduced quadratic equation by a (is it possible?), and to sign b / a = p and c / a = q, then we ll receive the reduced quadratic equation: x 2 + px + q = 0, roots of which are calculated by the formula: E x a m p l e. x 2 + 5x + 6 = 0. Here p = 5, q = 6. Then we have: hence, x 1 = 5 / / 2 = 2, x 2 = 5 / 2 1 / 2 3 Properties of roots of a quadratic equation. Viete s theorem Roots of quadratic equation. Discriminant. Viete's theorem. The formula shows, that the three cases are possible: 1) b 2 4 a c > 0, then two roots are different real numbers; 2) b 2 4 a c = 0, then two roots are equal real numbers; 3) b 2 4 a c < 0, then two roots are imaginary numbers. The expression b 2 4 a c, value of which permits to differ these three cases, is called a discriminant of a quadratic equation and marked as D. Viete s theorem. A sum of roots of reduced quadratic equation x 2 + px + q = 0 is equal to coefficient at the first power of unknown, taken with a back sign, i.e. x 1 + x 2 = p, and a product of the roots is equal to a free term, i.e. x 1 x 2 = q.

26 To prove Viete s theorem, use the formula, by which roots of reduced quadratic equation are calculated. Example

27 Power and Roots Powers and roots Operations with powers. 1. At multiplying of powers with the same base their exponents are added: a m a n = a m + n. 2. At dividing of powers with the same base their exponents are subtracted: 3. A power of product of two or some factors is equal to a product of powers of these factors: ( abc ) n = a n b n c n 4. A power of a quotient (fraction) is equal to a quotient of powers of a dividend (numerator) and a divisor (denominator): ( a / b ) n = a n / b n. 5. At raising of a power to a power their exponents are multiplied: ( a m ) n = a m n. All above mentioned formulas are read and executed in both directions from the left to the right and back. E x a m p l e. ( / 15 ) 2 = / 15 2 = 900 / 225 = 4. Operations with roots. In all below mentioned formulas a symbol root ( all radicands are considered here only positive ). means an arithmetical 1. A root of product of some factors is equal to a product of roots of these factors: 2. A root of a quotient is equal to a quotient of roots of a dividend and a divisor: 3. At raising a root to a power it is sufficient to raise a radicand to this power: 4. If to increase a degree of a root by n times and to raise simultaneously its radicand to the n- th power, the root value doesn t change:

28 5. If to decrease a degree of a root by n times and to extract simultaneously the n-th degree root of the radicand, the root value doesn t change: Widening of the power notion. Till now we considered only natural exponents of powers; but operations with powers and roots can result also to negative, zero and fractional exponents. All these exponents of powers require to be defined. Negative exponent of a power. A power of some number with a negative (integer) exponent is defined as unit divided by the power of the same number with the exponent equal to an absolute value of the negative exponent: Now the formula a m : a n = a m - n may be used not only if m is more than n, but also for a case if m is less than n. E x a m p l e. a 4 : a 7 = a 4 7 = a 3. If we want the formula a m : a n = a m - n to be valid at m = n we need the definition of zero exponent of a power. Zero exponent of a power. A power of any non-zero number with zero exponent is equal to 1. E x a m p l e s. 2 0 = 1, ( 5 ) 0 = 1, ( 3 / 5 ) 0 = 1. Fractional exponent of a power. To raise a real number a to a power with an exponent m / n it is necessary to extract the n-th degree root from the m-th power of this number a: About meaningless expressions. There are some expressions: Case 1. where a 0 doesn t exist.

29 Really, if to assume that where x some number, then according to the definition of a division we have: a = 0 x, i.e. a = 0, but this result contradicts to the condition: a 0 Case 2. is any number. Really, if to assume that this expression is equal to some number x, then according to the definition of a division: 0 = 0 x. But this equality is valid at any number x, which was to be proved. Case 3. If to assume, that rules of operations with powers are spread to powers with a zero base, then 0 0 is any number. S o l u t i o n. Consider the three main cases: 1) x = 0 this value doesn t satisfy the equation ( Why? ) ; 2) at x > 0 we receive: x / x = 1, i.e.1 = 1, hence, x any number, but taking into consideration that in this case x > 0, the answer is: x > 0 ; 3) at x < 0 we receive: x / x = 1, i.e. 1 = 1, and the answer is: there is no solution in this case. So, the answer: x > 0. Arithmetical root As we know, an even degree root has two values: positive and negative, so An arithmetical root of the n-th degree of a non-negative number a is called a non-negative number, the n-th power of which is equal to a. An algebraic root of the n-th degree of a given number a iscalled a set of all roots of this number. An algebraic root of an even degree has the two values:positive and negative, for instance:

30 An algebraic root of an odd degrees has a single value: either positive, or negative.for example, the arithmetical root Unlike this, the cube degree root: An arithmetical root is closely connected with the notion of an absolute value(modulus) of number, exactly: 1. Examples 2.

31 Power Functions and Their Graphs Power functions and their graphs Inverse proportionality. If variables y and x are inverse proportional, then the functionaldependence between them is represented by the equation: where k is a constant. y = k / x, A graph of an inverse proportionality is a curve, having two branches ( Fig.10 ). This curveis called a hyperbola. These curves are received at crossing a circular cone by a plane.as shown on Fig.10, a product of coordinates of a hyperbola points is a constant value, equal in this case to 1. In general case this value is k, as it follows from a hyperbola equation: x y = k. The main characteristics and properties of hyperbola: - the function domain: x 0, and codomain: y 0 ; - the function is monotone ( decreasing) at x < 0 and at x > 0, but it is not monotone on the whole, because of a point of discontinuity x = 0 (think, please, why? ); - the function is unbounded, discontinuous at a point x = 0, odd, non-periodic; - there are no zeros of the function. Quadratic function. This is the function: y = ax 2 + bx + c, where a, b, c constants, a 0. In the simplest case we have b = c = 0 and y = ax 2. A graph of this function is a quadratic parabola - a curve, going through an origin of coordinates ( Fig.11 ). Every parabola has an axis of symmetry OY, which is called an axis of parabola. The point O of intersection of a parabola with its axis is a vertex of parabola.

32 A graph of the function y = ax 2 + bx + c is also a quadratic parabola of the same shape, that y = ax 2, but its vertex is not an origin of coordinates, this is a point with coordinates: The form and location of a quadratic parabola in a coordinate system depends completely on two parameters: the coefficient a of x 2 and discriminant D = b 2 4ac. These properties follow from analysis of the quadratic equation roots. All possible different cases for a quadratic parabola are shown on Fig.. Show, please, a quadratic parabola for the case a > 0, D > 0. The main characteristics and properties of a quadratic parabola: - the function domain: < x < + ( i.e. x is any real number ) and codomain: ( answer, please, this question yourself!) ;

33 - the function is not monotone on the whole, but to the right or to the left of the vertex it behaves as a monotone function; - the function is unbounded, continuous in everywhere, even at b = c = 0, and non-periodic; - the function has no zeros at D < 0. ( What about this at D 0? ). Power function. This is the function: y = ax n where a, n constants. At n = 1 we receive the function, called a direct proportionality: y = ax ; at n = 2 - a quadratic parabola; at n = 1 - an inverse proportionality or hyperbola. So, these functions are particular casesof a power function. We know, that a zero power of every non-zero number is 1, thus at n = 0 the power function becomes a constant: y = a, i.e. its graph is a straight line, parallelto an x-axis, except an origin of coordinates ( explain, please, why? ).All these cases (at a = 1 ) are shown on Fig.13 ( n 0 ) and Fig.14 ( n < 0 ).Negative values of x are not considered here, because then some of functions:

34 If n integer, power functions have a meaning also at x < 0, but their graphs have different forms depending on that is n an even or an odd number. On Fig.15 two such power functions are shown: for n = 2 and n = 3. At n = 2 the function is even and its graph is symmetric relatively an axis Y ; at n = 3 the function is odd and its graph is symmetric relatively an origin of coordinates. The function y = x 3 is called a cubic parabola. On Fig.16 the function is represented. This function is inverse to the quadratic parabola y = x 2, its graph is received by rotating the quadratic parabola graph around abisector of the 1-st coordinate angle. (This is the way to receive a graph of every inverse function from its original function). We see by the graph, that this is the two-valued function(the sign ± before the square root symbol says about this). Such functions are not studied in an elementary mathematics, therefore we consider usually as a function one of its branches: either an upper or a lower branch.

35 Examples 1.

36

37 Trigonometric Functions and Their Graphs Trigonometric functions of an acute angle Trigonometric functions of an acute angle are ratios of different pairs of sides of a right-angled triangle ( Fig.2 ). 1) Sine: sin A = a / c ( a ratio of an opposite leg o a hypotenuse ). 2) Cosine: cos A = b / c ( a ratio of an adjacent leg to a hypotenuse ). 3) Tangent: tan A = a / b ( a ratio of an opposite leg to an adjacent leg ). 4) Cotangent: cot A = b / a ( a ratio of an adjacent leg to an opposite leg ). 5) Secant: sec A = c / b ( a ratio of a hypotenuse to an adjacent leg ). 6) Cosecant: cosec A = c / a ( a ratio of a hypotenuse to an opposite leg ). There are analogous formulas for another acute angle B ( Write them, please! ). E x a m p l e. A right-angled triangle ABC ( Fig.2 ) has the following legs: a = 4, b = 3. Find sine, cosine and tangent of angle A. S o l u t i o n. At first we find a hypotenuse, using Pythagorean theorem: c 2 = a 2 + b 2, According to the above mentioned formulas we have: sin A = a / c = 4 / 5; cos A = b / c = 3 / 5; tan A = a / b = 4 / 3. For some angles it is possible to write exact values of their trigonometric functions. The most important cases are presented in the table:

38 Although angles 0 and 90 cannot be acute in a right-angled triangle, but at enlargement of notion of trigonometric functions ( see below), also these angles are considered. A symbol the table means that absolute value of the function increases unboundedly, if the angle approaches the shown value. in Trigonometric functions of any angle To build all trigonometry, laws of which would be valid for any angles ( not only for acute angles, but also for obtuse, positive and negative angles), it is necessary to consider so called a unit circle, that is a circle with a radius, equal to 1 ( Fig.3 ). Let draw two diameters: a horizontal AA and a vertical BB. We count angles off a point A (starting point). Negative angles are counted in a clockwise, positive in an opposite direction. A movable radius OC forms angle with an immovable radius OA. It can be placed in the 1-st quarter ( COA ), in the 2-nd quarter ( DOA ), in the 3-rd quarter ( EOA ) or in the 4-th quarter ( FOA ). Considering OA and OB as positive directions and OA and OB as negative ones, we determine trigonometric functions of angles as follows. A sine line of an angle ( Fig.4 ) is a vertical diameter of a unit circle, a cosine line of an angle - a horizontal diameter of a unit circle. A sine of an angle ( Fig.4 ) is the segment OB of a sine line, that is a projection of a movable radius OK to a sine line; a cosine of an angle - the segment OA of a cosine line, that is a projection of a movable radius OK to a cosine line.

39 Signs of sine and cosine in different quarters of a unit circle are shown on Fig.5 and Fig.6. A tangent line ( Fig.7 ) is a tangent, drawn to a unit circle through the point A of a horizontal diameter. A cotangent line ( Fig.8 ) is a tangent, drawn to a unit circle through the point B of a vertical diameter. A tangent is a segment of a tangent line between the tangency point A and an intersection point ( D, E, etc., Fig.7 ) of a tangent line and a radius line. A cotangent is a segment of a cotangent line between the tangency point B and an intersection point ( P, Q, etc., Fig.8 ) of a cotangent line and a radius line. Signs of tangent and cotangent in different quarters of a unit circle see on Fig.9. Secant and cosecant are determined as reciprocal values of cosine and sine correspondingly. 8. Trigonometric functions. Building trigonometric functions we use a radian as a measure of angles. Then the function y = sin x is represented by the graph ( Fig.19 ). This curve iscalled a sinusoid. The graph of the function y = cos x is represented on Fig.20 ; this is also a sinusoid, received from the graph of y = sin x by its moving along an x-axis to the left for / 2.

40 From these graphs the following main characteristics and properties of the functions are obvious: - the functions have as a domain: < x < + and a codomain: 1 y +1; - these are periodic functions: their period is 2 ; - the functions are bounded ( y 1 ), continuous in everywhere; they are not monotone functions, but there areso called intervals of monotony,inside of whichthey behave as monotone functions ( see graphs Fig.19 and Fig.20); - the functions havean innumerable set of zeros. Graphs of functions y = tan x and y = cot x are shown on Fig. 21 and Fig. 22 correspondingly. The graphs show, that these functions are: periodic (their period is ), unbounded, not monotone on the whole, but they have the intervals of monotony (what intervals?), discontinuous functions (what points of discontinuity these functions have?). The domain and codomain of these functions:

41 Inverse Trigonometric Functions The relation x = sin y permits to find both x by the given y, and also y by the given x ( at x 1 ). So, it is possible to consider not only a sine as a function of an angle, but an angle as a function of a sine. The last fact can be written as: y = arcsin x ( arcsin is read as arcsine ). For instance, instead of 1/2 = sin 30 it is possible to write: 30 = arcsin 1/2. At the second record form an angle is usually represented in a radian measure: / 6 = arcsin 1/2. Definitions. arcsin x is an angle, a sine of which is equal to x. Analogously the functions arccos x, arctan x, arccot x, arcsec x, arccosec x are defined. These functions are inverse to the functions sin x, cos x, tan x, cot x, sec x, cosec x, therefore they are called inverse trigonometric functions. All inverse trigonometric functions are multiple-valued functions, that is to say for one value of argument an innumerable set of a function values is in accordance. So, for example, angles 30, 150, 390, 510, 750 have the same sine. A principal value of arcsin x is that its value, which is contained between / 2 and + / 2 ( 90 and +90 ), including the bounds: / 2 arcsin x + / 2. A principal value of arccos x is that its value, which is contained between 0 and ( 0 and +180 ), including the bounds: 0 arccos x. A principal value of arctan x is that its value, which is contained between / 2 and + / 2 ( 90 and +90 ) without the bounds: / 2 < arctan x < + / 2. A principal value of arccot x is that its value, which is contained between 0 and ( 0 and +180 ) without the bounds: 0 < arccot x <. If to sign any of values of inverse trigonometric functions as Arcsin x, Arccos x, Arctan x, Arccot x and to save the designations: arcsin x, arcos x, arctan x, arccot x for their principal values, then there are the following relations between them: where k any integer. At k = 0 we have principal values.

42 Basic relations for inverse trigonometric functions

43 Transforming of trigonometric expressions Transforming of trigonometric expressions to product

44 Double-, triple-, and half-angle formulas Signs before the roots are selected depending on the quarter, in which the angle is placed.

45 Addition and subtraction formulas Some important correlations The last three formulas are called a universal substitution; they are used at solving some of trigonometric equations and integrating of trigonometric functions.

46 Examples Proving of Trigonometric Identities. Solving of Trigonometric Equations Trigonometric equations 1. 2.

47 3. 4.

48 1. Examples Proving the identity 2.

49 Arithmetic and Geometric Progressions Arithmetic and geometric progressions Sequences. Let s consider the series of natural numbers: 1, 2, 3,, n 1, n,. If to replace each natural number n in this series by some number u n, subordinated to some law, we ll receive a new series of numbers: u 1, u 2, u 3,, u n - 1, u n,, called a numerical sequence. A number u n is called a general term of the numerical sequence. E x a m p l e s of numerical sequences: 2, 4, 6, 8, 10,, 2n, ; 1, 4, 9, 16, 25,, n², ; 1, 1/2, 1/3, 1/4, 1/5,, 1/n,. Arithmetic progression. The numerical sequence, in which each next term beginning from the second is equal to the previous term, added with the constant for this sequence number d, is called an arithmetic progression. The number d is called a common difference. Any term of an arithmetic progression is calculated by the formula: a n = a 1 + d ( n 1 ). A sum of n first terms of arithmetic progression is calculated as: E x a m p l e. Find a sum of the first 100 odd numbers. S o l u t i o n. Use the last formula. Here a 1 = 1, d = 2. So, we have: Geometric progression. The numerical sequence, in which each next term beginning from the second is equal to the previous term, multiplied by the constant for this sequence number q, is called a geometric progression. The number q is called a common ratio. Any term of a geometric progression is calculated by the formula: b n = b 1 q n - 1. A sum of n first terms of geometric progression is calculated as:

50 Infinitely decreasing geometric progression. This is the geometric progression, with q < 1. For it the notion of a sum of infinitely decreasing geometric progression is determined as a number, to which a sum of the first n terms of the considered progression unboundedly approximates at an unbounded increasing of number n. The infinitely decreasing geometric progression sum is calculated by the formula: E x a m p l e. Find the sum of the infinitely decreasing geometric progression: S o l u t i o n. Use the last formula. Here b 1 = 1, q = 1/2. So, we have: Converting of a repeating decimal to a vulgar fraction. Assume, that we want to convert the repeating decimal 0.(3) to a vulgar fraction. Consider this decimal in the more natural form: This is the infinitely decreasing geometric progression with the first term 3/10 and a common ratio q = 1/10. According to the above shown formula the last sum is equal to: Examples 1.

51 2.

52 1. Solving of Exponential Equations Examples 2.

53 3. 4.

54 5.

55 Logarithms Logarithms A logarithm of a positive number N to the base b ( b > 0, b 1 ) is called an exponent of a power x, to which b must be raised to receive N. The designation: & nbsp; &n bsp; This record is identical to the following one: E x a m p l e s : The above presented definition of logarithm may be written as the logarithmic identity : The main properties of logarithms. 1) log b = 1, because b 1 = b. b & nbsp; & nbsp; 2) log 1 = 0, because b 0 = 1. b 3) Logarithm of a product is equal to a sum of logarithms of factors: log ( ab ) = log a + log b. 4) Logarithm of a quotient is equal to a difference of logarithms of dividend and divisor: log ( a / b ) = log a log b. 5) Logarithm of a power is equal to a product of an exponent of the power by logarithm of its base:

56 A consequence of this property is the following: logarithm of a root is equal to the logarithm of radicand divided by the degree of the root: 6) If a base of logarithm is a power, then a value, reciprocal to this power exponent, may be carried out of the logarithm symbol: The two last properties may be united in the general property: 7) The transition module formula ( i.e. a transition from one base of the logarithm to another base ): In the particular case: N = a we have: Common logarithm is a logarithm to the base 10. It marks as lg, i.e. log 10 N = lg N. Logarithms of the numbers 10, 100, 1000,... are equal to 1, 2, 3, correspondingly, i.e. they have as many positive ones as many zeros are placed in the number after one. Logarithms of the numbers 0.1, 0.01, 0.001,... are equal to 1, 2, 3,, i.e. they have as many negative ones as many zeros are placed in the number before one ( including zero of integer part ). Logarithms of the rest of the numbers have a fractional part, called a mantissa. An integer part of logarithm is called a characteristics. Common logarithms are the most suitable for practical use.

57 Natural logarithm is a logarithm to the base е. It marks as ln, i.e. log e N = ln N. The number е is irrational, its approximate value is This number is a limit, which the number ( / n ) n approaches at unbounded increasing of n. Strange though it may seem, natural logarithms are very suitable at different operations in analysis of functions. Calculation of logarithms to the base е is executed quicker, than to any other base. Examples 1. Logarithmic equations

58 5.

59 Triangles Triangle is a polygon with three sides (or three angles). Sides of triangle are signed often by small letters, corresponding to designations of opposite vertices, signed by capital letters. If all the three angles are acute, then this triangle is an acute-angled triangle; if one of the angles is right, then this triangle is a right-angled triangle; sides a, b, forming a right angle, are called legs; side c, opposite to a right angle, called a hypotenuse; if one of the angles is obtuse, then this triangle is an obtuse-angled triangle. A triangle ABC is an isosceles triangle, if the two of its sides are equal ( a = c ); these equal sides are called lateral sides, the third side is called a base of triangle. A triangle ABC is an equilateral triangle, if all of its sides are equal ( a = b = c ). In general case ( a b c ) we have a scalene triangle. Main properties of triangles. In any triangle: 1. An angle, lying opposite the greatest side, is also the greatest angle, and inversely. 2. Angles, lying opposite the equal sides, are also equal, and inversely. In particular, all angles in an equilateral triangle are also equal. 3. A sum of triangle angles is equal to 180 deg. From the two last properties it follows, that each angle in an equilateral triangle is equal to 60 deg. 4. Continuing one of the triangle sides (, we receive an exterior angle BCD. An exterior angle of a triangle is equal to a sum of interior angles, not supplementary with it: BCD = A + B.

60 5. Any side of a triangle is less than a sum of two other sides and more than their difference ( a < b + c, a > b c; b < a + c, b > a c; c < a + b, c > a b ). Theorems about congruence of triangles. Two triangles are congruent, if they have accordingly equal: a) two sides and an angle between them; b) two angles and a side, adjacent to them; c) three sides. Theorems about congruence of right-angled triangles. Two right-angled triangles are congruent, if one of the following conditions is valid: 1) their legs are equal; 2) a leg and a hypotenuse of one of triangles are equal to a leg and a hypotenuse of another; 3) a hypotenuse and an acute angle of one of triangles are equal to a hypotenuse and an acute angle of another; 4) a leg and an adjacent acute angle of one of triangles are equal to a leg and an adjacent acute angle of another; 5) a leg and an opposite acute angle of one of triangles are equal to a leg and an opposite acute angle of another. Remarkable lines and points of triangle. Altitude ( height ) of a triangle is a perpendicular, dropped from any vertex to an opposite side( or to its continuation). This side is called a base of triangle in this case. Three heights of triangle always intersect in one point, called an orthocenter of a triangle. An orthocenter of an acuteangled triangle is placed inside of the triangle; and an orthocenter of an obtuse-angled triangle outside of the triangle; an orthocenter of a right-angled triangle coincides with a vertex of the right angle. Median is a segment, joining any vertex of triangle and a midpoint of the opposite side. Three medians of triangle intersect in one point O (always lied inside of a triangle), which is a center of gravity of this triangle. This point divides each median by ratio 2:1, considering from a vertex.