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ENGINEERING MATHEMATICS
The series is nite if n is a positive integer If n is negative or fractional, the a only series is in nite and will converge for b
6 Absolute Values The numerical or absolute value of a number n is denoted
by n and represents the magnitude of the number without regard to algebraic sign 3 3 3 For example,
7 Logarithms De nition of a logarithm: If N
bx, the exponent x is the logarithm of N to the base b and is written x logb N The number b must be positive, nite, and different from unity The base of common, or briggsian, logarithms is 10 The base of natural, napierian, or hyperbolic logarithms is 271 828 18 , denoted by e Laws of Logarithms logb MN logb M N logb M logb M m logb N m / r logb N logb N logb N logb1 logbb logb0 logb0 0 1 ,0 ,1 b b 1
logb Nm logb Nm Important Constants
log10e log10 x ln 10 ln x
0434 294 481 9 0434 3 loge x loge10 loge x 0434 3 ln x
2302 585 093 0 2302 6 log10 x
8 Permutations The number of possible permutations or arrangments of n difn n! (read: n factorial ) ferent elements is 1 2 3 If among the n elements there are p equal ones of one sort, q equal ones of another sort, r equal ones of a third sort, etc, then the number of possible per), where p q r n mutations is (n!) / ( p! q! r! 9 Combinations The number of possible combinations or groups of n ele-
ments taken r at a time (without repetition of any element within any group) is 1)(n 2)(n 3) (n r 1)] / (r!) (n)r If repetitions are allowed, [n(n so that a group, for example, may contain as many as r equal elements, then the number of combinations of n elements taken r at a time is (m)r, where mr n r 1 (n)n 2n 1 Note that (n)1 (n)2
CHAPTER THREE
10 Equations in One Unknown*
Roots of an Equation An equation containing a single variable x will in general be true for some values of x and false for other values Any value of x for which the equation is true is called a root of the equation To solve an equation means to nd all its roots Any root of an equation, when substituted therein for x, will satisfy 1)2 x2 the equation An equation which is true for all values of x, like (x 2 2 2x 1, is called an identity [often written (x 1) x 2x 1] Types of Equations 1 Algebraic equations Of the rst degree (linear), eg, 2x 6 0 (root: x 3) 2 2x 3 0 (roots: 1, 3) Of the second degree (quadratic), eg, x 3 2 6x 5x 12 0 (roots: 1, 3, 4) Of the third degree (cubic), eg, x 2 Transcendental equations 32 (no real root) Exponential equations, eg, 2x 32 (root: x 5); 2x Trigonometric equations, eg, 10 sinx sin3x 3 (roots: 30 , 150 ) Equations of First Degree These are linear equations Solution: Collect all the b, where a and b are terms involving x on one side of the equation, thus: ax known numbers Then divide through by the coef cient of x, obtaining x b / a as the root Equations of Second Degree These are quadratic equations Solution: Throw the equation into the standard form ax2 bx c 0 Then the two roots are x1 b b2 2a 4ac x2 b b2 2a 4ac
The roots are real and distinct, coincident, or imaginary, according as b2 4ac is x2 b / a; the product positive, zero, or negative The sum of the roots is x1 of the roots is x1x2 c / a px q, and plot the Graphical solution: Write the equation in the form x2 parabola y1 x2, and the straight line y2 px q The abscissas of the points of intersection will be the roots of the equation If the line does not cut the parabola, the roots are imaginary Equations of Third Degree The general cubic equation x3 bx2 cx d 0 y b / 3) to the form y3 vy w 0, where is reducible (substitute x v (3c b2) / 3 and w (2b3 9bc 27d) / 27 The roots of the reduced equation are
* This section is in part taken from Ref 1, Marks Standard Handbook for Mechanical Engineers, 9th 1987 Used by permission of McGrawed, by E A Avallone and T Baumeister III (cds) Copyright Hill, Inc All rights reserved
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