Examples of Elimination by Addition and Subtraction. 8. A person has two large pieces of iron whose weight is required. It is known that {ths of the first piece weigh 96 lbs. less than ths of the other piece; and that gths of the other piece weigh exactly as much as gths of the first. How much did each of these pieces weigh ? Ans. The first weighed 720, and the second 512 lbs. 9. $ 2652 are to be divided amongst three regiments, in such a way, that each man of that regiment which fought best, shall receive $1, and the remainder is to be divided equally among the men of the other two regiments. Were the dollar adjudged to each man in the first regiment, then each man of the two remaining regiments would receive $}; if the dollar were adjudged to the second regiment, then each man of the other two regiments would receive $ *; finally, if the dollar were adjudged to the third regiment, each man of the other two regiments would receive $1. What is the number of men in each regiment ? Ans, 780 men in the first, 1716 in the second, and 2028 in the third regiment. 10. To find three numbers such that if 6 be added to the first and second, the sums are to one another as 2 : 3; if 5 be added to the first and third, the sums are as 7:11; but if 36 be subtracted from the second and third, the remainders are as 6:7. Ans. 30, 48, 50. 10 Indeterminate Coefficients. CHAPTER IV. NUMERICAL EQUATIONS. SECTION I. Indeterminate Coefficients. 162. Theorem. If a polynomial A + B x + C 22 + D 2+E **+&c. is such, as to be equal to zero independently of x, that is, if it is equal to zero whatever values are given to x, it must always be the case that A=0, B=0, C=0, D=0, E=0, &c.; that is, that the aggregate of all the coeficients of each power of x is equal to zero, and also the aggregate of all the terms which do not contain x is equal to zero. Proof. Since the equation A + B x + C x2 + D x3 + &c.=0 is true for every value which can be given to x, it must be true when we make 2 = 0; in which case all the terms of the first member vanish except the first, and we have A=0. Indeterminate Coefficients. This equation, being subtracted from the given equation, gives Br + C + D23+ &c. = 0; and, dividing by 4, B + x + D 2? + &c. = 0; whence we may prove as above, that B = 0. C=0, D=0,E=0, &c. 163. Theorem. If two polynomials A+B x + C 2? + D 23 + E 24 + &c., A+ B'r + C 22 + D'x3 + E' x4 + &c. are identical, that is, equal, independently of x, it must always be the case that A= A', B=B', C=C", D=D', &c. Proof. For the equation A+B:+Ca2+&c.= A' +B'r+Cr2+&c. gives, by transposition, (A — A')+(B- B')x+(C—C") x2 + &c.=0; whence, by the preceding theorem, A—A'= 0, B- B' = 0, C-C'=0, &c.; that is, A=A', B=B', C=C", &c. ia A Function; its Variable, and Rate of Change. SECTION II. Derivation. 164. Definition. When quantities are so connected that their values are dependent upon each other, each is said to be a function of the others; which are called variables when they are supposed to be changeable in their values, and constants when they are supposed to be unchangeable. Thus if y = a + b y is a function of the a, b, and x; but if x is variable while a and b are constant, it is more usual to regard y as simply a function of x. 165. Definition. In the case of a change in the value of a function, arising from an infinitely small change in the value of one of its variables, the relative rate of change of the function and the variable, that is, the ratio of the change in the value of the function to that in the value of the variable, is called the derivative of the function. The derivative of the derivative of a function is called the second derivative of the function; the derivative of the second derivative is called the third derivative; and so on. 166. Corollary. The derivative of a constant is zero. 167. Corollary. The derivative of the variable, regarded as a function of itself, is unity; and the second derivative is zero. JU The Derivative of the sum of any Functions. or 168. Theorem. The derivative of the sum of two functions is the sum of their derivatives. Proof. Let the two functions be u and v, and let their values, arising from an infinitesimal change i in the value of their variable, be u' and v'; the increase of their sum will be (x + 0) – (u + o) u' -utv- 0, and therefore the derivative of the sum is u' v' + i which is obviously the sum of their derivatives. 169. Corollary. By reversing the sign of v, it may be shown, in the same way, that the derivative of the difference of two functions is the difference of their derivatives. V 170. Corollary. The derivative of the algebraic sum of several functions connected by the signs + and - is the algebraic sum of their derivatives. 171. Corollary. If, in this sum, any function is repeated any number of times, its derivative must be repeated the same number of times; in other words, if a function is multiplied by a constant its derivative must be multiplied by the same constant. Thus, if the derivatives of u, v, and w are respectively U, V, and W, and if a, b, c, and e are constant, the derivative of aut - cute is a U+6VcW. |