ing, whether the value of the letter found makes the terms multiplied by h2, h3, &c., a negative or positive number. 3. Divide a number a into two such parts, that the sum of their squares shall be the greatest possible. Let And x = one part, a-x the other, a2-2ax+2x2= the sum of their squares. Making xh one part, we have a2 −2 a (x + h) + 2(x + h)2 = the sum of their squares. Taking xh, we have a2-2 a (x-h)+2(x-h)2= the sum of their squares. a2. 2ax+2x2 Hence (1) 2x2 (2) a2-2 ax + 2 x2 + (2x-2 a) h-2h2 Does x = a make the function the greatest or least? The result gives one part a, and the other 0, which shows that the square of the whole line is greater than the sum of the squares of any two of its parts. 4. Find a number, such that its square subtracted from itself, will leave a greater remainder than the square of any other number subtracted from itself. Let Let x= the number, X x2= the function. (x + h) = the number, (x+ Then (x + h)-(x + h)2= the function. (x — h): = the number, (x — h) — (x — h)2 = the function. Let Then 5. Find a number, such that its square subtracted from twice itself shall be the greatest possible. 6. Find a number, such that its third power subtracted from its first power shall be the greatest possible. 7. Find a number, such that its third power subtracted from its second power shall be the greatest possible. 8. The sum of the length and breadth of a rectangle is 16. What must the length and breadth be, that it may contain the greatest possible area? 9. A man had a piece of work to do, which would employ 1 man 14 days. He employed two men to do it, who were to work separately, and promised to give each man as many shillings per day, as the other worked days. How many days must each one work, in order that they may obtain the greatest possible sum for their work, according to the bargain? 10. Find a number, such that its third power minus eighteen times its second power, plus twenty-six times the number, minus 20, shall be greater than the same function of any other number. It will be seen, by solving the preceding problem, that there are two values of the unknown quantity; that one will render the function the greatest, and the other the least possible. SECTION CXIV. Development of Functions. We have seen, in the solution of the preceding examples, that it was necessary to increase the letter in the function by some indefinite quantity, h, and to develop the expression according to the ascending powers of h; then put the term, multiplied by the first power of h, 0. This term is called the differential of the function. The coefficient of the h is called the differential coefficient: thus, in question first, (a-2x) h is the differential of the function ax - x2, and α 2 x is the differential coefficient of the function ax — Now it is convenient to know how to find the differential of any function, as soon as it is proposed, without the trouble of substituting xh for x, and developing, which is a tedious process. Each term which contains x in a function of x, will give a term multiplied by h in the development, and all these together will give the differential of the whole function. It is only necessary, then, to know how to find the differential of single terms. Let it be remembered, that in order to find the differential of any function of x, we substitute x+h for x in the function, and then develop the function, and the term multiplied by the first power of h is the differential of the function. 1. What is the differential of x? Ans. Substitute x + h for x, and it becomes x+h. Hence. h is the differential of x, and 1 is its differential coefficient. It is evident that the differential of -x would be - h. 2. What is the differential of ax? Ans. Substitute xh for x, and a x becomes a (x + h) = ax+ah Hence, ah is the differential of ax, and a its differential coefficient; -ah is the differential of 3. What is the differential of x2? Ans. x becomes ax. (x + h)2= x2+ 2 x h+h2 Hence 2 x h is the differential of x2, and — 2 x h is the dif ferential of x2. 4. What is the differential of x2? x2? x3? x®, &c.? Ans. x becomes (x + h)3 = x2+3x2h+3x h2+h® Then 3 x h is the differential of x3. x becomes (x + h) = x2+4x3h+6x2 h2 + 4x3 h2 + ha Hence 4 h is the differential of x1. Indeed we may see from the Binomial Theorem [PAGE 241,] that the differential of any power of x is equal to the continued product of the following factors: viz. the exponent of the power, the power less one of the x,and the h, or differential of x. 5. What is the differential of a x-x2? Ans. ah-2 xh, or (a-2x) h. 6. What is the differential of x3-x2? Of bx-cx2? Of x-x? 7. What is the differential of 4x2-6x3 ? Of x1-20 x3 +22-10? Of x^? If we wish to find what value of z will make these func tions the greatest or least possible, we have only to put these differentials equal to 0. 8. What is the differential of x? Ans. Substitute x + h for x, and it becomes h ferential coefficient. Now, may be written +h; 2x n 10. What is the differential of xm power of x? Ans. Substitute (x + h) for x, and we have We have, then, the following rule for finding the differential of any power of ≈ which has a fractional exponent: RULE.-Multiply together these three factors: viz. the exponent of the power, the power less one of the letter, and the increase of the letter. 11. What is the differential of xx xi? ax1? art -bx? It is sometimes necessary to find the differential of a function of a function, such as (a x—x2)3, (ax+x)*; (a+x)3, &c. Let it be proposed to find the differential of (a x-x3)3. Assume x+h for x. 12. Find the differential of (a x-x2)3. Substitute + h for x, and we have a (x + h) — (x + h)2)2 = (a x—x2 + (a− 2 x) h, &c. We may neglect all the increase of the expressions within the parentheses, except what contains the first power of h; since the terms which contain the first power of h form the differential. ((a x − x') + (a− 2 x) h)' = (a x — x3)* + 3 (a x — xo)2 (a-2x) h, &c. X Then the differential of (a x-x) is 3 (a x-x2) (a-2x) h. To find the differential of the function of a function, RULE.-Take the differential coefficient, as if the expression within the parentheses were a single letter, and multiply it by the differential of the expression within the parentheses. 13. What is the differential of (ax+x3)2? Ans. 2 (ax + x) (α — 2 x) k. |