Questions in Pure Mathematics Proposed at the B.A. and B.SC. Pass and Honours Examinations of the University of London; With Complete Solutions

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1882 Excerpt: ...the sign of the second differential coefficient when x=a(--1+ V 2). To find the two values of y, we have _x+a-1+V2 + 1--1 1+V2 2/-#2+a2-(i(3+l + 2v'2)2a(l+ V2) 2a' where the positive value is a maximum, and the negative a minimum. This might have been established by Algebra alone, thus:--Solving for y, 2.1-!/=1+ V1+4om/--4ahf =l±V(l + a2/)(l-/fy), where clearly the greatest value of y is and the least--; but a, (3 are respectively 2a(l+V2) and 2a (V 2--1): hence our result. In the second example we have y = a cot x + b cot 2x,.". =--a cosec2 a;--26 cosec2 2x ax =--j cosec2 x (2a+b sec2 Now, if a and 5 are of the same sign, or if b 2a numerically, this expression can never change its sign; there is therefore no maximum or minimum value of y in these cases. If 6 is less than or equal to 2a, numerically, and of opposite sign, will vanish when cos x =----; and in ax 2a this case y is a maximum or minimum, according to the quadrant in which x lies, and the sign of a. The values of y are readily found to be + V--b(2a + b). The whole problem may be solved far more simply by Algebra and Trigonometry, as follows:--y = a cot x + 6 cot 2x. Solving for tan x, we have otana;=--y± /y'+2ab + V. Thus y may have any value from--oo to +00 (since tana; may have any value) if 2ab + b2 is positive, i. e. if 6 and 2a+ b are of the same sign, which agrees with our previous condition. If, however, b(2a + b) is negative, the maximum and minimum values of y are respectively + V-6(2a + o) and--V--6(2a+6). 6. Integrate the expressions dx tan-i xdx; V2 V2 7. Find the values of the definite integrals dx 30 8. Show how to integrate J--L dx, where fx) andb(x)are rational algebraic functions of x, pointing out the different forms of integrals that mag take place. Williamson'...