50. Deduce from Taylor's Theorem, by putting h=-x, the (1) 2 π = cos20 52. Verify the following deductions from Ex. 51: 0+ cose.sino + sin20+ sin30+ cos30 cos40 sin40+... 4 sin sin 40 cos40 1 + ... sin.cos [EULER.] be a rational fraction in which the denominator has n factors, each equal to x-a, and the remaining factors are x - h, x-k, etc., so that F(x) = (x − a)”p(x) where Examine the closeness of the approximation in each case. 55. In the equation f(x+h) = f(x)+hf'(x + Oh), show that the limiting value of 0 as h is indefinitely diminished is. [Expand f'(x+Oh) in the above in powers of Oh, and also f(x+h) in powers of h, and compare the two series, remembering that itself, being a function of x and h, may be written =A+A2h+Agh2 + 0 where A, A1 ... are functions of x. The term 4 will be the limiting value of ✪ when h=0.] ... 56. In the equation f(x+h) = f(x)+hf'(x+Oh), if be expanded in powers of h, the first four terms will be 1 1 fsh+ 2 24 2 48 f22 + suffixes being used to denote differentiations. 57. Find by division the first six of Bernoulli's coefficients. They are 1 1 1 1 5 691 6' 30' 42' 30' 66' 2730 58. Prove by continuing the differentiations in Art. 130 that n(n − 1)(n − 2) B ̧ + ... = 1, n+1 19 a formula from which the values of the coefficients B1, B ̧... can be successively deduced by putting n = 2, 4, 6, etc. [DE MOIVRE.] [Differentiate expansion of cot 0, Art. 131.] 62. By taking the logarithmic differential of the expression for sin in factors and comparison of the expansion of the result with that of 0 cot 0 (Art. 131), show that B2n-1 where II (1-1) denotes the continued product of such factors for all integral prime values of r from 2 to ∞ . 63. Expand in powers of x. m 2 sin(m tan ̄1x)(1 + x2)1⁄2 CHAPTER VI. PARTIAL DIFFERENTIATION. 132. Functions of Several Independent Variables. Our attention has hitherto been confined to methods for the differentiation of functions of a single independent variable. In the present chapter we propose to discuss the case in which several such variables occur. Such functions are common; for instance, the area of a triangle depends upon two variables, viz., the base and the altitude; while the volume of a rectangular box depends upon three, viz., its length, breadth, and depth; and it is plain that each of these variables may vary independently of the others. 133. Partial Differentiation. If a differentiation of a function of several independent variables be performed with regard to any one of them just as if the others were constants, it is said to be a partial differentiation. called partial differential coefficients with regard to x, y, The meanings of the differential coefficients thus formed are clear; for if we denote u by f(x, y, z) the operation It will throw additional light upon the subject of partial differentiation if we explain the geometrical meaning of the process for the case of two independent variables. Let PQRS be an elementary portion of the surface z=f(x, y) cut off by the four planes Y=y, Y=y+dy [Capital letters representing X=x, X=x+S x S current co-ordinates], so that the co-ordinates of the corners P, Q, R, S are for P x, y, f(x, y), |