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OCR A LEVEL JUNE 2022 MATHEMATICS B PAPER 1,PAPER 2, PAPER 3 AND INSERT QUESTION PAPERS

Arithmetic series S = 1 n^a + lh = 1 n"2a +^n - 1hd, Geometric series Sn = a 1 - rn 1 - r a 3 1 - r for r 1 1 Binomial series ^a + bhn = an + nC1 an-1b + nC2 an... -2b2 +f+ nCr an-rbr +f+ bn n e N , where nC = C = JnN = L P n! r! n - r ! ^1 + xhn = 1 + nx + n^n - 1hx2 +f+ n^n - 1hf^n - r + 1hxr +f x 1 1, n e R 2! Differentiation f x r! f l x tan kx k sec2kx sec x sec x tan x cot x -cosec2x cosec x -cosec x cot x u dy v du - u dv Quotient Rule y = v , dx = dx dx v2 Differentiation from first principles f l^xh = lim f^x + hh - f^xh h"0 h Integration c f l^xh d f^xh dx = ln f^xh + c ; f l^xhaf^xhkn dx = 1 af^xhkn +1 + c Integration by parts ; u dv dx = uv - ; v du dx dx dx Small angle approximations sin i . i, cos i . 1 - 1 i2, tan i . i where i is measured in radians For a small angle x radians, the approximation sin x . x is valid. The curve y = sin x and the straight line y = x are shown in Fig. C1.1. Fig. C1.2 shows the curve y = x - sin x. Inspection of the graphs suggests that x is a reasonable approximation for sin x for -0.5 G x G 0.5 and also that y = x has the same gradient as y = sin x when x = 0. Calculating sin x Trigonometric functions, including sin x, are widely used so it is useful to be able to calculate the value of the sine of any angle accurately and quickly. This is easily done nowadays using a calculator but this was not possible in the past. The linear function, y = x, is only a reasonable approximation for y = sin x for values of x close to zero. Perhaps using a higher degree polynomial 10 would give a reasonable approximation for a wider range of values of x. Fig. C2.1 shows the curve y = sin x and the quadratic curve which goes through the points (0, 0), ar, 1k and (r, 0). The equation of this curve is y = 4x (r - x) - sin x. r2 y = 4x (r - x) . Fig. C2.2 shows the curve r2 The quadratic function seems to be a reasonably good approximation for sin x in the interval 15 0 G x G r. However, calculating percentage errors for selected values of x shows that the percentage errors made by using the quadratic function as an approximation to sin x are quite high for values of x close to zero or r. The spreadsheet in Fig. C3 shows values of x in column A, with the corresponding values of sin x and the quadratic function 4x (r - x) r2 in columns B and C. Columns D and E show the percentage 20 errors in using x and the quadratic as approximations for sin x. [Show More]

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