Download e-book for iPad: Curve Sketching by H. M. Kenwood, C. Plumpton (auth.)

By H. M. Kenwood, C. Plumpton (auth.)

ISBN-10: 0333348036

ISBN-13: 9780333348031

ISBN-10: 1349067091

ISBN-13: 9781349067091

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AEB) 20 Sketch the curve with polar equation r = a cos 6(2 - cos 6), where a > O. 3 2 2x- y-3=0 4 (a) tan - I (7/4), (b) 126/13 5 ( -2,4),(-3,1);(4,2),(3, -I) 6 (1,2). 5 1 (a) (i) x + 2 = 0, (ii) y ~ 0; (b) (i) x - 2 = 0, (ii) y ~ - 8 ; (c) (i) x = 0, (ii) y ~ 4; (d) (i) x = 0, (ii) y ~ 4 ; (e) (i) 2x - I = 0, (ii) y ~ -15/4 5 p = 4/9, q = - 3, r = 6. 6 3 (a) even, (b) odd, (c) none of these, (d) periodic, period 3 4 f(x) = x for 0 ~ x < I, f(x) = 4 - 2x for I ~ x < 3, f(x) = f(x + 3) for x E IR 5 (a) odd, (b) even, (c) even.

Since x 2e - x ' is an even function, the curve is symmetrical about the y-axis. = 2x(1 dy dx _ x 2)e- X ' 2 d y = 2(1 _ 5x 2 dx 2 0 1 0 e- I -1 e- I As x --+ ± 00 , y --+ 0. d2y Since dx 2 = when x = ° dx ° when x = 0, ± l. + 2x 4)e- x ' . <5 ± J17)], there are four points of inflexion. The curve is shown in Fig . 7(c) . Example 8 Sketch, for x ~ 0, the curve given by y = e- X sin x. +. This implies that the curves y = _e- X and y = e- X sin x and the curves y = e- X and y = e- X sin x have common points.

X = 2, y is a 26 max. 32 (i) x - 2 = 0, y - I = 0 ; (ii) x + 2y = 0 33 1/3 ~ A. ~ 3. 36 (b) odd , neither, even 37 2x + 8(1 - X)- 2, 2 + 16(1 - X)-3 , min. , (3, 5) 38 (i) (0,2), (2, -4) ; (ii) (-2,0), (4, -2) 40 4a 3 + 2ab + a 2 y ' + 2by ' = 0, 12a2 + 2b + 4ay' + a 2y" + 2y'2 + 2by" = O. ; (I , -2) and (-I, -2) min. 3 11 (a) n/4, (b) -n/6, (c) n, (d) -n/6, (e) n/6, (f) -n/6. -)1/3 , min . J3/4), (I , 0) 13 -I/e Answers 53 25,16'26°; 25, -25, 180no + 73·74° - x + 2x 3 /3; x + y = 0 -2(1 - X 2) 1/2 Max .

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Curve Sketching by H. M. Kenwood, C. Plumpton (auth.)


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