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

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

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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 . ; n 25 (i) f(x) ~ 1 + In 2, (ii) f-1(x) = (ex - 1 - 2)/3, (iii) x ~ 1 + In 2 27 (I, e- 1) ; 0, n : n even, min.

Sketch the graph of f for (i) a > b, (ii) a < b. 10 Sketch, in the same diagram, the graphs of y = e", y = e-", y = cosh x, y = sinh x and y = arcosh x , showing their correct relative positions. Transcendental curves 39 11 Sketch, for -7t/2 < x < 7t/2, the curve y = In (sec x) . 12 Sketch, for - 27t < X < 27t, the curves y = e" sin 2 x and y = e-" sin 2 x. 13 Sketch the curves whose equations are (a) y = In (1 - x 2 ) , (b) y = In (x 2 - I), (c) y = In (I + x 2 ) . 14 Sketch the graph of y = In (x 2 + x- 2 ) and show that the curve has two points of inflexion.

Are not necessarily unique in Example 1 Under a strict application of the definition we have just described, the polar equation (J = (X represents the half-line shown in Fig. 1(b), because 44 Curve sketching conventionally r ~ O. Similarly the polar curve represented by the equation r = 1 + 2 cos () is only defined for - 2n/3 ~ () ~ 2n/3 when the convention r ~ 0 is assumed, because outside this interval r takes negative values. Example 2 The equiangular spiral, whose polar equation is r = aecot ..

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