By H. M. Kenwood, C. Plumpton (auth.)
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For each such value, determine whether the corresponding value of y is a maximum, a minimum or neither. Sketch the graph of y against x for 0 ~ x ~ 2n. (C) 10 Sketch the curve whose equation is y = In (I - 3x) , for x < t. 11 Functions g and h are defined on the set of real numbers by g(x) =2- hex) = 2 + sin~ . (L) cos x, For each function, state (a) the period, (b) whether the function is odd , even or neither . Sketch graphs of these two functions for 3n 3n - - < x <- . 2 2 (L) 12 A curve joining the points (0, 1) and (0, -I) is represented parametrically by the equations x = sin 8, y = (1 + sin 8) cos 8, where 0 ~ 8 ~ n.
E 6 nl6 nl4 nl3 r -a o a/~3 nl2 a The directions of the tangents to the curve at the pole 0 are given by 2 sin 0 - cosec 0 = 0; that is, 0 = n/4 or 0 The curve, a strophoid, is shown in Fig. 3(b). The cartesian equation of the curve is y(x 2 + y2) Example 5 = + a(x 2 - 3n14. y2) = O. Sketch the rose-curve r = 4a sin 30 and find the area of a petal. Since sin 3(} = 3 sin (} - 4 sin? 0, the curve is symmetrical about the line = n/2. The curve lies within the circle r = 4a and touches the circle where (} = n16, 5nl6 and 3n/2 (see chapter 2: Example 8, page 37).
24 Curve sketching (iii) Show that [x> I] => [m(x) > I] and that [x < 0] (iv) Sketch the graph of Y = m(x). 28 Find the equations of the asymptotes of the curve Y = (2x 2 - 2x + 3)/(x 2 => - [m(x) < I]. (C) 4x). Using the fact that x is real, show that y cannot take any value between -I and 5/4. Sketch the curve, showing its stationary points , its asymptotes and the way in which the curve approaches its asymptotes . (L) 29 Given that the function y = f(x) has a stationary value at x = a, show that the value will be a minimum ifd 2y/dx2 > Oatx = a and a maximum ifd 2y/dx2 < 0 atx = a.