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

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15 Sketch the curves given by the equations (b) y = x 2 In x, (a) y = x In x, (c) y = x ln x" , (d) y = x 2lnx2 • Miscellaneous Exercise 2 1 Sketch the curve y = I + sin 2x for 0 (L) ~ x ~ 7t. 2 Sketch, in a single diagram, graphs of the functions f and g given that f: x f-+ 2 cos x , where x E IR and -7t/2 ~ x ~ 7t/2, g: x f-+ e", where x E IR and -7t/2 ~ x ~ 7t/2. Write on your diagram the coordinates of each of the points where the curves meet (L) the coordinate axes. 3 Sketch a graph of the function f: x f-+ 2" where x E IR.

F(x) == 2 • Sketch the curve y = 2 indicating clearly any x -2x-3 x -2x-3 asymptotes. (0) 21 Find the equation of the tangent to the parabola y 2 = tx at the point (2t 2, t). Hence, or otherwise, find the equations of the common tangents of the parabola y2 = and the ellipse 5x 2 + 20y2 = 4. Draw a sketch to illustrate the results. (0) X2 - 7x + 10 . di . I I . I , In icatmg c ear y any vertica asymptotes an d 22 Sketc h t h e curve y = tx x- 6 23 turning values. Draw a sketch-graph of the curve whose equation is (0) y = x 2 (2 - x).

49. Example 9 Sketch the curve given by y = (In xl/x. No part of the curve exists for x < O. As x -+ +0, Y -+ -00 . As x -+ 00 , Y -+ +0. When x = 1, y = O. dy = (l - 10 x)/x 2 dx d2 ~ = (2 In x - 3)/x 3 and dy = 0 when dx x = e. d 2y and - = 0 when X = e 3/2. dx 2 The point (e, e- l ) is a maximum, the point (e 3/2, ~e-3/2) is a point of inflexion. The curve is shown in Fig . 7(e) . 6 1 Sketch the curves given by the equations (a) y = elx, (b) y = e xl2 , (c) y = ell", (d) y = ell"', (e) y = e- 2" , (f) y = e- l/" ' .