# Elementary Analysis. Volume 1 by K. S. Snell, J. B. Morgan, W. J. Langford and E. A. Maxwell

By K. S. Snell, J. B. Morgan, W. J. Langford and E. A. Maxwell (Auth.)

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Additional resources for Elementary Analysis. Volume 1

Example text

Find the centroid of the three points A, B, C of no. 5. Compare this with the answers to no. 5. 8. Find the centroid of the four points 0, A, B, C where O is (0, 0), A is (—4, 5), B is (2, 1), C is (2, —1). Is this point the same as (i) the mid-point of either diagonal of OABC, or (ii) the mid-point of either of the lines joining the mid-points of OA, BC or AB, CO! 9. Show by shading in a figure the loci: (i) A = {(*, y)\y < * } , (ii) B = {(*, y)\x + y < 4}, (iii) A u 5, (iv) A n B. Describe the locus A n B n C, where C = {(*, y)\y > 0}.

C 58 ELEMENTARY ANALYSIS VOLUME 1 2 Since + V(* + y2) is the distance of the point (x, y) from the origin, this locus is the set of points P(x, y) such that OP = 1. The graph of the locus, or of the relation x2 + y2 = 1, is the circle with centre O and radius 1. If an equation y = f(x) gives rise to a certain graph, we say that y =f(x) is the equation of the graph; thus y = \/(l — x2) is the equation of the semi-circle with centre O and radius 1 that lies in the first and second quadrants. Conversely, if a graph contains all the points (x, y) which satisfy a certain equation, and no other points, we say it is the graph of the equation.

10. Construct the graph of the locus of (*, y) when (i) 0 < x < 2 and - 1 * < 1. Ex. 11. What is the locus whose graph is shown as the shaded area OAB in Fig. 13? Yl , I 1 3 0 fill M Ul / iiiini% 2 A 3 < 5 , X Fig. 13 EXAMPLE. Find the equation of the graph, shown in Fig. 14, consisting of the lines OA and OB. —- By" \ ~4 -3 -2 i K/ -I 0 -A / ;> 5 *\ X Fig. 14 When x is positive the graph is the line OA, whose equation is y = x. When x is negative the graph is the line OB, whose equation is y = —x.