[...]... \ t' 4 9 \ I -3 i 4 - I 3 - \ 4 2 I:; PW'l I -2 -1 I I I 5 - \ -1 -2 -3 f I I 8 \ \ \ 0 I "I 1 I 1 - 1 2 3 x Fig 3-2 I 113 It4 -1 /4 -1 13 -1 12 -1 -2 -3 3 4 -4 -4 1 -2 -1 1 -3 I 1 -3 -2 - -1 -1 12 -1 13 -1 - -2 - -3 - -4 Fig 3-3 I /- 1 1 \ \ \ 1 -2 1 1 -1 1 ' Fig 3-4 -1 - I \ * , 1i X /' 0 16 GRAPHS OF EQUATIONS [CHAP 3 Circles For a point P(x, y) to lie on the circle with center C(a, b) and radius r,... in the upper right-hand corner (see Fig 2-4 ) Quadrant I1 consists of all points with negative x-coordinate and positive y-coordinate; quadrants I11 and IV are also shown in Fig 2-4 8 CHAP 23 4Y (-2 .3) T 3 - 0 2 - (-4 ,I) 1 1 - (2.2) + I (5.2) - 1 0 (-3 -2 ) 2- I 2 A 3 (3.0) 1 1 5 I 4 9 (3*2) 1- %+* 3 x ‘ 4 5 x 1 - 1 - + -2 (C -3 3- 4 1 I 1 2 -I 0 0 4 - 4 I (3.3) I - 1 -3 4Y (-2 .4) 4 - -4 9 COORDINATE SYSTEMS... graph of y = x2 (Fig 3-2 ) by switching the x- and y-coordinates 18 GRAPHS OF EQUATIONS 3 - 4 - ,, 2 - ' -/ -2 -3 3.3 [CHAP 3 - -4 4 Identify the graphs of: + 1=0 + 120 = 0 + + + (c) 3x2 3y2 - 6~ - y x2 y2 20x - 4y (a) (6) x2 + y2 - 8~ + 16y + 80 = 0 First, divide both sides by 3, (U) x2 1 1 + y2 - 2x - - y + - = 0 3 3 Complete the squares, 1 (x-1)2+ or ( y 1 1 12 = I + - - -1 - + - - - = =... 1 -3 4Y (-2 .4) 4 - -4 9 COORDINATE SYSTEMS IN A PLANE (0, -2 ) 4 - 0 (4, -3 ) -2 , 1 - (-1 ,-3 ) -3 4Y 11 I 3- - (- +) (-1 .2) 2 0 (+* +) I’ I -3 0 1 -2 I 0 (2.1) I 0 -1 (-3 ,-1 ) -1 - 1 I 1 2 3 Fig 2-4 The points having coordinates of the form (0, b) are precisely the points on the y-axis The points having coordinates (a, 0) are the points on the x-axis If a coordinate system is given, it is customary to... 1 1 1 1 1 2 - b b I I 3 \4\ 1 I 4- \ 1 I 5- \ \ \ 1- I I 6- \ i I I I I I \ I I I I \ \ \ \ 9 I / / / # t \ \ \ \ \ 1 1 -5 -4 I\Jl -3 -2 I ? ;-3 /2 - 1- -1 0 - t t Fig 3-8 x 20 36 GRAPHS OF EQUATIONS [CHAP 3 Find the graphs of: (a) x = 0) - 2)2; (b)x = ( y + 2)2 The graph of x = (y - 2)2 is obtained by raising the graph of x = y2 [Fig 3-7 (b)] by two units [see Fig 3-9 (a)] The argument is analogous... graphs of: (a) - - - - 1; 9 (a) If (b) 4 y2 x2 -= 9 4 1 X2 x2 y2 X2 - - - - 1, then - = - + 1 2 1 So - 2 1 and, therefore, I x I 2 3 y2 9 4 Hence, there are no points 9 4 9 (x, y) on the graph within the infinite strip - 3 < x < 3 See Fig 3-ll(b) for a sketch of the graph, which is a hyperbola (b) Switching x and y in part (a), we obtain the hyperbola in Fig 3-1 2 Fig 3-1 1 Fig 3-1 2 Supplementary Problems. .. AC = J [- 1 - ( -3 )12 + (2 - 6)2 = = J G Z = JZ BC = J[4 - (-3 )]' + (7 - 6)2 = J7T+1z = JZGT = Jw' Jso Jso Since AB = E the triangle is isosceles , 2.2 Determine whether the triangle with vertices A( - 5, - 3), B( - 7, 3), C(2,6) is a right triangle Use (2.2) to find the squares of the sides, AB2 = ( - 5 BC2 = (-7 A C 2 = (-5 Since AB2 + + 7)2 + (-3 - 3)2 = 22 + (-6 )2 = 4 + 36 = 40 - 2)2 + (3 - 6)2 =... and (2, 8); (b) (3, 1) and (3, -4 ); (c) (4, 1) and (2, 1); (4( - 3,4) and (54) 2.7 Draw the triangle with vertices A(4, 7), B(4, - 3), and C - 1, 7) and find its area ( 2.8 If ( - 2, - 2), ( - 2,4), and (3, - 2) are three vertices of a rectangle, find the fourth vertex eF E Fig 2-8 13 COORDINATE SYSTEMS IN A PLANE CHAP 2) 2.9 If the points (3, 1) and (- 1,O) are opposite vertices of a rectangle whose... center (1,i) and radius 2 (6) Complete the squares, + (y + 8)2 + 80 - 16 - 64 = 0 or (x - 4)2 + (y + 8)' = 0 Since (x - 4)2 2 0 and (y + 8)2 2 0, we must have x - 4 = 0 and y + 8 = 0 Hence, the graph consists (x - 4)2 of the single point (4, - 8) (c) Complete the squares, (x + 10)2+ (y - 2)2 + 120 - 100 - 4 = 0 or (x + 10)2 + (y - 2)2 = -1 6 This equation has no solution, since the left-hand side is... the equal quantities (y+:) -~ B2 EXAMPLE Let us find the graph of the equation x2+y2+4x-2y+ 1= o Completing the squares, replace x2 + 4x by (x + 2)2 - 4 and y2 - 2y by (y - 1) 2- 1, (x+2) 2-4 +(y-1) 2-1 +1=o (x + 2)2 + (y - 1)2 = 4 This is the equation of a circle with center ( - 2, 1) and radius 2 Solved Problems 31 Find the graph of: (a) the equation x = 2; (b) the equation y -3 = (a) The points satisfying . SERIES Schaum's Outline of Theory and Problems of Beginning Calculus Second Edition Elliott Mendelson, Ph.D. Professor of Mathematics Queens College City University of New York To the memory of my father,. Maureen B. Walker Library of Congress Cataloging-in-Publication Data Mendelson, Elliott. Schaum's outline of theory and problems of beginning calculus / Elliott Mendelson, 2nd ed. p. cm His principal area of research is mathematical logic and set theory. Schaum's Outline of Theory and Problems of BEGINNING CALCULUS Copyright © 1997,1985 by The McGraw- Hill Companies,