Download e-book for iPad: Analytic geometry with calculus by Robert Carl Yates
1 and sin 0 0 A serviceable corollary is Limit 0 0 1-cos9=0 0 39 VARIABLES, FUNCTIONS, LIMITS, CONTINUITY Sec.
It is tangent to the line y = x at the origin. Following the method of Par. 4 (2) and (3), we locate the maximum and minimum points. Writing x, X2 y= x2 +x2+1' we have xl x2 xi+x2+ = 0 or xix2 + x1 - x21x2 - x2 (xi + 1) (x2 + 1) - = U. The numerator here is the zero factor, thus (x, - x2) (1 - x,x2) = 0, and, for xi x2, 1 - xix2 = 0. xi --+ x, this approaches 1 - x2 = 0, from which x As 1, -1, the critical x2 values. Maximum and minimum values of y are yat = #, Y. = - I. EXERCISES 13. Use reciprocal ordinates to sketch each of the following.
3 or (PQ) 2 = (OP)2 + (OQ) 2 - 2(OP) (OQ) cos 8 (xl - x2)2 + (y1 - Y2)2 = (X12 + yl) -I- (x2 + y,) - 2r1r2 cos B. 22 + y1 y2 r1 r2 r1 r2 29 MEASURES Sec. e. X1 y1 ri r1 - = cos a1, = cos,61 X2 Y2 r2 r2 and - = cos a2, = cos 02 Accordingly, cos 0 = Cos a1 COS a2 + cos 01 cos 132 The particular product-sum form of the right member will be referred to as the scalar product of the pairs [ cos a,, cos,6,] and [ cos a2, cos 02]. 4. Parallel and Perpendicular Line Segments It is at once evident that if two line segments are parallel their direction angles as well as their direction cosines are equal and their direction numbers are proportional; and conversely so.