- 0. Then devise a bisection process to approximate the trisection of angle 0. 4. The Derivative We introduce here the derivative concept, one of the most important and certainly most useful ideas that ever occurred to man.
Slope of a Curve y =f (x) We display here the geometry involved in the process of finding Dxy and give interpretations of utmost importance. Consider y = x2 the parabola shown. Let P(x, y) be some particular but unspecified point on the curve. Then (x + Ax, y + Ay) is a neighboring point Q. Since Q also lies on the curve, y + Ay = (x + Ax)2 = x2 + 2x(Ax) + (Ax)2 or, with y = x', AY = 2x(Ax) + (Ax)'. Then 47 THE DERIVATIVE Sec. 2 y = 2x + (Ox). Ox Now if Ax - 0, then Ay - 0 and Q follows the curve toward P.
Accordingly, P is the right-angle vertex of triangles with the fixed hypotenuse OD. The locus is thus a circle through the pole with radius 2 and center (2, 30°). Fig. 22 Fig. 23 THE REAL NUMBER SYSTEM-GRAPHS Sec. 7 21 Consider the form r = a cos 0 + b sin 0 and let P(r, 0) be representative of all points of the locus. If we multiply and divide the right member by 1/a2 + b2, r= a2 a __ )2 cos 9 + - - az +22 sin B b2 [-\/-aT } ; we may set a a2+ )2 = COS a, b a2 + b2 = sin a as shown. Thus, r = -\/a2 + b2 (cos a cos 0 + sin a sin 0) or r=1/a2+b2cos(0-a).