80 



PRACTICAL MATHEMATICS 



made the more nearly correct does the approximation become. 

 Thus, if we can find the slope of an infinitely small chord, one 

 extremity of which is at the point, then we have found the actual 

 slope of the curve at that point. 



Let P be a point on a curve (Fig. 26), PQ a chord, and PT the 

 tangent to the curve at the point P. Let PT make an angle 

 with the axis of x and the chord PQ make an angle a with PT. 



Then the slope of the chord PQ = tan (0 + a). 



As the point Q approaches P, the chord PQ becomes smaller 

 and smaller, and so does the angle a, and when the chord PQ is 

 made infinitely small the angle a becomes negligible in com- 

 parison with 0. 



Thus the slope of the infinitely small chord == tan 0. 



It follows, therefore, that the slope of the infinitely small chord 



FIG. 26. 



PQ, which gives the actual slope of the curve at the point P, 

 is the same as that of the tangent to the curve at the point P. 



We can now take the slope of a curve at a certain point to be 

 given by tan 0, where is the angle which the tangent to the 

 curve, at that point, makes with the axis of x. 



This provides us with a graphical way of finding the slope of 

 a curve. We can draw the tangent to the curve at the required 

 point, take two points on this line as far removed as the paper 

 permits, make that part of the line between these two points the 

 hypotenuse of a right-angled triangle, measure the perpendicular 

 of this triangle to the vertical scale and the base to the hori- 

 zontal scale, and 



A perpendicular 



the slope of the curve = tan = - . 



base 



We cannot obtain the true value of the slope of a curve in this 

 way, because we have no definite construction for drawing the 

 true tangent to the curve ; we can only draw what appears to 

 be the true tangent. If the supposed tangent is inclined to the 

 axis cf $ at an angle gfcghtly smaller than that of the true tangent,. 



