80 



GEOMETRICAL ILLUSTRATIONS. 



104. The following is another geometrical illustration of 

 Art. 101. 



If y = F(x) be the equation to a curve, then F' (#) is the 

 trigonometrical tangent of the 

 angle between the axis of x 

 and the tangent to the curve 

 at the point (x, y). See Art. 38. 



Let 

 then 



OM=a, MN=h, 



F(a + h)-F-(a] 



is the tangent of the inclination of the chord PQ to the axis 

 of x. Hence Art. 101 amounts to asserting that at some 

 point R between P and Q the tangent RT to the curve is 

 parallel to PQ. 



We call this an illustration. When, however, the student 

 has -sufficiently considered the nature of the tangent to a 

 curve, it may amount to a proof of the proposition in 

 question. 



105. The following is an illustration of the general pro- 

 position in Art. 98. 



Let there be two curves APQ and apq. Let F(sc} denote 

 the area contained between 

 the 'first curve, the axes of x 

 and y, and an ordinate to 

 the abscissa x\ then y=F'(x) 

 is the equation to this curve. 

 Let f(x) denote a similar area 

 with respect to the second 

 curve ; then y =/' (x) is the 

 equation to this curve. 



Let OH = a, MN= k. 

 Then F(a+h)-F(a)= area PMNQ, 



f(a + h)-f(a} = 

 Hence the equation 



F(a-+Ji)-F(a} _ 

 ~ -/() = 



