6 APPLIED MECHANICS 
R=radius of the circumscribing circle of a triangle. 
vr =radius of the inscribed circle. 
__ @ _ abe PRN sie 
“San A 4A" atb+e 
8. Differential and Integral Calculus. —The curve APB (Fig. 1) is 
the graph of the equation y= 202 — 2x73. Let # and y be the co-ordinates 
of the point P on the curve. Take another point Q on the curve near 
to P, and let its co-ordinates be « + dx and y + dy. Then for P, 
y= 20. — 2°, and for Q 
y + dy = 20 (a+ dx) — 2 (w+ dx)8 
= 20x + 208x — 2a — 6x?dx — 6(8z)? — 2(dx)8, 
fhetefore dy = 208a — 6xda — 6x(dx)? — 2(dx)°, 
and rd = 20 — 6x? — Gade — 2(8n)?. 
Now let Q approach nearer 
and nearer to P so that dz and 
dy get smaller and smaller, “ 
then the terms 6x32 and 2(dx)? ATS 
will get smaller and smaller, jg 
8 
approaches Ls 4 
nearer and nearer to 20 — 622. 
In the limit when Q is inde- 5 , 
finitely near to P the ratio A 
oy - ; dy 
Sr 18 written da? 
to 20 — 6x. 
dy 
The ratio ae is evidently a measure of the slope of the curve or 
2 
(25 
.* 
i\ 
and the value of 
me f -|— SS Lf 4 
ty 
—) 
and is equal 
tangent at any point whose co-ordinates are « and y. Also, - is a 
x 
measure of the rate of increase of y with respect to x. 
The ratio a is called the differential coefficient of y with respect 
to x. 
At the highest point B of the curve APB, y has its maximum value, 
and the slope of the tangent CB is zero. Hence where y is a maximum, 
= = 20 —62?=0, or x= a =1:826 nearly, and the maximum value 
of y is 20 x 1°826 — 2 x 18263 = 24°34 nearly. 
The process of finding a differential coefficient is called differentiation. 
Now let ue 20 — 6x? be plotted as shown by the curve A’P’B’ 
in Fig. 1. Then, when dx and dy are very small, ual very nearly, 
and wx = dy very nearly. But wd is the area of the shaded strip very 
nearly when 6x is very small, and when dx is indefinitely small wdx= dy. 
