216 
CALCULUS 
The function cf)(x) thus represented or defined admits a derivative, 
obtained as follows. Since 
*o+* 
<K»o + h ) ~ <K x o) = I /(0 dty 
Xq 
we have, on applying the Law of the Mean, 
x 0 +h 
j f(t) dt = hf(xo + 6h), 0 < 0 < 1. 
Xq 
Hence 
^ X -° + = fixo + dh) and lim ( K X " + 
ri /¡=o h 
Thus we have proved the theorem that 
x 
f x jf(t)*=>(»)• 
EXERCISE 
O 
Prove that ^ / /(i) di = — /(îc). 
Fig. 49 
2. Continuation: Curved Paths. Suppose the particle describes a 
curved path C in a plane, and that the force, varies in magnitude 
and direction in any continuous manner. What will be the work 
done in this case ? 
Suppose the path C is a right line and the 
force, though oblique to the line, is constant in 
magnitude and direction; Fig. 50. Resolve 
the force into its two components along the 
line and normal to it. Surely, we must lay 
down our definition of work so that the work done by g is equal to 
the sum of the works of the component forces. How, the work done 
by the component along the line has already been defined, namely, 
FI cos ifj, where F = g | is the intensity of the force. 
It is an essential part of the idea of work that the force overcomes 
resistance through distance (or is overcome through distance). How, 
the normal component does neither; it merely sidles off and side 
steps the whole question. It is natural, therefore, to define it as