DIFFERENTIAL CALCULUS. 



CHAPTER I. 



DEFINITIONS. LIMIT. INFINITE. 



1. SUPPOSE two quantities which are susceptible of 

 change so connected that if we alter one of them there is 

 a consequent alteration in the other, this second quantity 

 is called a function of the first. Thus if x be a symbol to 

 which we can assign different numerical values, such ex- 

 pressions as ic 2 , 3*, log x, and sin x, are all functions of x. 

 If a function of x is supposed equal to another quantity, 

 as for example sin x = y, then both quantities are called 

 variables, one of them being the independent variable and 

 the other the dependent variable. An independent vari- 

 able is a quantity to which we may suppose any value 

 arbitrarily assigned ; a dependent variable is a quantity the 

 value of which is determined as soon as that of some in- 

 dependent variable is known. Frequently when we are 

 considering two or more variables it is in our power to fix 

 upon whichever we please as the independent variable, but 

 having once made our choice we must admit no change 

 in this respect throughout our operations ; at least such 

 a change would require certain precautions and transfor- 

 mations. 



2. We generally denote functions by such symbols as 

 F(x}> f(x], <j>(sc), ty (#), and the like, the variable being 

 denoted by x. Such an equation as y = <f> (x) implies that 

 the dependent variable y is so connected with the independent 

 variable x, that the value of y becomes known as soon as 

 that of x is given, and that if any change be made in the 

 numerical value assigned to x, the consequent change in y 

 can be found. 



T. D. c. n 



