Elementary CALCULUS
Function:
If the value of a variable quantity (y) depends on another variable quantity (x) through some relation, such that there exists only one finite value of y for each value of x then ‘y’ is called a function of ‘x’. It is denoted as
Type of elementary functions
- Constant function y = C.
- Linear function y = ax + b
- Quadratic Function y = ax2 + bx + c
- Power function: y = A.xn
Where n is a constant real number. For n = 0 power function is a constant quantity y = A
- Trigonometric function : y = sin Ax, y = cos2 x etc.
- Exponential function: y = ax
- Logarithmic function: y = log x2
What is Function? Basic Idea.
(In Hindi + English Mix)
DIFFERENTIATION:
In simple words,
“if y is a function of x i.e. if , , then the rate of change of y (dependent variable) with respect to x (the dependent variable), for small change in x, is called derivative (differentiation) of y with respect to x”.
So, derivative of y w.r.t. x is
,
Where, 
is the change in the value of y when the value x is changed from x to 
.
Graphically, 
represents the slope of the curve y, for a given value of x.
Differential Calculus[part-1]
( In Hindi + English Mix)
Derivatives of Standard Functions
Maximum and Minimum
If a quantity y depends on another quantity x in a manner shown in fig. it becomes maximum at x1 and minimum at x2.
At these points the tangent to the curve is parallel to x axis and hence its slope is zero at these points. Since, slope of the curve equals the rate of change 
. Thus, at a position of maximum or minimum 
.
Just before the maximum the slope is positive, at the maximum it is zero and just after the maximum it is negative. Thus dy/dx decreases at a maximum and hence the rate of change of 
is negative at a point of maximum i.e., at x = x1 
,
Conversely, at the position of minima (x = x2), 
and 
.