Calculus For Physics
Differentiation
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? (Optional)
(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 [Introduction]
( In Hindi + English Mix)
Derivatives of Standard Functions
Differentiation [part 2]
(In Hindi + English Mix)
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 
.
Differentiation [part 3]
(In Hindi + English Mix)
Integration
Mathematically, “The process of finding a function, whose differential coefficient is known (given), is called integration.”
Hence, integration is the inverse process of differentiation. Integration and differentiation are two operations which are inverse to each other.
Let 
. Then, by definition, integration of 
with respect to x is 
. We shall use the notation
………….[1]
- The symbol
is called the symbol of integration. 
is called the notation of integration of with respect to x.- x is called the variable of integration (independent variable).
Physical Meaning of Integration
Physically, since 
Or 
Taking summation of all the differentials, we have 
Or 
…………………….[2]
When dx approaches to zero, the summation is replaced by integration and we write 
………………..[3]
It means, physically, the method of integration is used to sum up the effect of a continuous varying function.
It is important to note that the sign 
is used for summation of discrete values, while sign is used for summation of continuous function.
How to use integration in physics?
(In Hindi + English Mix)
Indefinite Integration
Since, differentiation of a constant factor is zero i.e. 
We get, 
So, by Eq [1], we get
……………..[4]
Where, C is constant independent of x. The value of constant C can be determined using some limiting conditions.
Definite Integration
When a function is integrated between definite limits, the integral is called definite integral. Usually, these limits are the initial and final value of the independent variable x. In this case no constant is used and we write eq [4] as
………………. [5]
Thus, physically 
gives the change in the value of for x = a to x = b.