Physics CALCULUS FOR PHYSICS

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CALCULUS FOR PHYSICS

DIFFERENTIATION

The purpose of differential calculus is to study the nature (i.e., increase or decrease) and the amount of variation in a quantity when another quantity (on which first quantity depends) varies independently.

`text(Quantity :)`
Anything which can be measured is called a quantity.

`text(Constant :)`
A quantity, whose value remains unchanged during mathematical operations, is called a constant quantity. The integers, numbers like,e, etc are all constants.

`text(Variable :)`
A quantity which can have any numerical value between certain specified limit is called as variable.

`text(Function :)`
A quantity y is called a function of a variable x, if corresponding to any given value of x, there exists a single definite value of y. The phrase "y is function of x" is represented as ,`y=f(x)`

For example, consider that y is a function of the variable x which is given by

`y = 3x^2 +7x +2`

If `x = 1,` then `y = 3(1)^2 + 7(1) +2 =12`

and when `x=2,y=3(2^2) +7(2) +2 = 28`

Therefore, when the value of variable x is changed, the value of the function y also changes. But corresponding to each value of x, we get a single definite value of y. Hence, `y = 3x^2 +7x +2` represents a function of `x` .

`text(Meaning of limit)`

Rabbit A wants to reach rabbit B, which sits stationary. A jumps half the distance remaining between them every second. How soon does the rabbit A reach its goal?

Never! (Assuming both rabbits to be point objects)

The gap remaining between them becomes infinitesimal ( i.e. very very small) after a long time.

- It is not a number, that can be expressed. It is smaller than the smallest positive number, that you can think of. The gap `Deltax->0` or it is dx. (dx denotes a very very small change in x.) and is represented as `lim_(x->0)` (read as limit of delta x tends to zero)

- Rabbit A tends to rabbit B.

- Rabbit A's limit is rabbit B.

`text(Slope of secant)`

Consider a curve where the variation of a function `y = f(x)` is plotted with respect to variable x. let P and Q be two points on the curve. The segment PQ is called as secant.

The slope of the secant: slope `= tan theta = (Delta y)/(Delta x)`

(Slope is defined as tan of angle between line and positive x -axis taken counter - clockwise )

`text(Geometric Meaning of Derivative)`

- let P go closer to Q. When the gap between P and Q becomes infinitesimal (very very small), the secant can be approximated as tangent

Hence, `(dy)/(dx)` (or `f'(x)` ) at any point represents the rate of change of y (or f(x)) with respect to x at that point and is also known as derivative of y w.r.t. x.

Physical Meaning of : `(dy)/(dx)`

1. The ratio of change in the function y to change in variable x is called the average rate of change of y

w.r.t.x. For example, if a body covers a distance `Delta s` in
time `Delta t` then average velocity of the body, `v_(Delta t) =(Delta s)/(delta t)` Also,
if the velocity of a body changes by an amount `delta v` in
small time `Delta t` then average acceleration of the body,
`a_(Delta v) = (Delta v)/(Delta t)`

2. The differentiation of a function w.r.t. a variable
implies the instantaneous rate of change of the
function w.r.t. that variable.

Thus, instantaneous velocity of the body, `lim_(Delta t -> 0) (Delta s)/(Delta t) = (ds)/(dt)`

and instantaneous acceleration of the body, `a= lim_(Delta t->0)(Delta v)/(delta t) = (dv)/(dt)`

`(dy)/(dx)` `text(of Functions and their Properties)`

We have found the derivative of , 'with respect to x.
Like wise we can also find derivatives of other functions.Some standard derivatives are as given in the table

`text(Note :)`
In the above table, n and a are constant.

Mathematical operations for derivatives :

`(dK)/(dx) = 0`

`(dK_u)/(dx) = K (du)/(dx)`

`d(u pm v)/(dx) = (du)/(dx) pm (dv)/(dx)`

`(d(uv))/(dx) = u (dv)/(dx) + v (du)/(dx)`

`d(u/v)/(dx) = (v(du)/(dx) - u(dv)/(dx))/v^2`

(Here, K is a constant and u and v are functions of x.)

`text(Increasing and Decreasing Functions)`

NOTE :
A function may not be continuously increasing or decreasing during the entire range (e.g. `sin x, cos x`).

Then we say function is increasing in the range where
`(dy)/(dx) > 0` and decreasing where `(dy)/(dx) < 0`

`text( Local Maxima and Local Minima)`

For local maxima or minima, slope of the curve where it lies must be zero.

So, `(dy)/(dx) =0,` at the point s of local maxima or local minima

Note 2:
Local maxima/minima does not mean that the function has the highest / lowest value at that point. It only means that the function was increasing before and decreasing after that point in case of local maxima and vice-versa in case of minima

`text(Criterion for Local Maxima and Local Minima)`

Note 1:
When we talk of local maxima I minima, we are talking of x: while when we say local maximum value or local minimum value, we are talking of value of y corresponding to that particular local maxima or minima

Note 2:
Local maxima/minima does not mean that the function has the highest / lowest value at that point. It only means that the function was increasing before and decreasing after that point in case of local maxima and vice-versa in case of minima

INTEGRATION

In integral calculus, the differential coefficient of a function is given. We are required to find the function. Thus, integration is the reverse of differentiation.

`int` sign is used for integration. If `I` is integration of f(x) with respect to x then `I = intf(x) dx` and it is read as integration of `f(x)` w.r.t. x is `I`

For example, let us proceed to obtain integral of `x^n` w.r.t. x. We already know that `d/dx(x^(n+1)) = (n+1)x^n`

Since the process of integration is the reverse process of differentiation,

`int(n+1)x^n dx = x^(n+1)` or `intx^n dx = (x^(n+1))/(n+1)`

The above formula holds for all values of `n`, except `n = -1`.
`int 1/x dx = log _e x`

It is because, for `n = -1, int x^-1dx = int 1/x dx ` since `d/(dx) (log_e x)=1/x`

`therefore int 1/x dx =log_e x`

Similarly, the formula for integration of some other functions can be obtained if we know the differential coefficients of various functions.

Basic Integration Formulas

`1.` `int[f(x) pm g(x)]dx = int f(x) dx pm int g(x)dx`

`2.` `intx^n dx = (x^(n+1))/(n+1) +C, n ne -1`

`3.` `int(dx)/x = ln|x| + C`

`4.` `inte^x dx = e^x +C `

`5.` `int sin x dx =-cos x + C`

`6.` `int cos x dx = sin x +C`

`7.` `int tan x dx = ln |sec x| +C`

`8.` `int cot x dx = ln | sin x| +C`

`9.` `int secx dx = ln |sec x +tan x| +C`

`10.` `intcosec x dx = - ln |cosec x + cot x| + C`

`11.` `int sec^2x dx = tan x + c`

`12.` `int cos e c^2x dx = -cot x + C`

`13.` `int sec x tan x dx = sec x +c`

`14.` `int cos e c x cot x dx = -cosx secx +C`

Integration by Substitution�

Suppose we represent an integral by `int f(u)du = F(u) +c` (i.e., `F(u)` is the integral of `f(u)` w.r.t. `u` ).

Now suppose u is a function of `x` given by `u = g(x)` , then `du = g'(x)dx` (where `g'(x)` is `(d (g(x)))/(dx)`) and the above integral can be written as `int f(g(x)), g(x)dx =F (g(x)) +C`

Thus if we have to find the integral given in the form of a function along with its derivative, we can use the above formula to find the integral.

An important result: if `intf(x)dx = F(x) + C `
then

`intf(ax +b) dx = (F(ax +b))/b +C`

Definite Integral

Consider the curve as shown. The area under the curve (the area bounded by the curve and the x-axis) can be found by dividing this area into infinitesimal areas and adding them up.

Consider this area to be divided into n parts, where each part can be assumed as a rectangle if n is very large. The length of each such part at `x=x_L` will be equal to `y_i = f(x_i)` while the breadth will be equal to `Delta x`
where `Delta x = (b-a)/n`

Area of each rectangle `= A_i = y_i Delta x`

The total area will be the sum of all these areas and will be given by

`Asum_(i=1)^(i=n) y_i Delta x`

if `Delta x -> 0,` the same area is represented by `int_(x=a)^(x=b) y dx` or `int _(x=a)^(x=b) f(x) dx`

This integral is known as definite integral of the curve `y = f(x)` between `x = a` to `x = b`, where `a` and `b` are known as the lower and upper limits of the integral respectively.

Fundamental Theorem of Integral Calculus

If `intf(x)dx =F(x)+C, ` then `intf(x)dx= F(x)_a^b = F(b)-F(a)`

Application of Calculus in Kinematic

The problems in kinematics can be solved using the differential and integral calculus, in addition to the already known equations which are given as under

`V = (dx)/(dt) ;`

`a= (dv)/(dt) = (dv)/(dx) xx (dx)/(dt) = v (dv)/(dt) ; a= (dv)/(dt) = d/(dt)((dx)/(dt)) = (d^2x)/(dt^2)`

( Where 'x' is displacement, V is velocity and 'a' is acceleration at time t)
Similarly, for circular motion

`omega =(d theta)/(dt)`

`alpha = (d omega)/(dt) = (d omega)/(d theta) xx (d theta)/(dt) = (omega d omega)/(d theta)` or `alpha =(d omega)/(dt) = d/(dt)(d theta)/(dt) = (d^2 theta)/(dt^2)`

( Where `theta` is angular displacement, `omega` is angular velocity and `alpha` is angular acceleration at time t)

Also by the definition of integral

`Delta x = int vdt , Delta v = int a dt`

`Delta theta = int omega dt , Delta omega = int alpha dt`

NOTE :

`(dx)/(dt)` is called as instantaneous velocity, it is the velocity in small time dt. Average velocity over a period of time `Delta t` can be given as `v = (Delta x)/(Delta t)`