Today very important topic calculus in physics calculus is the important branch of mathematics, greatest invention for mankind and invented by Issac Newton when he was stuck at a point where normal mathematics was not supporting to carry out his experiment and derivation after rigorous hard work for two years Newton finally invented Calculus and you know without knowledge of mathematics we can’t understand physics well mathematics is tool of physics, or backbone of physics so we need to study here calculus is amazing branch of mathematics simple and complex both but we are studying here physics so we will not study here in depth as per our need we will study here see picture below.
Calculus important for Physics
some time you see Delta ‘∆’
meaning of this ∆→ change take an example suppose in your class room there are 50 students. some more 10 student is added in class room then what will be change in number of student
change in number of student = 60 – 50 = 10 now in short we can write this ∆ no of student = 10 we have use here ∆ means change suppose you have 500 rupees and your mom given you 300 more then ∆ in rupees = 300 hence delta term is frequently use for any type of change.
Concept of infinity
-5 -4 -3 -2 -1 0 1 2 3 4 5 …….
Suppose this is number line and i told your to draw largest number on the number line number will come 5 10, 100 , 1000, 10000, 1000000 and so on infect you can’t tell the largest number on number line so here concept of infinity comes in picture, we can think infinity is the largest possible number so what is exactly infinite number we can’t tell not possible to know exact value
suppose i tell you i am writing largest number 1000000000000 is this infinity big no infinity is larger than this number so add some more zero 1000000000000000000 is this infinity big no, infinity is larger than this number keep on adding zero but infinity will be larger and larger hence we can’t know exact value of infinity so infinity is biggest number of the world hence we have written infinity is a concept can’t not write exact value so infinity is denoted by ∞very very big number we can’t write vague concept but don’t worry you will understand whatever you have digested is enough .
Concept of infinitesimally small
this concept is opposite of infinity concept perhaps you have understand we want to write very smallest positive number tell me what we can write ? like .00000001 is this smallest number of the world big no then .000000000000000001 is this smallest number of the world big no smallest number can be less than above number keep adding zero but we can’t write exact smallest number of the world, you will write any smallest number but less than that number is possible to write hence smallest number of world is not possible to write. take an example to clear our concept more suppose we have a rod length of x m, x may be any length 1 m or 2 m now attached one 1 m length rod to x m rod and divide 1 m rode in two parts again divide one part of 1/2 m rod peace into two parts see picture below
← x ← 1 m →
───────────────── change in length ∆x = 1m
───────────── change in ∆x =1/2 m
─────────── change in ∆x =1/4 m
──────────change in ∆x =1/8 m
───────── change in ∆x =1/16 m
every time 1 m red rod is divided into two parts.
now 1/8 m red rod part given someone to divide it and again given to other person to divide it, again given someone to divide it again given to someone divide it and continue to given every body of this world divide it just imagine how smallest it will be now the change in rod can’t be written it will be infinitesimally small and hence it is written as dx now change in length is dx it will not be ∆x why ? because ∆x is for large change in x whenever dx is written anywhere you must understand it is smallest change in x of the world.
Time interval and instant
difference between time interval and instant need to understand
time interval suppose you are driving a car going from one city to other city ↓ ↓
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t = 1000 s t = 0 time from 0 to 1000 s is time interval means span of time or period of time an event has happen if we want to calculate change in time ∆t = 500 – 0 = 500 s .
now instant of time means moment of time seen black car some moment of time car will be there that will be instant of time just after that car will be ahead of this point and just before back at position ↓ is exact instant of time exact car will be at that position if we say to calculate here ∆t then it will be zero why ? because it is instant of time, span of time here time has not spend so ∆t = 0 take an example when you do videography there may be 5 m, 10 m duration that is interval of time but when you do photography in this case an instant is capture here ∆t = 0 but during videography an interval is capture.
Rate of change
it is very interesting we use this term in day to day life.
here 20- 20 match score card is given in cricket we generally calculate run rate so we want to calculate whole match run rate during 0 to 20 overs hence run rate = (180 – 0)/(20 – 0) = 180/20 =9 hence 9 run per over this run rate is whole match but if we want to calculate last five overs run rate then we can calculate as run rate (20 – 15) = (180 – 120)/(20 – 15) = 60/5 = 12 run per over see here how we are calculating run rate = change in runs/change in overs hence we can write this run rate = ∆runs/∆overs this run rate is actually rate of change of runs with overs so when we talk about rate of change then we actually dividing change of two physical quantities.
when we study physical quantities from one quantity other quantity is made we have already studies in previous post but let me explain here when we change position one point to other point in between second quantity is made that is displacement because change of position is called displacement similar change in displacement with respect to time made second quantity velocity again change in velocity with respect to time made second quantity acceleration so this types of quantity in calculus is taken in pair in which one quantity is changing continuously so other quantity has to change due to change of first quantity one is called dependent variable and other is called independent variable example when our motorbike is moving here as time is changing our displacement is changing hence displacement depend upon time and time is changing here independently so displacement is dependent variable and time is independent variable if we want to know how much displacement done then our first question will be after what time so displacement depend upon time now there is third quantity which is called constant in this story there may be many quantity which are not changing or these two quantity do not depend upon that those quantity are called constant example you are going on motorbike you accelerated you bike speed increases distance increases after 60 s what do you think about your mass ? answer is mass not changes after 60 s hence mass is constant it is important to recognize dependent and independent variable in every story suppose you have fill 4 litter of petrol in your bike and bike is just standing for two days find dependent and independent variable see bike is standing time is passing petrol volume is independent of time because after two days petrol volume is same but when you start bike and travelling distance as long of distance you will travel that much of petrol volume will be consumed hence independent variable is distance and dependent variable is petrol volume because as much of petrol will be available in bike that much of long distance you can travel now how it is written in mathematical for y depends upon x or y is a function of x here y is dependent and x is independent now in mathematics it is written as
y = f(x) important this is not equation of mathematics this is just an information which simply imply that y depend upon x.
example suppose we start running with speed of 10 m/s then tell me which is function of what ? answer analyse the situation we start running with speed 10 m/s its mean that in first second 10 m distance will be travel in next second 20 m distance will be travel in third second 30 m distance will be travel hence we can write
distance = f(t) here distance is function of time so distance is dependent variable so we can write as s = f(t) here s is distance so it is first language of calculus here f indicate the exact relation between two variable when we replace f then for example it is written as y = 2x + 3 so in this equation as value of x is changing value of y will change hence y depend upon x now you will study in mathematics details here value of x will be not an imaginary number except imaginary you can put any value important point calculus is applicable for only one variable change what does it mean take an example y = f(x) = 2x² +z -1 then tell me here y is function of x answer is yes as value of x change value of y will change then what about z ? here it is given y = f(x) so z is constant if x and z both change together then calculus will not apply here so only for single variable calculus is applicable.
Suppose a car is going from city A to city B
────────────────────── x₂, t₂ x₁,t₁
here velocity of car will be written as v = (x₂ – x₁)/(t₂ – t₁)
so here in differentiation we added two operation one is difference and other is division of difference of two physical quantities combined effect of this is called differentiation and as above already discussed regarding change so we can write v = ∆x/∆t but we know that ∆ is for lager change but for smallest change we write v = dx/dt similarly for acceleration a = ∆v/∆t for smallest change we can write a = dv/dt.
now in next topic we will continue i hope you enjoyed learning this topic thanks for reading