Skip to main content

What are Function and how its work on Calculus?

What are Function ?

    Introduction with beautiful example

    Here's a plant, and what you see here is it's  shadow. Can you list the things that the length of the shadow is dependent on. One, it's dependent on the position of the source of light. Anything else that you can think of. If the height of the plant grows then the shadows length will also change, right. So the length of the shadow is dependent on the position of the source of light, and the height of the plant too. So we can say that the length of the shadow is a function of the following two things. The output is dependent on these two things, which could be considered as the inputs. That's a very simple way to understand functions. Could you think of more inputs, this output is dependent on here, tell us yours answers in the comment section below. 

    How do Function work in calculus ?

    That's what we'll see in this topics. Previously, we saw an idea to find the instantaneous speed of an object. Do you remember average speed, and limits.

    Average Speed 

    Let's say we want to find the speed at this instant (t=20 sec). For this, we find the average speed of the car in small intervals near this position (x=400). 

    As we find the average speeds here, we see that these speeds approach a number (40), which will be at this instant (t=20 sec). But let me ask you a question. How do we know for sure that these two sequences approach average speed 40 and not 40.05 or any random number between 39.9 and 40.1 . Well we can find the average speeds at even closer time intervals. Let's say we get the average speed between t=20 and t=20.1 as 40.001 so 40.05 can't be the answer. But then how do we know that 40.0005 is not our answer? How can we be so confident that it's 40. Any solultion for this? Think about what information we need to find the speed at an instant.

    Distance and Time Relation

    We should know the distance travelled by the car in various time intervals, very close to t=20s . Let's say, we knew that this distance d and time followed a nice relationship like d=t². Notice what this relation tells us. It gives us information about the car position at each instant of time near the point.

    Let's see how Powerful this Relation in figuring out the speed at this instant (t=20s). From this relation, we get that, at 20 sec the distance travelled is 20²= 400 metres. Now after h (where h tends to 0) second, the distance travelled will be (20+h)² metres. So what will be the average speed between at 20 second and 20+h second instances of time? The distance travelled will be (20+h)² - 20² and the time duration will be (20+h)-20. And by doing some elementary algebra, we will get the average speed to be 40+h. So now I guess you realise what this relation has achieved for us. h here signifies some duration of time after t=20 seconds. As we consider small values of h , that means you're calculating the average speed between smaller and smaller time intervals. Then we see that as h approaches to 0, the average speed approaches the number 40 exactly. If we want to find the average speed between t=20 seconds position and the positions before it. We just have to substitute -h in place of +h. And we would get the average speed as 40-h. Then again we could get that as h tends to 0, the average speed approaches the number 40. So the speed at this instance is exactly 40 metres per second. Isn't knowing such a relationship helpful? Wait there is more. Let say we want to find the speed at an arbitrary instance we denote that by a variable t'. Now instead of 20, We can substitute t' everywhere in place of 20.

    We get the average speed as 2(t')+h. Then we will get that the speed at time (t') = 2(t'). Take a moment and go through these calculation. Interesting right? Is this formula correct. We can verify it with the results we got earlier. At time t=20 seconds , we get the  instantaneous speed as 2×20, which is 40. In just one attempt, we would know the speed of the car at any instant. But this ia all possible if you know this relationship d=t². This is a Mathematical representation of a function.

    Function- Relation between two variables

    We know that when an object is in motion, it mean it's position changes with respect to time. So we have two variables, time and distance, and they are related to each other. Related to each value of the variable 't', there is one value of the variable 'd'. Such a type of relation between two variables is called a function. So the distance travelled by an object is a function of the time elapsed. And how the two variables are literally related, is represented by an equation as we saw earlier. This equation 'd=t²' tells us one specific relation between the time and distance. 

    It's just an example that we took. If the car was traveling with a different speed and acceleration, we will get different relationships. But now let's look at the beauty to this concept. We earlier found out that according to this relation d=t², at any instance t', the car speed will be two time t'. Let's say we slightly modify this Relation d=t² to d=kt².Here k is a constant. Now if we go back and do the calculations, we will get the speed at any instant t' will be k(2t'). In the 17th century, Galileo discovered that this relation is followed by any object falling freely towards the ground. So can you tell me the speed of any object falling towards the ground at any instant of time. That's correct, it will be equal to k(2t'). We see that knowing that an object moves according to this relationship d=kt², we can instantly find its speed at any instant. But wait.. the story of this equation doesn't end here. Let's continue this in the next part. 

    Part 2

    Types of variable

    Consider a square, if we increase the length of it's side, then we see the area of the square increases. So the area of the square is the function of the length of it's side. Can you tell me what this function will be?  If 'l' represent the length of the square side and 'A' represent s the area, then the function can be represented by ' A=l² ' equation. But notice this is the exact relation we saw earlier. We only have to take k=1 here. So we saw two different functions represented by the same equation. In y=x², here x can be time and y be distance. Or x can be the length is squares side and y be the Area. As we put different value of x (y=x²), we will get different value of y. So, 'x' is called the  independent variable and y is called the dependent variable.

    Denotation of Function and variable

    Now whenever we talk about functions to distinguish between these two variables ( y & x), we denote a function by a symbol like y(x). So, these two functions (A=l² and d=kt²) can be represented like A(l) and d(t). Therefore we can clearly tell which are the dependent variables and which are the independent ones. We will see the advantages of this notation (y=x²), As we further explore this concept in our upcoming topics. Earlier we found out what the speed will be at any instant for an object following this relation (d=kt²). We know that similar to the distance travelled, the speed of an object also depends on the time elapsed. In other words, the speed is also a function of time. 

    Differentiation

    In this case, we saw that the speed of the object will follow this relationship (2kt) . Let's denote the speed by a new variable 'S'. So we see that from this function, using the process of differentiation, we get a new function. But now what does this speed function mean, in the case of a square. Here the speed is the rate of the change in the distance traveled with respect to time. So, here as the length of the squares side changes, this function tells us the rate of the change in the area with respect to this length (r(t)=2l). So it's the same thing here!. That means, once we know that the function between two variables follows this relation (y=kt²). We instantly know the rate, this function tells rate of the change of the dependent variable with respect to the independent variable. So until now, we saw what a function is. It's a relation between two variables such that related to each value of a independent variable, there is one value of the dependent variable. We also saw how the concept of functions, makes our idea of finding the rate of change powerful. In the next topic, we will further explored this concept. But before leaving, Let me ask you a question. Do you think this example of a function we saw here has anything to do with a flashlight or a dish satellite. Let's see if you can find this out. Share your thoughts in the comments section below.


    Labels:

    Comments

    Post a Comment

    Popular posts from this blog

    Indefinite Integration - Notes, Concept and All Important Formula

    INDEFINITE INTEGRATION If  f & F are function of \(x\) such that \(F^{\prime}(x)\) \(=f(x)\) then the function \(F\) is called a PRIMITIVE OR ANTIDERIVATIVE OR INTEGRAL of \(\mathrm{f}(\mathrm{x})\) w.r.t. \(\mathrm{x}\) and is written symbolically as \(\displaystyle \int \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}\) \(=\mathrm{F}(\mathrm{x})+\mathrm{c} \Leftrightarrow \dfrac{\mathrm{d}}{\mathrm{dx}}\{\mathrm{F}(\mathrm{x})+\mathrm{c}\}\) \(=\mathrm{f}(\mathrm{x})\) , where \(\mathrm{c}\) is called the constant of integration. Note : If \(\displaystyle \int f(x) d x\) \(=F(x)+c\) , then \(\displaystyle \int f(a x+b) d x\) \(=\dfrac{F(a x+b)}{a}+c, a \neq 0\) All Chapter Notes, Concept and Important Formula 1. STANDARD RESULTS : (i) \( \displaystyle \int(a x+b)^{n} d x\) \(=\dfrac{(a x+b)^{n+1}}{a(n+1)}+c ; n \neq-1\) (ii) \(\displaystyle \int \dfrac{d x}{a x+b}\) \(=\dfrac{1}{a} \ln|a x+b|+c\) (iii) \(\displaystyle \int e^{\mathrm{ax}+b} \mathrm{dx}\) \(=\dfrac{1}{...
    Post a Comment
    Read more

    Logarithm - Notes, Concept and All Important Formula

    LOGARITHM LOGARITHM OF A NUMBER : The logarithm of the number \(\mathrm{N}\) to the base ' \(\mathrm{a}\) ' is the exponent indicating the power to which the base 'a' must be raised to obtain the number \(\mathrm{N}\) . This number is designated as \(\log _{\mathrm{a}} \mathrm{N}\) . (a) \(\log _{a} \mathrm{~N}=\mathrm{x}\) , read as \(\log\) of \(\mathrm{N}\) to the base \(\mathrm{a} \Leftrightarrow \mathrm{a}^{\mathrm{x}}=\mathrm{N}\) . If \(a=10\) then we write \(\log N\) or \(\log _{10} \mathrm{~N}\) and if \(\mathrm{a}=e\) we write \(\ln N\) or \(\log _{e} \mathrm{~N}\) (Natural log) (b) Necessary conditions : \(\mathrm{N}> \,\,0 ; \,\, \mathrm{a}> \,\,0 ; \,\, \mathrm{a} \neq 1\) (c) \(\log _{a} 1=0\) (d) \(\log _{a} a=1\) (e) \(\log _{1 / a} a=-1\) (f) \(\log _{a}(x . y)=\log _{a} x+\log _{a} y ; \,\, x, y> \,\,0\) (g) \(\log _{a}\left(\dfrac{\mathrm{x}}{y}\right)=\log _{\mathrm{a}} \mathrm{x}-\log _{\mathrm{a}} \mathrm{y} ; \,\, \mathrm{...
    Post a Comment
    Read more

    Hyperbola - Notes, Concept and All Important Formula

    HYPERBOLA The Hyperbola is a conic whose eccentricity is greater than unity \((e>1) .\) 1. STANDARD EQUATION & DEFINITION(S): Standard equation of the hyperbola is \(\dfrac{\mathbf{x}^{2}}{\mathbf{a}^{2}}-\dfrac{\mathbf{y}^{2}}{\mathbf{b}^{2}}=\mathbf{1},\) where \(b^{2}=a^{2}\left(e^{2}-1\right)\) or \(a^{2} e^{2}=a^{2}+b^{2}\)    i.e.    \(e^{2}=1+\dfrac{b^{2}}{a^{2}}\) \(=1+\left(\dfrac{\text { Conjugate Axis }}{\text { Transverse Axis }}\right)^{2}\) (a) Foci : \(\mathrm{S} \equiv(\mathrm{a} e, 0) \quad \& \quad \mathrm{~S}^{\prime} \equiv(-\mathrm{a} e, 0) .\) (b) Equations of directrices: \(\mathrm{x}=\dfrac{\mathrm{a}}{e}\quad \) & \(\quad \mathrm{x}=-\dfrac{\mathrm{a}}{e}\) (c) Vertices: \(A \equiv(a, 0)\quad \) & \(\quad A^{\prime} \equiv(-a, 0)\) (d) Latus rectum: (i) Equation: \(\mathrm{x}=\pm \mathrm{ae}\) (ii) Length:  \(\begin{aligned} &=\dfrac{2 b^{2}}{a}=\dfrac{(\text { Conjugate Axis })^{2}}{(\text { Transverse Axis ...
    Post a Comment
    Read more