Transcription
Here’s a plant, and what you see here is its shadow. Can you list the things that the length of the shadow is dependent on? One, it is dependent on the position of the source of light. Anything else you can think of? If the height of the plant grows, then the shadow’s 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 your answers in the comments section below! Also, how do functions work in Calculus? That’s what we will see in this video!
Previously, we saw an idea to find the instantaneous speed of an object. Do you remember average speed and limits? Let’s say we want to find the speed at this instant. For this, we find the average speed of the car in small time intervals near this position. As we find the average speeds here, we see that these speeds approach a number; which will be the speed at this instant. But let me ask you a question? How do we know for sure that these two sequences approach ‘forty’ and not ‘forty point zero five’; or any random number between these two numbers? Well, we can find the average speeds at even closer timer intervals. Let’s say we get the average speed between these points as ‘forty point zero zero one’. So forty point zero five cannot be the answer! But then how do we know that ‘forty point zero zero zero five’ is not our answer? How can we be so confident that it’s 40? Any solution for this?
Think about what information we need to find the speed at an instant. We should know the distance travelled by the car in various time intervals very close to this instant. Let’s say we knew that this distance and time followed a nice relationship like this one. Notice what this relation tells us. It gives us information about the car’s position at each instant of time near this point. Let’s see how powerful this relation is in figuring out the speed at this instant. From this relation, we get that, at ‘twenty seconds’ the distance travelled is ‘four hundred meters’. Now after ‘h’ seconds, the distance travelled will be this. So what will be the average speed between these two instances of time? The distance travelled will be this and the time duration will be this. And by doing some elementary algebra, we will get the average speed to be ‘forty plus h’. So now I guess you realise what this relation has achieved for us. ‘h’ here signifies some duration of time after ‘t equal to twenty seconds’. As we consider smaller values of ‘h’; that means we are calculating the average speed between smaller and smaller time intervals. Then we see that as ‘h’ approaches zero, the average speed approaches the number ‘40’ EXACTLY. If we want to find the average speed between this position and the positions before it; we just have to substitute ‘negative h’ instead of ‘h’. And we would get the average speed as ’40 minus h’. Then again we would get that as ‘h’ tends to zero; the average speed approaches the number ‘40’. So the speed at this instant is exactly ‘forty metres per second’. Isn’t knowing such a relationship helpful?
Wait… there is more. Let’s say we want to find the speed at an arbitrary instant. We denote this by a variable ‘t dash’. Now instead of ‘twenty’, we can substitute ‘t dash’ everywhere here. We get the average speed as ‘2 times t dash’ plus ‘h’. Then we will get that the speed at time ‘t dash’ to be ‘two times t dash’. Take a moment, and go through these calculations. Interesting, right? Is this formula correct? We can verify it with the results we got earlier. At time ‘t’ equal to ’20 seconds’, we get the instantaneous speed as ‘2 times 20’ which is 40. In just one attempt, we would know the speed of the car at ANY instant. But this is all possible if we know this relationship. This is a mathematical representation of a function.
We know that when an object is in motion, it means its 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 tells us one specific relation between the time and distance. It’s just an example that we took. If the car was travelling with a different speed and acceleration, we will get different relationships. But now let’s look at the beauty of this concept. We earlier found out that according to this relation, at any instant ‘t dash’, the car’s speed will be ‘two times ‘t dash’. Let’s say we slightly modify this relation to this. Here ‘K’ is a constant. Now if we go back and do the calculations, we will get that the speed at any instant ‘t dash’ will be ‘k times two t dash’. In the seventeenth 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 this. We see that knowing that an object moves according to this relationship, we can instantly find its speed at any instant. But wait… the story of this equation does not end here. Let’s continue this in the next part.
Consider this square. If we increase the length of its side, then we see that the area of the square increases. So the area of the square is the function of the length of its side. Can you tell me what this function will be? If ‘L’ represents the length of the square’s side and ‘A’ represents the area, then the function can be represented by this equation. But notice that this is the exact relation we saw earlier. We only have to take ‘k’ equal to one here. So we saw two different functions represented by the same equation. Here ‘X’ can be time and ‘Y’, distance. Or ‘X’ can be the length of square’s side and ‘Y’ the area. As we put different values of ‘X’ here, we will get different values of ‘Y’. So ‘X’ is called the Independent variable and ‘Y’ is called the dependent variable. Now whenever we talk about functions, to distinguish between these two variables, we denote a function by a symbol like this. So, these two functions can be represented like this. Therefore we can clearly tell which are the dependent variables and which are the independent ones. We will see the advantage of this notation as we further explore this concept in our upcoming lessons.
Now we earlier found out what the speed will be at any instant for an object following this relation. We know that similar to the distance travelled, the speed of an object also depends upon the time elapsed. In other words the speed is also a function of time. In this case, we saw that the speed of the object will follow this relationship. 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 change in the distance travelled with respect to time. So, here as the length of the square’s side changes, this function tells us the rate of the change in the area with respect to this length. So it’s the same thing here! That means Once we know that the function between two variables follows this relation, we instantly we know the rate. This function tells the rate of change of the dependent variable with respect to the independent variable. So until now, we saw what a function is. It is a relation between two variables, such that related to each value of 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 lesson we will further explore 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. And hey, what’s this we see in the screen? Yes, you can subscribe to our channel by clicking here!