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    Growing

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Glossary

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6th class > > Growing

Growing

Growing

In the grand world of Calculus, everything is in motion, constantly

growing

or

changing

Here, we encounter two tribes: the steadfast Constants and the ever-changing Variables.

Who do you think is more easy to deal with?

Constants

The Constants, known for their unwavering values, are like the ancient guardians of this world. They are marked by symbols from the beginning of the alphabet, like a, b, or c, representing their fixed nature.

Do constants change their values?

Variables

On the other side, we have the Variables, the true adventurers of this world. These brave souls are always on the move, constantly shifting and growing as the journey unfolds. We recognize them by symbols from the end of the alphabet, such as x, y, z, u, v, w, or even t

Do variables change their values?

Dependent variables

The fun starts when dealing with more than one variable at once and thinking of the way in which one variable depends on the other. For example, consider a rectangle. Lets say it has a fixed area. If we increase the length of the rectangle, what will happen to it's breadth if it's area has to be fixed? It will .

Review

Since the length and breadth are changing, they are .

So let's say length, should be represented by .

Area is a constant, so it should be represented by .

In general we call one of the variables x, and the other that depends on it y.

So in above example, length is and breadth is

A little bit of change

Suppose we change x by adding a little bit to it. What do we call a little bit of x? .

We are thus causing x to become x+dx. Then, because x has changed, y will also change by a little bit, and will have become y+.

Here the bit dy may be in some cases positive, in others negative; and it won't (except by a miracle) be the same size as dx.

Let x represent, in above figure, the horizontal distance, from a wall, of the bottom end of a ladder, BC, of fixed length; and let y be the height it reaches up the wall. Now y clearly depends on .

It is easy to see that, if we pull the bottom end a bit further from the wall(change x a little), the top end will come down a little lower(y will change a little).

Let us state this in our new vocabulary. If we increase x to x+dx, then y will become y

That is, when x receives a positive increment, the increment which results to y is negative.

Let's put some concrete numbers and check. Suppose the ladder was so long that when the bottom end was 19 inches from the wall the top end reached just 15 feet from the ground. Now, if you were to pull the bottom end out 1 inch more, how much would the top end come down?

Let's put it all into inches: x=19 inches, y= inches.

Now the increment of x which we call dx, is inch

So x+dx=20 inches.

How much will y be diminished? The new height will be y.

The length of the ladder is 1802+192= inches.(Pythagoras theorem)

So new height y−dy=

(y−dy)^2=(181)^2−(20)^2

  • set: (y−dy)^2=32761−400,,By Pythagoras theorem
  • set: (y−dy)^2=32361
  • set: y−dy=sqrt(32361)
  • set: y−dy=179.89

y−dy=179.89

  • set: 180-dy=179.89,,Because y is 180.
  • set: dy=180-179.89
  • set: dy=0.11 inch

So we see that making dx an increase of 1 inch has resulted in making dy a decrease of inch.

And the ratio of dy to dx= dydx=−0.111.

It is also easy to see that (except in one particular position) dy will be of a different size from dx

Now right through the differential calculus we are hunting, hunting, hunting for a curious thing, a mere ratio, namely, the proportion which dy bears to dx when both of them are indefinitely small.

It should be noted here that we can only find this ratio dydx when y and x are related to each other in some way, so that whenever x varies y does vary also.

For instance, if the distance x of the foot of the ladder from the wall be made to increase, the height y reached by the ladder in a corresponding manner, slowly at first, but more and more rapidly as x becomes greater.

In these cases the relation between x and y is perfectly definite, it can be expressed mathematically, x2+y2=l2 (where l is the length of the ladder) respectively, and dydx has the meaning we found.

Now let:

x=distance of the foot of the ladder from the wall(as before)

y=number of bricks in the wall

A change in x change in y

In this case dydx has no meaning whatever, and it is not possible to find an expression for it. So in the calculus world, we are not worried with variables with no relation.

Functions

Whenever we use differentials dx, dy, dz, etc., the existence of some kind of relation between x, y, z, etc., is implied, and this relation is called a function in x, y, z, etc.; For example, from above, x2+y2=l2, is a function of x and y.

yx=tan30°

Such expressions contain implicitly (that is, contain without distinctly showing it) the means of expressing either x in terms of y or y in terms of x, and for this reason they are called implicit functions in x and y.

Choose an implicit function:

Implicit functions are in a way like the boy and girl in the class who every one knows are in love, but have not expressed it to each other yet.

Implicit functions can be put into the forms like

y=xtan30° or x=ytan30°

y=l2−x2 or x=l2−y2

These expressions state explicitly (that is, distinctly) the value of x in terms of y, or of y in terms of x, and they are for this reason called functions of x or y.

This is like someone solves the untold love between the boy and the girl in the class and they update their status as "in a relationship".

Implicit-Explicit

x2+3=2y−7 is an implicit function in x and y

It may be written as y=x2+102 (explicit function of x)

Or as x=2y−10(explicit function of ).

We see that an explicit function in x, y, z, etc., is simply something the value of which changes when x, y, z, etc., are changing, either one at the time or several together.

Variables

The value of the explicit function is called the dependent variable, as it depends on the value of the other variable quantities in the function

The other variables are called the independent variables because their value is not determined from the value assumed by the function.

For example, in y=x2+102 the dependent variable is and the independent variable is

If u=x2sinθ, and θ are the independent variables, and is the dependent variable.

y = 2x + 3
y = √(x + 1)
y = 3e^x
x^2 + y^2 = 25
xy = 10
Implicit Functions
Explicit Functions

Unknown Function

Sometimes the exact relation between several quantities x, y, z either is not known or it is not convenient to state it; it is only known, or convenient to state, that there is some sort of relation between these variables, so that one cannot alter either x or y or z singly without affecting the other quantities. Like we know there is some relation between Tom and Sarah, but is it friendship, is it love, or is it something altogether different.

The existence of a function in x, y, z is then indicated by the notation F(x,y,z) (implicit function) or by x=F(y,z), y=F(x,z) or z=F(x,y) (explicit function). Sometimes the letter f or ϕ is used instead of F, so that y=F(x), y=f(x) and y=ϕ(x) all mean the same thing, namely, that the value of y depends on the value of in some way which is not stated.

We call the ratio dydx the differential coefficient of y with respect to x

It is a solemn scientific name for this very simple thing. But we are not going to be frightened by solemn names, when the things themselves are so easy. Instead of being frightened we will simply pronounce a brief curse on the stupidity of giving long crack-jaw names; and, having relieved our minds, will go on to the simple thing itself, namely the ratio dydx.

In ordinary algebra which you learned at school, you were always hunting after some unknown quantity which you called x or y; or sometimes there were two unknown quantities to be hunted for simultaneously. You have now to learn to go hunting in a new way; the fox being now neither x nor y. Instead of this you have to hunt for this curious cub called dy/.

The process of finding the value of dydx is called differentiating. But, remember, what is wanted is the value of this ratio when both dy and dx are themselves indefinitely small. The true value of the differential coefficient is that to which it approximates in the limiting case when each of them is considered as infinitesimally minute.

Let us now learn how to go in quest of dydx.