Calculus is essentially the study of rates of change. We're all familiar with rates of change in everyday life, even if we don't always realise it. E.g. speed is a rate of change - its the rate of change of distance with time.
Gradient (or slope) is another example of a rate of change - its the rate of change of height(y) with distance travelled horizontally(x).
The mechanics of obtaining a rate-of-change function from the function itself is not inherently difficult. There are simple rules which apply. Here are some examples for functions which are powers of x :-
1. If a curve has equation y = x², its gradient at any point will be 2x. [e.g. if x = 3, then y = 9 and the gradient will be 6]
2. If the curve equation is y = x³, its gradient at any point will be 3x².[e.g., if x = 2, then y = 8 and the gradient will be 12]
3. If y = 5x³ + 7x² - 2x + 19, then the gradient = 15x² + 14x - 2.
(Note here that x changes to 1 and pure constants, such as 19, change to 0. So -2x + 19 became just -2]
4. For an equation y = x^7, the gradient is given by 7x^6, and so on.
In general, for a function x^n, the gradient will be given by nx^(n-1)
[ Note - x^n is read as x to the power n, or simply x to the n.
Similarly, nx^(n - 1) is read as nx to the (n - 1), but remember that here its only the x which is raised to the power, as the n in front is just a multiplying number.
The rule for powers of x is simply to bring the power down in front of the x and reduce the power of x itself by 1.
For the sake of mathematical correctness at this stage I should point out that this process of deriving a rate-of -change function is called 'differentiation'. You may see books referring to it as 'finding the derivative' or 'finding the derived function'.
There is more than one way of expressing problems of diffferentiation.
Take the equation y = x³, for which we saw dy/dx = 3x².
A problem based on this could be worded in any of the following ways:-
a. If y = x³, find dy/dx.
b. If f(x) = x³ , find f '(x)
c. Find the derived function of y = x³.
d. Differentiate y = x³
In example b the derived function is written f '(x), read as 'f dashed x'.
In the same way, if we were dealing with some other function such as g(x), its derived function could be written as g'(x), read as 'g dashed x'.
The notation dy/dx is read as 'dee-y by dee x' (which I suggest you think of as a 'diminutive bit of y' divided by a 'diminutive bit of x')
So differentiation of powers of x decreases the powers.
Some other kinds of function:-
The derived function of sinx is cosx ;
" " " " " " cosx is - sinx;
" " " " " tanx is -sec²x
" " " " lnx is 1/x
" " " " " e^x is e^x (no change)
(The reverse process, which we'll come to later, is called 'integration', and this process increases the powers.)
In the first paragraph above we began by mentioning speed as an example of a rate of change. If it is speed we are dealing with, the letters s and t are used to denote distance and time, and the sort of equation to be differentiated might look like this :- s = 2t + t³.
The answer required is ds/dt = 2 + 3t².
A personal view :- Just about every explanation of calculus I've seen in books begins with text and diagrams which involve 'strange' notation, such as ðx and ðy, and talk about 'increments shrinking to zero'. This is, of course, perfectly correct, but the wisdom of introducing such mathematical rigour at the very start of a new topic has to be questionable.
So my approach would be to get the mechanics of differentiation established first. The theoretical explanation of where the rules come from and why it works can come a bit later, after the students have gained some confidence in handling these hitherto unfamiliar processes.
Now for a brief word about the process known as 'integration', which is the reverse of differentiation. It is in fact the process of finding 'what has been differentiated' to result in the given function.
So if differentiation changes x² into 2x, integration will say that the integral of 2x is x².
Actually the full answer is x² + c, where c is a numerical constant.
This would be written as ∫2x.dx = x² + c.
Why do we have to write the '+ c' part? Well, because there could have been a number added on to the x² before it was differentiated, as this number would have disappeared under differentiation, because a number doesn't have a gradient. (The graph of y = 4, for example, is a horizontal line, which, like a flat road, has no gradient. The same goes for y = any other number. This other number is the c we've tagged on to the x².)
Whenever so-called indefinite integration is done you automatically add a c in this way. In particular cases you may be given some extra information to enable the actual value of c to be calculated, but otherwise just leave it as c.
Integration is called definite when numerical limit values are given, attached as small numbers at the bottom and top of the integration sign. What happens in such cases is that you first of all do the integration process, then evaluate your answer for each of the two given numbers, and subtract one valuation from the other (lower from upper).
Integration is even more powerful than differentiation, as it can be used to find areas, volumes, centres of gravity, and in many other ways..
Gradient (or slope) is another example of a rate of change - its the rate of change of height(y) with distance travelled horizontally(x).
The mechanics of obtaining a rate-of-change function from the function itself is not inherently difficult. There are simple rules which apply. Here are some examples for functions which are powers of x :-
1. If a curve has equation y = x², its gradient at any point will be 2x. [e.g. if x = 3, then y = 9 and the gradient will be 6]
2. If the curve equation is y = x³, its gradient at any point will be 3x².[e.g., if x = 2, then y = 8 and the gradient will be 12]
3. If y = 5x³ + 7x² - 2x + 19, then the gradient = 15x² + 14x - 2.
(Note here that x changes to 1 and pure constants, such as 19, change to 0. So -2x + 19 became just -2]
4. For an equation y = x^7, the gradient is given by 7x^6, and so on.
In general, for a function x^n, the gradient will be given by nx^(n-1)
[ Note - x^n is read as x to the power n, or simply x to the n.
Similarly, nx^(n - 1) is read as nx to the (n - 1), but remember that here its only the x which is raised to the power, as the n in front is just a multiplying number.
The rule for powers of x is simply to bring the power down in front of the x and reduce the power of x itself by 1.
For the sake of mathematical correctness at this stage I should point out that this process of deriving a rate-of -change function is called 'differentiation'. You may see books referring to it as 'finding the derivative' or 'finding the derived function'.
There is more than one way of expressing problems of diffferentiation.
Take the equation y = x³, for which we saw dy/dx = 3x².
A problem based on this could be worded in any of the following ways:-
a. If y = x³, find dy/dx.
b. If f(x) = x³ , find f '(x)
c. Find the derived function of y = x³.
d. Differentiate y = x³
In example b the derived function is written f '(x), read as 'f dashed x'.
In the same way, if we were dealing with some other function such as g(x), its derived function could be written as g'(x), read as 'g dashed x'.
The notation dy/dx is read as 'dee-y by dee x' (which I suggest you think of as a 'diminutive bit of y' divided by a 'diminutive bit of x')
So differentiation of powers of x decreases the powers.
Some other kinds of function:-
The derived function of sinx is cosx ;
" " " " " " cosx is - sinx;
" " " " " tanx is -sec²x
" " " " lnx is 1/x
" " " " " e^x is e^x (no change)
(The reverse process, which we'll come to later, is called 'integration', and this process increases the powers.)
In the first paragraph above we began by mentioning speed as an example of a rate of change. If it is speed we are dealing with, the letters s and t are used to denote distance and time, and the sort of equation to be differentiated might look like this :- s = 2t + t³.
The answer required is ds/dt = 2 + 3t².
A personal view :- Just about every explanation of calculus I've seen in books begins with text and diagrams which involve 'strange' notation, such as ðx and ðy, and talk about 'increments shrinking to zero'. This is, of course, perfectly correct, but the wisdom of introducing such mathematical rigour at the very start of a new topic has to be questionable.
So my approach would be to get the mechanics of differentiation established first. The theoretical explanation of where the rules come from and why it works can come a bit later, after the students have gained some confidence in handling these hitherto unfamiliar processes.
Now for a brief word about the process known as 'integration', which is the reverse of differentiation. It is in fact the process of finding 'what has been differentiated' to result in the given function.
So if differentiation changes x² into 2x, integration will say that the integral of 2x is x².
Actually the full answer is x² + c, where c is a numerical constant.
This would be written as ∫2x.dx = x² + c.
Why do we have to write the '+ c' part? Well, because there could have been a number added on to the x² before it was differentiated, as this number would have disappeared under differentiation, because a number doesn't have a gradient. (The graph of y = 4, for example, is a horizontal line, which, like a flat road, has no gradient. The same goes for y = any other number. This other number is the c we've tagged on to the x².)
Whenever so-called indefinite integration is done you automatically add a c in this way. In particular cases you may be given some extra information to enable the actual value of c to be calculated, but otherwise just leave it as c.
Integration is called definite when numerical limit values are given, attached as small numbers at the bottom and top of the integration sign. What happens in such cases is that you first of all do the integration process, then evaluate your answer for each of the two given numbers, and subtract one valuation from the other (lower from upper).
Integration is even more powerful than differentiation, as it can be used to find areas, volumes, centres of gravity, and in many other ways..
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