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Abstract
Introduction
1. DIFFERENTIATION,
1.1 Modern Approach to Differentiation,
1.2 Vedic Approach to Differentiation,
1.2.1 Limits,
1.2.2 Secant of a Parabola,
1.2.3 Gradient of a Curve,
1.2.4 Extending,
1.3 Ratio of Gradients,
1.3.1 Proof that the Ratio of Gradients = n,
1.3.2 Classroom Approach,
2. AREAS UNDER CURVES,
2.1 Modern Approach to Areas under Curves,
2.2 Vedic Approach to Areas under Curves,
2.2.1 Ratios of Areas,
2.2.2 Area of a Strip,
2.2.3 Extending,
2.2.4 Explanation of Ratio of Areas = n for y = x2,
2.2.5 GENERAL PROOF THAT Ratio of Areas = n,
3. SUMMARY and Concluding Remarks,
Reference.
1.
TEACHING CALCULUS
Kenneth Williams
Director, Vedic Mathematics Academy, UK
Calculus comes under the Calana Kalanābhyām Sutra of Vedic Mathematics.
Though it is a subject usually taught later in the school career, Sri Bharati Krishna
Tirthaji tells us that in the Vedic system ‘Calculus comes in at a very early stage’.
This paper aims to show how Calculus may be taught to quite young children.
Gradients of curves can be found without ‘differentiation’ in the usual sense, and
areas under curves can be found without the usual integration process.
Furthermore this approach uses a similar method for both thereby unifying these
two aspects of Calculus. After the idea of a limit is introduced, in a simple way,
and with a little algebra and geometry, we arrive at an easy technique that gives
exact gradients of curves and areas under curves. There is no need for the
complex notation that is normally associated with this subject and which adds to
its perceived difficulty in school (though this notation can be introduced at the
teacher’s discretion).
Calculus is usually taught towards the end of a school career because its sophisticated
arguments require a good grounding in various other areas and its complex notations tend to
baffle. Swami Tirtha makes it clear in the Prolegomena of his book1 that the topics in Vedic
mathematics can be taught in any order, and also that ‘Calculus comes in at a very early
stage’ in the Vedic system.
Since Calculus is such an important and useful subject it makes sense that it should be
introduced in a simple way early on. Other topics, like geometry, graph work, statistics and
so on are introduced early on and developed from there. Why can’t we do the same with
Calculus? What we need is a simple approach that builds on what pupils know and keeps the
notation within bounds.
This article shows that this can be done. If pupils know a little algebra and understand basic
graphs and the idea of gradients, then they can take the approach to Calculus outlined here.
First we will look at how differentiation is approached traditionally before introducing the
Vedic method. Also we will see how a simple relationship (that for y = axn a certain ratio of
gradients is equal to n) can also be used to get gradients. In the second section we see a
certain ratio of areas that is also equal to n, and which can be used to easily obtain areas
under curves. The area under the curves traditionally comes under integration in calculus.
Thus the current paper covers both differentiation and integration, which form the core of the
subject of calculus.
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2.
1. DIFFERENTIATION
1.1 Modern Approach to Differentiation
The modern approach to teaching differentiation is to introduce function notation and define
a tangent gradient of a curve by:
f ( x x) f (x)
f (x) lim
x 0 x
So in addition to the function notation we have the special notation for limits and the notion
of vanishing quantities like x, formed, confusingly, by two inseparable symbols.
Then a diagram shows the set-up for a particular case, invariably y = x2.
Next the definition is applied, giving: y
y = x2
f (x x) f (x) Q(x+x, y+y)
f (x) lim
x 0 x
(x x) x 2
2
y
lim
x 0 x
P(x, y)
2 x x ( x)2
lim x
x 0 x 0 x
lim (2 x x)
x 0
2x
This may work well for the older, and more mathematically sophisticated child, and of course
the definition is general and so applies to all functions, but the alternative approach given
next is entirely suitable for younger children and introduces the concepts of limits and
gradients much more simply.
1.2 Vedic Approach to Differentiation
1.2.1 Limits
Children know about limits in their everyday life – they know there is a limit to the distance
they can throw a ball or the height they will grow to.
B
From here we can give mathematical
examples: like what is the limit of
the line AB as B moves towards,
and merges into, C:
A C
2
3.
Or the limit of the secant
secant
line PQ as Q approaches
P P around the circle,
and merges into P:
Q
1.2.2 Secant of a Parabola
Suppose we have the simplest of mathematical curves: the parabola, described by the
y = x2.
We can take two points on the curve, say where x = 2 and x = 5, and we can find the gradient
of the secant that passes through them. Since the difference of the y-coordinates divided by
25 4
the difference of the x-coordinates is 5 2 we find the gradient to be 7.
Further examples lead to the observation of a simple relationship between the two
x-coordinates: that the sum of those x-coordinates gives the gradient. In this case 2 + 5 = 7,
the gradient of the secant. We just use the Sutra By Addition.
We can go on to prove this result. Given points on the parabola with x = p and x = q we
expect the gradient of the secant to be p + q.
We have: P(p, p2) and Q(q, q2) and
p2 q2 p – q p q
the gradient of PQ = p q = pq = p + q.
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4.
This approach can then be developed to find secant gradients for:
y = ax2,
y = ax2 + bx + c,
y = x3 and so on.
The formula for the difference of two squares used above is well-known and very useful. It
also leads to further formulae for the difference of two cubes etc., which can be made use of
when finding secant gradients for other curves.
We will stay with y = x2 for the moment.
1.2.3 Gradient of a Curve
We can discuss with pupils the idea of the gradient of a curve: that it varies, and that the
gradient of the curve is the gradient of the tangent at that point.
Coming back to the secant through PQ, suppose we let Q approach P around the curve. Then
the secant ultimately becomes the tangent at P and the value of q approaches the value of p,
and ultimately becomes equal to p.
That means that since the secant gradient is p + q, the gradient at P is p + p = 2p, or 2x since
p is an x-coordinate.
Pupils need to understand the generality and limit of this result: that it is true for all points on
this particular parabola, but not necessarily true for any other curve. And they can confirm
graphically that the gradient is always twice the x-coordinate.
This “secant-tangent method” (in which we find the secant gradient and then allow one of the
points defining the secant to merge into the other) is extendible and can lead to the general
result that for any polynomial the gradient formula can be found in the conventional way.
i =n i =n
d
That is, that: dx ai x i =
i 0
a ix
i 0
i
i -1
.
p–q
In the last step of the derivation of (p + q) for the secant gradient, the cancelling of pq
is not
valid when p = q, but this can be explained as appropriate in terms of limits.
This secant-tangent approach does not extend to other functions such as trigonometrical and
exponential functions, but these can be tackled with alternative Vedic methods.
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5.
1.2.4 Extending
After studying the graphs of:
y = ax2,
y = ax2 + bx + c (here we demonstrate the distributive law for differentiation holds),
y = ax3 etc.
in a similar way, we can look at negative indices:
1
y= x ,
a
y= x ,
a
y= x2 etc.
And then fractional indices:
e.g. y2 = 5x3,
which can lead into the chain rule, implicit differentiation and parametric equations.
All of these can be treated in the way described above.
1
To find the gradient for y = x , for example, we find the difference of the y and x-coordinates
1 1
for P(p, p ) and Q(q, q ).
1
y y= x
1/q Q
P
1/p
0 q p x
p q
1 q1
p pq 1
The secant gradient is p q = – p q = – pq .
1 1
Then in the limit, as QP, the tangent gradient is – p2 =– x2 .
1
So simple curves like y = x2, y = x etc. have tremendous potential for teaching the
fundamentals of differentiation – finding gradients of curves using a limiting process.
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6.
1.3 Ratio of Gradients
Here we have an alternative relationship that we will later connect with a similar ratio for
areas under curves.
There are in fact two gradients associated with any point on a curve:
1) the gradient of the tangent
to the curve at that point,
2) the gradient of the line
joining the point to the origin.
And for y = axn, where n is rational and a is a real number:
The ratio of these gradients is n.
dy /dx dy y
So we have: ratio of gradients y / x = n, which means that dx = n × x
.
1.3.1 Proof that the Ratio of Gradients = n
dy
Given yu = axv, where u, v , and a , we have uyu-1 dx = avxv-1.
y ax v -1 dy v y
And since x = u-1 , then dx = u × x .
y
This result comes under the Sutra If One is in Ratio the Other is Zero. It can also be derived
using the “secant-tangent method” given earlier. It
Example: Find the gradient of y = 2x3 at the point (3, 54).
dy 54
We note n = 3 and so: dx = 3 × 3 = 54.
Example: And to find the gradient of y2 = 4x3 at the point (4, 16),
3
we note n = 2 and so:
dy 3 16
dx = 2 × 4 = 6.
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7.
For comparison with the contemporary method the corresponding solution to the last
example above would be:
dy
2y dx = 12x2
dy 6x 2
dx = y
dy 96
dx = 16 = 6.
1.3.2 Classroom Approach
This ratio of gradients can be approached in the classroom:
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a) graphically, for example for y = x the ratio of gradients is -1,
y
y=
1
For y = x
0 x
the Ratio of Gradients = -1
b) by algebraic calculation of the ratio for specific cases,
c) algebraically to verify the general conclusion.
It is very motivating to see a conjecture suggested from a graph verified algebraically or vice
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8.
2. AREAS UNDER CURVES
2.1 Modern Approach to Areas under Curves
First, integration is defined as the reverse of differentiation. Then various methods are used to
show that an integral (or anti-derivative) of a function defines an area under that function.
We may argue as follows.
Let A(x) = the area under y = f(x)
from x = a to x = x, where A(a) = 0.
Then the area of the shaded strip
on the right = A(x+x) - A(x) x f(x).
A(x x) A(x)
Therefore f(x).
x
A(x x) A(x)
But lim = A( x) , therefore in the limit A( x) = f(x).
x 0 x
x
And if A( x) = f(x) then A = a
f (x)dx .
The full argument is not given here due to lack of space, but we see that we can find areas
under y = f(x) by integrating f(x), substituting the limits and subtracting.
Next we look at an alternative approach that can be easily grasped by young children.
2.2 Vedic Approach to Areas under Curves
2.2.1 Ratios of Areas
Take a pair of axes and from the
origin draw a straight line as shown,
with any gradient.
Select any point on the line and drop
perpendiculars onto the two axes, as also shown.
The two triangles generated will be
equal in area, and this will apply for
any straight line chosen: the ratio of
areas will be 1:1.
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9.
Now suppose we wish to draw a line, starting at the origin, but the requirement is that the
ratio of areas similarly generated will be 2:1 rather than 1:1.
The line must clearly be a curve, rather than a straight line.
Also its gradient must be increasing if the area to the left of the line is to be double the area
under it.
What will be the equation of the curve that has this special property?
y
y = x2
Well, it turns out that the
answer is the parabola given by y = x2. 2
In fact any of the parabolas given 1
by y = ax2 will have this property.
We will demonstrate this result shortly. 0 x
This means that the area indicated by ‘1’ above is one third of the total area, and area ‘2’ will
be two thirds of the total rectangle.
Example: Suppose we want the area under y = x2 from x = 0 up to x = 4.
y
y = x2
We can find the area of the
rectangle and find one third of that.
The height of the rectangle is
2
42 = 16, so the rectangle area is 4 × 16 = 64. 1
0 4 x
1 64
And the required area will therefore be 3 × 64 = 3 .
In fact we can see that the area is one third of the cube of the x-coordinate.
This is just simple Proportion.
2.2.2 Area of a Strip
Similarly we can find the area of a strip A, like the area from x = 2 to x = 3 shown below.
y
y = x2
Since we can find the area
from the origin up to x = 3
and also the area up to x = 2,
A
9
0 2 3 x
10.
we can find the difference of these,
which will be the required area.
So we need
33 23 27 8 19
3 – 3 = 3 – 3 = 3 .
19
We get the area as 3 .
We simply subtract the cubes of the two x-values, and divide by 3.
2.2.3 Extending
We can also use symmetry in various ways to extend our range of techniques.
And we can extend to:
y = ax2 + bx + c where we have a difference of cubes,
difference of squares and difference of x-coordinates,
y = ax3 where the ratio of areas is 3:1,
y2 = ax3 where the ratio of areas is 3:2.
a
y= , x 0, where the ratio will be 2:1 (since areas are always positive).
x2
And in general for y = axn, n , a , the ratio is n:1.
So for y = axn:
The ratio of these areas is n.
2.2.4 Explanation of Ratio of Areas = n for y = x2
We can show that the ratio of areas for y = x2 is 2:1 as follows.
B
That is, show that A = 2,
where A is the value of the area between
the curve, the x-axis and the lines x = p
and x = q.
And B is the value of the corresponding area,
as shown.
The explanation follows similar lines
to that for the gradient of a curve inasmuch
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11.
as we take two points P(p, p2), Q(q, q2), express the ratio of areas approximately and then let
We approximate areas y
y = x2
A and B with the rectangles shown p2 P
The ratio of these areas is
B Q
q2
B q ( p q )
2 2
A = ( pq )q2 A
pq
= q , 0 q p x
2p
which in the limit goes to p = 2.
This shows that as the horizontal and vertical strips get narrower and narrower the ratio of the
areas approaches 2:1, and ultimately equals 2:1.
And since the ratio of areas is independent of x, it applies to all such strips.
Now any area to the left of y = x2 and
bounded by two horizontal lines and
the y-axis (like area B) can be split
into a large number of strips as shown.
And since the ratio of areas of two corresponding strips (one horizontal and one vertical) is
ultimately 2:1, therefore the ratio of the whole area consisting of a collection of strips on the
left, to the corresponding area below the curve will also be 2:1.
This shows that any area B in y = x2 is double the corresponding area A.
2.2.5 GENERAL PROOF THAT Ratio of Areas = n
Given y = axn, where n , a, x , take points on the curve P(p, apn) and Q(q, aqn),
p, q .
p
p p
ax n+1
ax
n
ydx = dx , n -1*.
n1
Then A = =
q q
q
11
12.
apn
an y 1n 1
n n 1
ap ap
y
= n1 a , n -1.
n
And B = n
x dy = n a dy
aq aq
aq n
n nn1 n nn1
B n
p q pn+1 qn+1
So A =
pn+1 - qn+1 = pn+1 qn+1 .
n
pn+1 qn+1
And lim n n+1 n+1 = n.
p q
q p
* The case where n = -1 is straightforward.
3. SUMMARY and Concluding Remarks
We have seen that we can use limiting arguments to obtain the gradient of curves given by
y = axn and to obtain areas under them. Compared to traditional approaches this has certain
advantages, notably that calculus can be taught to quite young students.
The secant-tangent method, and the memorable result that Ratio of Gradients = Ratio of
Areas = n enable us to easily get gradients of a range of curves and areas under them.
By restricting the study to curves of the form y = axn pupils can be exposed to the main
principles of calculus at a young age, so that the understanding and development of this
subject becomes a lot smoother, and a good foundation is prepared for the more advanced
work which may follow.
This way pupils get the opportunity to develop and apply skills in algebra, limits etc. The
necessary notation can be introduced as appropriate at the discretion of the teacher. ‘Vedic’
methods for more advanced work (like the product rule, chain rule, differential equations etc.)
are also available so this approach can be developed further, and merged with the
conventional approach at any point.
[1] Bharati Krsna Tirthaji Maharaja, “ Vedic Mathematics”, Motilal Banarasidas Publisher,
Delhi, 1994.
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