Transcript “Teach A Level Maths” Vol. 1: AS Core Modules 15: The Gradient of the Tangent as a Limit © Christine Crisp The Gradient of the.

“Teach A Level Maths”
Vol. 1: AS Core Modules
15: The Gradient of the
Tangent as a Limit
© Christine Crisp
The Gradient of the Tangent as a Limit
Module C1
Module C2
AQA
MEI/OCR
Edexcel
OCR
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The Gradient of the Tangent as a Limit
The Gradient of a Tangent
We found the rule for differentiating by noticing a
pattern in results found by measuring gradients of
tangents.
However, if we want to prove the rule or find a
rule for some other functions we need a method
based on algebra.
This presentation shows you how this is done.
The emphasis in this presentation is upon understanding
ideas rather than doing calculations.
The Gradient of the Tangent as a Limit
Consider the tangent at the point A( 1, 1 ) on
(2,4)
yx
2
A(1,1)

y  x2

Tangent at A
As an approximation to the gradient of the
tangent we can use the gradient of a chord from
A to a point close to A.
e.g. we can use the chord to the point ( 2, 4 ).
( We are going to use several points, so we’ll call
this point B1 ).
The Gradient of the Tangent as a Limit
Consider the tangent at the point A( 1, 1 ) on
B1(2,4)
yx
2
Chord
AB1
A(1,1)

y  x2

Tangent at A
The gradient of the chord AB1 is given by
y 2  y1
m
x 2  x1
41
 m
3
21
We can see this gradient is larger than the
gradient of the tangent.
The Gradient of the Tangent as a Limit
To get a better estimate we can take a point B2
that is closer to A( 1, 1 ), e.g. B2 (1  5, 2  25)
Chord
AB2
yx
2
B1
B2 (1  5, 2  25)

A(1,1)


Tangent at A
The gradient of the chord AB2 is
2  25  1
m
 25
1 5  1
The Gradient of the Tangent as a Limit
We can get an even better estimate if we use
the point B3 (1  1, 1  21).
Chord
AB3
yx
2
B1

B2
A(1,1) B 3


Tangent at A
We need to zoom in to the curve to see more
clearly.
The Gradient of the Tangent as a Limit
We can get an even better estimate if we use
the point B3 (1  1, 1  21).
B3 (1  1, 1  21)

Chord AB3
Tangent at A
y  x2

A(1,1)
The gradient of AB3 is
1  21  1
m
 21
11  1
The Gradient of the Tangent as a Limit
Continuing in this way, moving B closer and closer to
A( 1, 1 ), and collecting the results in a table, we get
Point
B1
B2
B3
B4
x
2
1 5
11
1 01
1 001
y (  x2 )
4
2  25
1 21
1 0201
1 002001
y1
3
1 25
0  21
0  0201
0  002001
x1
1
05
01
0  01
0  001
Gradient
of AB
3
25
21
2  01
B5
2  001
As B gets closer to A, the gradient approaches 2.
This is the gradient of the tangent at A.
The Gradient of the Tangent as a Limit
As B gets closer to A, the gradient of the chord
AB approaches the gradient of the tangent.
We write that the gradient of the tangent at A
 lim
as B  A
( gradient of the chord AB )
The gradient of the tangent at A is “ the limit of
the gradient of the chord AB as B approaches A ”
The Gradient of the Tangent as a Limit