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Презентация на тему по математике Коэффициент наклона прямой, 10 класс

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
Made by ALEXP1 Chapter 6.2«The Gradient of the Tangent as a Limit» We found the rule for differentiating by noticing a pattern in results e.g. we can use the chord to the point ( 2, 4 The gradient of the chord AB1 is given by B1A(1,1)  (2,4) The gradient of the chord AB2 is We need to zoom in to the curve to see more clearly. The gradient of AB3 is Continuing in this way, moving B closer and closer to A( 1, As B gets closer to A, the gradient of the chord AB We need a general notation for the coordinates of B that suggests So, the gradient of the chord AB is m where But, the gradient of the tangent gives the gradient of the curve, so Solution: The gradient of the curve is given by the gradient of the tangent, so
Слайды презентации

Слайд 2 We found the rule for differentiating by noticing

We found the rule for differentiating by noticing a pattern in

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.


Слайд 3 e.g. we can use the chord to the

e.g. we can use the chord to the point ( 2,

point ( 2, 4 ).
( We are going to

use several points, so we’ll call this point B1 ).

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.


Слайд 4 The gradient of the chord AB1 is given

The gradient of the chord AB1 is given by B1A(1,1) (2,4)

by
B1
A(1,1)
(2,4)
Tangent at A
We can see

this gradient is larger than the gradient of the tangent.

Chord AB1



Слайд 5 The gradient of the chord AB2 is

The gradient of the chord AB2 is

Слайд 6 We need to zoom in to the curve

We need to zoom in to the curve to see more clearly.

to see more clearly.


Слайд 7 The gradient of AB3 is

The gradient of AB3 is

Слайд 8 Continuing in this way, moving B closer and

Continuing in this way, moving B closer and closer to A(

closer to A( 1, 1 ), and collecting the

results in a table, we get

As B gets closer to A, the gradient approaches 2. This is the gradient of the tangent at A.


Слайд 9 As B gets closer to A, the gradient

As B gets closer to A, the gradient of the chord

of the chord AB approaches the gradient of the

tangent.

Слайд 11 We need a general notation for the coordinates

We need a general notation for the coordinates of B that

of B that suggests it is near to A.


Слайд 12 So, the gradient of the chord AB is

So, the gradient of the chord AB is m where

m where


Слайд 13
But, the gradient of the tangent gives the

But, the gradient of the tangent gives the gradient of the curve, so

gradient of the curve, so


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