| We have said that the gradient to a curve, at a particular point is given by the gradient of the tangent to the curve at that point |
| The problem is, how do we find the gradient of the tangent? |
| We could approximate the tangent by drawing a straight line from our
point, |
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| to a point a little further along the curve | |
|
| Lets say we travel
a further distance |
|
| This would increase
the y value by |
|
| |
| The gradient of this
line would be |
|
|  | |
| Although this is a reasonable approximation, it is clearly not the tangent |
| If if we
reduce | |
| our line becomes closer to the tangent |
| |
| |
|  | |
| If we
reduce | |
| further still, our line starts to become
very close to the tangent. If we continue this, the line will
eventually have the same gradient as the
tangent.
|
|  | |
| How does this help us find a formula for the gradient? |
| Consider the example of the curve |
| at the point |
| |
| Suppose that we draw a line to the point. |
| |
| Since this point if formed from the same
rule, we get |
|
|
| Which expands to |
| |
| but |
| |
| so if we subtract the equations, we get |
| |
| To find the gradient, we
divide by | |
|
| |
| In order for our line
to be tangent, we
must say |
|
| To signify this
change, we use the
notation |
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|
|
| This is the gradient function
for | |
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| | |
| A Second Example | | |
| Differentiate the function |
| |
| If we take our first point to be | | |
| then the second point must satisfy |
| |
| which expands to |
| |
| Subtracting the original equation gives |
| |
| which re-arranges to |
| |
| At this point we say that |
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| and |
| |
| which gives |
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| | |
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