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 Mathematics - Functions 4
In order to find the tangent to the graph of a function, this graph must, of course, be a curve. The simplest function rule, which had a curve to the graph, was in the chapter Functions 1
In addition to f(x) = x2, which starts from zero, we have added a constant three times in the diagram so that the next higher curve is generated:
f(x) = x2 + 1
f(x) = x2 + 2
f(x) = x2 + 3 |
No matter which of the three we use for our first calculation of the slope of a tangent, e.g. at the point 1/1, 1/2, 1/3 or 1/4, the constant has no effect on it. This also proves the calculation of the first derivative of one of
these functions:
f(x)
| = x2 + 1
|
f'(x)
| = (x + Δx)2 + 1 - (x2 + 1)
|
| = x2 + 2 Δx · x + 1 -x2 - 1
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| 2Δx · x + Δx2 |
| = |  |
| Δx |
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| = 2x +Δx
|
And as Δx is to become zero, the first derivative of f (x) = x2 + 1:
If we insert for x 1, the straight line 1 with the slope 2 touches the graph of the function at point 2/1, we set for x 2, touches the degrees 2 with the slope 4 at the point 1/5.
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