h
0
 h |
Start with a point (a, f(a)) on the graph of a function f. If the curve is smooth at that point -- that is, if f'(a) exists -- then we have
| f'(a) | = | lim h 0 |
|
This means that the smaller h becomes, the closer (f(a+h) - f(a))/h approximates f'(a). Thus, for small values of h (close to zero),
| f'(a) |  |
h |
Multiplying both sides by h and solving for f(a+h) gives
f(a+h) - f(a),
f(a) + hf'(a).
Now, since h is small, a+h is a number close to a. It is more useful for our purposes to call this number x, so x = a+h. This also gives us h = x-a. Substituting gives us the approximation
f(a) + (x-a)f'(a).
The function given by the right-hand side,
is the desired linear approximation (its graph is the tangent line to the curve at (a, f(a)).
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