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Let us assume I have a parabola, or some kind of arbitary function.

Now, my question is: How can I geometrically construct the tangent line to a part of the function?

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Above is just an example graph. I want to know how I can construct a tangent line through A without knowing the expressions =)

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"geometrically construct" - as in ruler/straightedge? I don't believe there's a general method; try looking up the methods for constructing tangents to conics, for instance. –  J. M. Feb 8 '12 at 22:59
    
Whoops, quite the error in my previous comment; I of course intended to say "compass/straightedge". Silly me... –  J. M. Feb 8 '12 at 23:23
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1 Answer 1

The slope of the tangent line at $A = (a, f(a))$ is the derivative of the function $f$ at $a$. Approximations can be obtained by using the slopes of secant lines from $A$ to points on the curve near $A$. Better approximations can be obtained by using secant lines between points equidistant from $A$ on both sides. That is, take a small circle centred at $A$, cutting the curve at $B$ and $C$, and draw the line through $A$ parallel to the line $BC$.

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