Question

How do I draw tangent lines?

Answer 1

See explanation...

Explanation:

Given a function `f(x)` which is smooth enough in the neighbourhood of `x=a`, the tangent is a line through `(a, f(a))` which touches the graph of `f(x)` at `(a, f(a))`, but does not cross the graph within a short distance of that point.

The tangent line has the same slope as the function at that point.

What do we mean by slope of a function at a point?

If it exists, then it is the limit of the slope of lines through `(a, f(a))` and `(a+delta, f(a+delta))` as `delta->0`.

So given the graph of `f(x)` in the neighbourhood of `(a, f(a))`, you can place a ruler against `(a, f(a))` and rotate it about that point until it no longer cuts the curve of the function. Then the ruler indicates the tangent:

graph{(y-x^3+2x)(x-y-2.001) = 0 [-0.591, 1.909, -1.52, -0.27]}