Question

Can you determine the equation of the tangent line of a point on a function by solely using the function's graph?

Answer 1

I would say, that we cannot "determine", but we can estimate.

Explanation:

Unlike a function given by an expression (or expressions), the graph of a function can never be real with 100% accuracy.

The best we can ever do is estimate.

If we are only estimating the function, we can only estimate the tangent lines.

For example, here is the graph of a function:

graph{.9995x^2 [-10, 10, -5, 5]}

And supppose we are asked for the equation of the tangent line at the point where `x = 2`.

It is unlikely that anyone will correctly see that the point on the graph has coordinates `(2.3.998)` and the tangent line has equation `y=3.998x-3.998`

We could instead concoct an example in which there is a tiny cusp at `(2,4)`. In this case there would be no tangent line at `(2,4)`

`f(x) = {(x^2,"if",x <= 2),(0.995x^2+0.002,"if",x > 0):}`