The tangent line is the best straight-line copy of a curve at a point — it
gets the tilt right. It cannot bend, so it is wrong the moment the
curve does. Add one more term and you get the best parabola, which
gets the tilt and the bend. That extra term is where
f″ lives.
Slide the point. Watch the red line keep the tilt and
the green parabola keep the bend as well.
Try x^3-3x, sin(x), x^2,
e^x, ln(x), 1/x.
What's left over is — the
curve with its tangent line subtracted away, so only the bending remains.
It touches zero at the point and opens upward where f″ > 0,
downward where f″ < 0. Mind the vertical scale: this panel
has its own, and it is usually far finer than the one above.
f(x) tangent — the tilt best parabola — tilt and bend