How to interpret $y = f(a) + f'(a)(x-a) + f''(a)(x-a)^2$

Question

As you may know we can define the equation of a tangent line of a differentiable function at any point $a$ is given by: $$y = f(a) + f'(a)(x-a)$$

However how can I interpret this equation? $$y = f(a) + f'(a)(x-a) + f''(a)(x-a)^2$$

This would be very useful to me. Looks like a Taylor expansion at the point $a$. However I can't see this geometrically.

If this doesn't have an answer, is there any geometric meaning to the third derivative of a function?

The first derivative, is the slope of the graph at the point. The second derivative describes "how fast the slope is changing", or "the rate of change" of the graph at that point. This is indeed the Taylor expansion, just cut short, and so it is, in rough terms, the best polynomial which approximates the function around $a$. — Norse, Dec 16 '19 at 18:07
This represents the local behavior of the function. The derivative is a linearization of a differentiable function at a point. Adding successive terms, in theory, makes for a better approximation. — Cameron L. Williams, Dec 16 '19 at 18:08
If, as Mohammad suggests, you mean $$y=f(a)+f'(a)(x-a)+\tfrac12f''(a)(x-a)^2$$ then this is the second-degree polynomial that matches $f(x)$ in value, first derivative, and second derivative at $x=a$. Geometrically, it's the equation of a parabola that matches the graph of $y=f(x)$ in position, slope, and curvature at $(a,f(a))$. (Note: Curvature isn't given by the second derivative alone, but is calculated from $f'$ and $f''$.) — Blue, Dec 16 '19 at 18:48
By the way, the meaning of the third derivative is discussed in this question. A site search for "third derivative geometrically" yields a number of additional results. — Blue, Dec 16 '19 at 18:55

score 4 · Answer 1 · answered Dec 16 '19 at 18:13

You mean $$y = f(a) + f'(a)(x-a) + \frac {f''(a)}{2}(x-a)^2$$

which is a quadratic approximation to the function around the point $(a,f(a))$ instead of the linear approximation which is the tangent line.

This is a better approximation due to the second derive at the point $ (a,f(a))$ which involves the concavity of the graph as well as the slope at the point of tangency.

score 1 · Answer 2 · answered Dec 16 '19 at 18:50

The equation $y = f(a) + f'(a) (x-a)$ represents the only 1st degree polynomial function that has the same value and the same first derivative at $x=a$ as the function $f$. Geometrically, this is the equation of the unique line that is the best fit or best approximation to the graph of $y=f(x)$ at the point $(a,f(a))$.

The equation $y = f(a) + f'(a) (x-a) + \frac{f''(a)}{2} (x-a)^2$ represents the only 2nd degree polynomial function (quadratic function) that has the same value, the same first derivative, and the same second derivative at $x=a$ as the function $f$. Geometrically, this is the equation of the unique parabola that is the best fit, or best approximation, to the graph of $y=f(x)$ at the point $(a,f(a))$.

Want to guess what the equation $y = f(a) + f'(a) (x-a) + \frac{f''(a)}{2} (x-a)^2 + \frac{f'''(a)}{6}(x-a)^3$ represents?

How to interpret $y = f(a) + f'(a)(x-a) + f''(a)(x-a)^2$

2 Answers2