I am trying to understand what the change in angle of the slope of a curve means. It is hard to explain with words so here's an image that should help.
The red curve has had its derivative approximated at three points. The tangent at each point is also shown in black. These slopes can be compared to one another, and using
tan(z)=(m1+m2)/(1+m1*m2) we can calculate the angle between the slope at one point and the next.
As shown in the image, the angle become smaller at the curve straightens out. What I'm trying to figure out is what this change in angle represents. My first instinct is that it can be thought of as the rate of change of the slope, therefore the second derivative should be analogous to the angle change. However the angles are changing due to the curvature of the curve, so I also feel like this is a sort of rate of change of curvature. If the curve was a circle, the angle would always be the same, therefore the change in curvature would be zero.
Which, if any, of these two ways makes more mathematical sense?
Thanks for the help.
Best Answer
One definition of curvature is the rate of change of the angle with respect to the length of the curve. However, you seem to be discussing the rate of change of the angle with respect to the $x$ coordinate. Acceleration uses the $x$ coordinate but measures the rate of change of the slope, not the angle.
You seem to be discussing something new, neither curvature nor acceleration, though curvature seems to be closer due to your discussion of the circle. If the function is $f(x)$ then you seem to be discussing
$$\frac{d}{dx}\left(\tan^{-1}\left(\frac{df}{dx}\right)\right)$$
since the angle of the slope of the curve is the arctangent of the derivative. Of course, you can use the chain rule to simplify this to
$$\frac{\frac{d^2f}{dx^2}}{1+\left(\frac{df}{dx}\right)^2}$$