We study gradient fields when studying functions of more than one variable, but never reduce this to the 1D level or relate these to slope fields. This applet tries to do this.
Answers for the given function: f(x)=x+2sin(x).
* Maximums - The vectors point towards each other at x≈2.5, x≈8.5, ...
* Minimums - The vectors point away from each other at x≈4.5, x≈10.5, ...
The slider v=Vectorscale is so that the vectors do not overlap, i.e. you can increase it if change is slow, decrease it if change is fast. When v=1, the magnitude of the vectors drawn is their actual magnitude; otherwise divide by v to get actual magnitude.
list1 is the list of the vectors of the 1D gradient field. At each point x, it calculates fDer(x) and then draws a vector on the x-axis, centered at (x,0) with magnitude fDer*v with direction determined by the sign of fDer(x). (Actually list1 is segments and list1a and list1b are arrow segments.)
list2 is the list of points (x,f(x))
list3 is the list of segments of length v centered at the function points whose slope is fDer, i.e. the magnitude of the vectors of the 1D gradient field.
