How do you explain what a Rate of Change is?

>>16733830 (OP)

If I were teaching the idea (of a derivative) to a group of people/kids for the very first time, I would start with graphs of elementary functions, and very minimal numerical information (positive, negative, zero), focusing on the more squishy, qualitative question: "okay is curve/line going up or down here, and does it look like it's going up or down faster, etc", to draw the derivative curves. And we'd spend most of the very first class just doing that, talking through what the various curves are doing. Store that, then start to (re-)introduce the standard function stuff on Day 2, then call back to the Day 1 visual exercises and saved drawings/pictures from day 1 as appropriate.

>>16733830 (OP)
Amount by which a quantity changes with respect to a given change in another quantity.

>>16733830 (OP)
in the first year 2 more people entered your tribe
in the second year, 2 more people entered your tribe
in the third year, 2 more people entered your tribe
every year, 2 more people enter your tribe
the change in your tribe is the same every year
this is a constant rate of change

contrast with
in the first year, 2 people entered your tribe
in the second year, 4 people entered your tribe
in the third year, 6 people entered your tribe
the amount of people entering your tribe every year is increasing
this is a positive rate of change

in the first year 2 people joined your tribe
in the second year, 1 person joined your tribe
in the third year, nobody joined or entered your tribe
in the fourth year, 1 person left your tribe
in the fifth year, 2 people left your tribe
every year, fewer people join your tribe or more people leave your tribe
this is a negative rate of change

>>16733838
God you are either so fucking stupid or you completely misinterpret due to heavy autism what questions like the OP are actually asking about.
>muh insisting on intuitive explanations
Intuitive explanations don't exist. What exists are intuitive *parts* of explanations. Which you didn't include.

My IP range is not allowed to upload images for some dumb reason. So my immaculate MSPaint skills will sadly go unrecognized,

Imagine you're driving at a constant 30mph. If you graph this as a function of speed over time, you just get a horizontal line that corresponds to the function: y=30 where y is speed and x is time (because the speed is invariant, x does not show up in the equation)

To find the distance travelled, you multiply the speed travelled at by the time travelled for. This can be thought of as calculating the rectangular area underneath the function previously described.

This provides a new function: y=30x where y is the distance from the starting point and x is still time.
We see that the constant in the earlier graph (speed) became the coefficient of x in the distance graph.
If the speed was higher in the first graph, the slope would be steeper in the second graph. The inverse is obviously true as well. So speed is the rate that the distance changes.

Put more simply: the faster you're going, the further you're going to travel in a given amount of time.

>>16733830 (OP)
A rate is a frequency, a cyclic thing. 1 per second, 1 per minute, etc.
Of change, means that you are measuring a change in something (your measured value) often with dimensions like meters.
Your rate of change in a function plotted between x being the independent variable and y being the dependant, is analogous to measuring something dependent on point in time.
I went from y = y_1 at x = x_1 to y = y_2 at x = x_2
If we want to give it interesting dimensions think about an airplane's ascent rate.
How much did the airplane ascend above ground level between t = 1 sec to t =10 sec?
You can show a graph, and they can find the slope of the line.

[math] \displaystyle

\\ \text{Continuous compounding}
\\ \displaystyle P(t)=P_{0} \, e^{rt}
\\ \text{ } P_0 \; \, \text{initial value}
\\ \text{ } r \quad \text{rate of growth}
\\ \text{ } t \quad \text{time}
\\ ~~~~----- \\
P(t_2) = 2P_0 \Rightarrow 2P_0 = P_{0} \, e^{rt_2} \\
2 = e^{rt_2} \\
e^{ln(2)} = e^{rt_2} \\
ln(2) = rt_2 \\
t_2 = \frac{ln(2)}{r} \approx \frac{70\%}{100r\%} \\
\\
P(t_{10}) = 10P_0 \Rightarrow 10P_0 = P_{0} \, e^{rt_{10}} \\
10 = e^{rt_{10}} \\
e^{ln(10)} = e^{rt_{10}} \\
ln(10) = rt_{10} \\
t_{10} = \frac{ln(10)}{r} \approx \frac{230\%}{100r\%}

[/math]

>>16733845
Tpbp

>>16733840
retard award goes to you

>>16733830 (OP)
just use elevation as a function of distance (like a graph for a hiking trail), then rate of change in elevation at a given point of distance is the steepness of terrain at that point