Graphical Intro to
Derivatives and Integrals
Derivatives and Integrals have a two-way relationship!
Let's start by looking at sums and slopes:
Example: walking in a straight line
- Walk slow, the distance increases slowly
- Walk fast, the distance increases fast
- Stand still and the distance won't change
A distance increase of 4 km in 1 hour gives a speed of 4 km per hour
Or, walking at 4 km per hour for 1 hour increases the distance by 4 km
Speed is the rate of change of distance
Change in distance is the sum of the speed over time
It will make more sense when you play with it below: change the distance line, or the speed line, to see its affect on the other:
Play with that a little and get comfortable with the two-way relationship. Try zero speed, or negative speed.
The slope of the distance line gives us the speed line, like this:
The "area" under the speed line gives us the increase in distance, like this:
Many things have that same two-way relationship:
- Wealth and income
- Volume and flow rate
- Energy and power
- lots more!
Here is the same app as above, but you can choose different topics:
Integrals and Derivatives also have that two-way relationship!
Try it below, but first note:
- Δx (the gap between x values) only gives an approximate answer
- dx (when Δx approaches zero) gives the actual derivative and integral*
*Note: this is a computer model and actually uses a very small Δx to simulate dx, and can make erors.
For true derivatives refer to Derivative Rules, and for integrals refer to Introduction to Integration