# 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