32 lines
1.3 KiB
Plaintext
32 lines
1.3 KiB
Plaintext
# Exponential Graphs
|
|
=> differenciation.gmi You should understand differentiation before reading this section
|
|
Exponential graphs are those in the form of y = k^(x)
|
|
they should look like this:
|
|
y
|
|
|
|
|
| |
|
|
| |
|
|
| |
|
|
| |
|
|
| |
|
|
| |
|
|
| _-
|
|
| ___--
|
|
|---------___________ x
|
|
As you can see, the line curves upwards, increasing in gradient costantly.
|
|
It's also interesting to note that all exponential graphs (without a specified y intercept), intersect y at 1 as n^0 === 1
|
|
If you draw the gradient graph of an exponential, you get another exponential graph, they are usually slightly different to the original graph,
|
|
for instance, the gradient graph of 2^x is slightly shallower than 2^x
|
|
and the gradient graph of 3^x is slightly steeper than 3^x,
|
|
As you can see, the gradient graph and initial graph must cross at a certain value between 2 and 3
|
|
that value is about 2.718, the gradient graph 2.718^x is exactly the same as 2.718^x, we call this value e.
|
|
Similar to pi, the number e actually goes on forever, I just rounded it to 2.718, the first 10 digits are 2.718281828.
|
|
So e is a special number because if you differenciate it, you get the same value:
|
|
```
|
|
y = e^x
|
|
dy/dx = e^x
|
|
|
|
y = e^kx
|
|
dy/dx = ke^kx
|
|
```
|