Unit 6: Growth in proportion to size, and the number e
About this unit
An exponential grows at a rate proportional to its own size, and e is the base for which that proportion is exactly 1, so e^x is its own derivative.
By the end, you can: See why the slope of 2^x is 2^x times a fixed constant, why e is the base that makes that constant exactly 1, and differentiate e^(kx) and a^x.
Modules
5 modulesLearn
- Growth that feeds on itselfArticle · A doubling culture's slope is its size times a fixed constant: (2^h − 1)/h settles on about 0.693, so d/dx 2^x ≈ 0.693 · 2^x.Not started
- The base that makes the constant 1Article · The constant grows with the base; at e ≈ 2.718 it is exactly 1, so (e^x)′ = e^x, (e^(kx))′ = k e^(kx) and a^x = e^(x ln a).Not started
- Slope equals heightExperiment · Drag along e^x until its slope is 5 and see that the height there is 5 too.Not started
- Exponentials in one pageReview · Rate proportional to size; the constant ln a; e makes it 1; the k in e^(kx) comes out in front.Not started