Growth rates and the exponential function

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====== Growth rates and the exponential function - Tutorial in R======


{{section>en:ecovirt:roteiro:math:exponencial_base#growth_rates_and_the_exponential_function}}

===== From discrete to continuous time =====

{{section>en:ecovirt:roteiro:math:exponencial_base#from_discrete_to_continuous_time}}

== If you want to reproduce the plot ==

Open R and paste the code:

<code>
time=1:10
nbact= c(2,4,8,16,32,64,128,256, 512,1024)
plot(time, nbact, main="Bacterial growth", ylab= "number of bacteria", pch=16, col="red",cex.main=1.5, cex=1.5,cex.lab=1.5, cex.axis=1.5,bty="l")
</code>

{{section>en:ecovirt:roteiro:math:exponencial_base#continuacao_tempo_discreto}}

==== Intuitive notion for derivatives ====

{{section>en:ecovirt:roteiro:math:exponencial_base#Intuitive_notion_for_derivatives}}

<code>
## t=1
time=1
##let's create a vector with the delta t values decreasing
dt=c(0.5,0.1,0.01,0.001,0.0001,0.00001,0.000001)
tdt=time+dt
Nt=time^2 
Ntdt=tdt^2
(Ntdt-Nt)/dt
</code>


{{section>en:ecovirt:roteiro:math:exponencial_base#continua_nocao_de_derivada}}

===Definition of a derivative===

{{section>en:ecovirt:roteiro:math:exponencial_base#definicao_de_uma_derivada}}

===== Exponential growth =====

{{section>en:ecovirt:roteiro:math:exponencial_base#crescimento_exponencial}}

==== The exponential function ====


{{section>en:ecovirt:roteiro:math:a_funcao_exponencial}}


<code>
##########################
#### Continous growth ####
##########################

n<- c(0:100, 200, 500,1000, 10000, 100000,1e+10)
N0 <- 2
rd1 <- 1  
N1<-N0* (1+ rd1/n)^n
N1_N0= N1[length(N1)]/N0
plot(1:103, N1[1:103]/N0, type="l")
text(x=50, y=2.5, labels= paste("Maximum = ", N1[length(N1)/N0]))
N1_N0
</code>

{{section>en:ecovirt:roteiro:math:continua_funcao_exponencial}}


==== Duplication time ====

{{section>en:ecovirt:roteiro:math:exponencial_base#tempo_de_duplicacao}}


===== Exercises =====

{{section>en:ecovirt:roteiro:math:exponencial_base#exercises}}

===== To know more =====
 {{section>en:ecovirt:roteiro:math:exponencial_base#to_know_more}}


{{tag>R maxima uma_população crescimento_exponencial tempo_discreto tempo_contínuo equação_diferencial}}