* [[en:ecovirt:roteiro:math:exponencial|{{:ecovirt:logcalc.jpg?20|}}]]
* [[en:ecovirt:roteiro:math:exponencialr|{{:ecovirt:rlogo.png?30|}}]]
====== Growth rates and the exponential function - Tutorial in spreadsheets ======
{{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}}
** Let's build the plot **
- Open your spreadsheet program ((Excel or LibreOffice for example)) and follow the steps:
* write **//time//** in the first column **A**
* write a sequence form 1 to 10 to represent the time intevals
* write **//nbact//** in the first cell of the column **B**;
* write the formula //=2^A2// in the **B2** cell and press //ENTER//;
* paste the formula to the other cells below
{{ :ecovirt:roteiro:den_ind:excel02.png?500 |}}
{{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}}
** time = 1 **
- Open the spreadsheet program and follow the steps:
* write //t// in the first cell of the column (**A**) and 1 (one) in the following seven lines below in the same column;
* in the second column (**B**), write ''dt'' (delta time) in the firs cell;
* write the following numbers in the second column: 0.5,0.1,0.01,0.001,0.0001,0.00001,0.000001
* in the third column (**C**), write ''tdt'' (delta time + time);
* include the formula ''=A2+B2'' in C2 cell;
* in the fourth column **D** write ''Nt'' (population size in time t);
* include the formula ''=A2^2'' in D2 cell;
* in the fifth column (**E**) write ''Ntdt'' (population size in time t + dt);
* include the formula ''=C2^2'' in E2 cell;
* in the last column **F**, write (Ntdt-Nt)/dt and include the formula ''=(E2-D2)/B2'' in the cell **F2**;
* drag the **C2** until **F2** cells until the line 8 to copy the formulas
- Redo the same procedure to times 2 to 5, copying side-by-side the columns just modifying the time column.
- Observe what happens with the growth rates as time interval decreases.
{{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}}
- In the spreadsheet, write ''N0'' in A1 cell, and ''2'' in B1 cell.
- Write ''r'' in A2 cell and 1 in B2 cell.
- Write ''n intervals'' in the A4 cell and ''Nt'' in the B4 cell.
- From cell A5 to A24, write: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 50, 100, 200, 500, 1000, 10000, 100000, 1000000;
- In cell B5, write the formula ''=$B$1*(1+$B$2/A5)^A5''
- In cell C4, write "Nt/N0", and in cell C5 write the formula ''=B5/$B$1''
- Copy the formulas in cells B5 and C5 and paste it until line 24;
- In cell D1, write''e^r'', and write ''=exp(B2)'' in cell E1.
{{:ecovirt:roteiro:math:planilha_exponencial.png?700 |}}
{{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}}