✎ Growth rates and the exponential function

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====== 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}}


<box 80% red | Exercises>
**  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
</box>

{{  :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}}


<box 80% red | Decreasing $\Delta t$ >
**  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.
</box>


{{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}}

<box 80% red | Exercises>
    - 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.
</box>



{{: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}}
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