# Math Insight

### Discrete exponential growth and decay exercises

These exercises allow you to practice how to solve discrete dynamical systems involving exponential growth and decay.

#### Exercise 1

Write a solution equation for the following discrete dynamical systems. In each case, compute the solution at time $t=100$. \begin{align*} {\rm{a.}} \hspace{2mm} x_0 & = 1,000 & x_{t+1} - x_t & = 0.2 \times x_t \\ {\rm{b.}} \hspace{2mm} x_0 & = 138 & x_{t+1} - x_t & = 0.05 \times x_t \\ {\rm{c.}} \hspace{2mm} B_0 & = 138 & B_{t+1} - B_t & = 0.5 \times B_t \\ {\rm{d.}} \hspace{2mm} y_0 & = 1,000 & y_{t+1} - y_t & = -0.2 \times y_t \\ {\rm{e.}} \hspace{2mm} P_0 & = 1,000 & P_{t+1} & = 1.2 \times P_t \\ {\rm{f.}} \hspace{2mm} z_0 & = 1000 & z_{t+1} - z_t & = -0.1 \times z_t \\ {\rm{g.}} \hspace{2mm} c_0 & = 1,000 & c_{t+1} & = 0.9 \times c_t \end{align*}

#### Exercise 2

The equation, $x_{t} - x_{t-1} = r x_{t-1}$, carries the same information as $x_{t+1} - x_{t} = r x_{t}$.

1. Write the first four instances of $x_{t} - x_{t-1} = r x_{t-1}$ using $t=1$, $t=2$, $t=3$, and $t=4$.
2. Obtain an expression for $x_4$ in terms of $r$ and $x_0$.
3. Write solutions to and compute $x_{40}$ for \begin{align*} {\rm{(i.)}} \hspace{2mm} x_0 & = 50 \hspace{5mm} & x_{t} - x_{t-1} & = 0.2 \times x_{t-1} \\ {\rm{(ii.)}} \hspace{2mm} x_0 & = 50 & x_{t} - x_{t-1} &= 0.1 \times x_{t-1}\\ {\rm{(iii.)}} \hspace{2mm} x_0 & = 50 & x_{t} - x_{t-1} &= 0.05 \times x_{t-1}\\ {\rm{(iv.)}} \hspace{2mm} x_0 & = 50 & x_{t} - x_{t-1} &= -0.1 \times x_{t-1} \end{align*}

#### Exercise 3

Suppose a population is initially of size 1,000,000 and grows at the rate of 2% per year. What will be the size of the population after 50 years?

#### Exercise 4

The polymerase chain reaction is a means of making multiple copies of a DNA segment from only a minute amount of original DNA. The procedure consists of a sequence of multiple cycles. During the course of one cycle, each DNA segment present is duplicated. Introduce notation and write a discrete dynamical system with initial condition from which the amount of DNA present at the end of each cycle can be computed. Suppose you begin with 1 picogram = 0.000000000001 g of DNA. How many grams of DNA would be present after 30 cycles.

#### Exercise 5

Write a solution to the dynamic system you obtained for growth of V. natriegens in growth medium of pH 7.85 in exericse 5 from the bacteria growth exercises. Use your solution to compute an estimate of $B_4$.

#### Exercise 6

There is a suggestion that the world human population is growing exponentially. Shown below are the human population numbers in billions of people for the decades 1940 - 2010.

World Human Population
YearIndex, $t$Human population (billons)
194002.30
195012.52
196023.02
197033.70
198044.45
199055.30
200066.06
201076.80
1. Test the equation $$P_t = 2.2 \times 1.19^t$$ against the data where $t$ is the time index in decades after 1940 and $P_t$ is the human population in billions.
2. What percentage increase in human population each decade does the model for the equation assume?
3. What world human population does the equation predict for the year 2050?