Calculate $y_{1} =$ $\times y_0 =$ $y_{2} =$ $\times y_1 =$ $\times y_0 =$ $y_{3} =$ $\times y_2 =$ $\times y_0 =$
This means that to calculate $y_{3}$, you start with $y_0$ and multiply by how many times?
Continuing the pattern $n$ times, to calculate $y_n$, you start with $y_0$ and multiply by how many times?
The formula for $y_n$ is therefore: $y_n =$ $\times y_0 =$ (When answering online, you can use ^ for exponentiation, so enter $a^b$ as a^b.)
Calculate $y_{17}=$ (Keep at least 6 significant digits in your response. Online, you can enter a number like $1.2352 \times 10^{-5}$ as either 1.2352*10^-5 or 1.2352E-5.)
Calculate $z_{1} =$ $z_{2} =$ $z_{3} =$
Continuing the pattern, calculate: $z_t =$ (When answering online, you can use ^ for exponentiation, so enter $a^b$ as a^b.)
Calculate $z_{33}=$
If we start with the tiny initial condition $P=7.4 \cdot 10^{-9}$, calculate $z_{47} =$ (Keep at least 6 significant digits in your response.)
$u_n =$ $u_5 =$ $u_{15} =$ $u_{25} =$
Consider the dynamical system where the state variable $y$ decreases by 60% each time step: \begin{align*} y_{ t+1} - y_t &= -0.6 y_t\\ y_0 &= 19. \end{align*} Each time step, the change in $y$, i.e., $y_{ t+1} - y_t$, is -60% of the old value of $y$.
Converting to function iteration form is easy, we just need to add $y_t$ to both sides of the equation so we get $y_{ t+1}$ alone on the left side of the equation. The evolution rule for the dynamical system becomes
The function iteration form makes it clear that after each time step, the new value of $y$ is % of the previous value of $y$. In other words, at each time step, we multiply by .
The result is a little uglier than the previous problems, but not bad.
The dynamical system would look a lot prettier in function iteration form if we could write is as \begin{align*} y_{ t+1} &= K y_t\\ y_0 &= L. \end{align*} If we wanted to transform our system into this prettier form, what must we make $K$? (It must depend on our original parameter $k$.)
$K = $
With this definition of $K$, what is the solution to the dynamical system? (Your answer shouldn't have $k$ in it, only $K$.)
$y_t = $
Calculate $m_{1} =$ $m_{2} =$ $m_{3} =$
Continuing the pattern, calculate: $m_t =$
If $c=3.2$, calculate $m_{70}=$ (Keep at least 6 significant digits in your response.)
Write the dynamical system both in difference form as
The function iteration result is easy to understand because the number of cells in one day is times the number from the previous day. To understand the difference form result, you need to deduce that for any given day, the change in the number of cells from the previous day to that day is equal to times the number from the previous day.
One tricky part of setting up the dynamical system is deciding whether it is easier to start with the difference form (such as $c_{t+1}-c_t =$ something) or the function iteration form (such as $c_{t+1}=$ something). If you choose to start with the difference form, then on the right hand side, you need to put the change in the number of cells. (And, if the number of cells is doubling, the change is not twice the previous number of cells.) If you choose to start with the function iteration form, then on the right hand side, you need to put an expression that gives the total number of cells after a day. (And, if the number of cells is doubling, the total number is indeed twice the previous number of cells.)
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$c_t = $
Since the increase in the number of cells is proportional to the number of the previous day (i.e., we have a linear dynamical system), we will get exponential growth. In each day, we multiply the number of cells by what number?
How does this number compare to the number of atoms in the observable universe, which is estimated to be about $10^{80}$? The number is about 1,000 10,000 100 100,000 10 times smaller larger than the number of atoms in the observable universe.
Do you believe the results of the model? If not, what must be true of the rate of cell division? Can the cells continue to divide into two each day for the whole course of the pregnancy? The model is wrong. The rate of cell division must speed up dramatically The model is fine. The rate of cell division can continue at the same speed. The model is wrong. The rate of cell division must slow down a little bit. The model is wrong. The rate of cell division must slow down dramatically. The model is fine. Scientists must have underestimated the number of atoms in the universe. The model is fine. These calculations are irrelevant. The model is wrong. The rate of cell division must speed up a little bit. The model is fine. Humans are just larger than we realize when being born.
Since you solved the system in the previous step, you just need to plug in the number of days of a pregnancy into the solution formula. If the number is ridiculously large, then what can you conclude about the rate of cell division? It is at all possible that this model is correct, i.e., that the rate of cell division continues at a constant rate of one division per day for the whole pregnancy? If the model is clearly bad, then what must be happening to the rate of cell division?