Question

1. Consider the difference equation $x_{t+1} = rx_t(1 - x_t)$. a) Show that the equilibria are $x = 0$ and $x = \frac{r-1}{r}$. b) Use the theorem from class to determine the values of $r$ for which $x = 0$ is a stable equilibrium. c) Use the theorem from class to determine the values of $r$ for which $x = \frac{r-1}{r}$ is a stable equilibrium. d) Suppose that $x_0 = 2$, $r = 0.8$. Based on your responses in parts (b) and (c), what do you expect $x_t$ to approach as $t$ gets large? Explain your reasoning. e) Let $x_0 = 2$, $r = 0.8$, use Matlab to plot $x_t$ for $0 \le t \le 50$. Make a rough sketch of the plot (or paste your plot in the space below), and specify what $x_t$ approaches as $t$ gets large. f) Suppose that $x_0 = 2$, $r = 2.5$. Based on your responses in parts (b) and (c), what do you expect $x_t$ to approach as $t$ gets large? Explain your reasoning. g) Let $x_0 = 2$, $r = 2.5$. Use Matlab to plot $x_t$ for $0 \le t \le 50$. Make a rough sketch of the plot (or paste your Matlab plot) in the space below, and specify what $x_t$ approaches as $t$ gets large.

          1. Consider the difference equation $x_{t+1} = rx_t(1 - x_t)$. 
a) Show that the equilibria are $x = 0$ and $x = \frac{r-1}{r}$.
b) Use the theorem from class to determine the values of $r$ for which $x = 0$ is a stable equilibrium.
c) Use the theorem from class to determine the values of $r$ for which $x = \frac{r-1}{r}$ is a stable equilibrium.
d) Suppose that $x_0 = 2$, $r = 0.8$. Based on your responses in parts (b) and (c), what do you expect $x_t$ to approach as $t$ gets large? Explain your reasoning.
e) Let $x_0 = 2$, $r = 0.8$, use Matlab to plot $x_t$ for $0 \le t \le 50$. Make a rough sketch of the plot (or paste your plot in the space below), and specify what $x_t$ approaches as $t$ gets large.
f) Suppose that $x_0 = 2$, $r = 2.5$. Based on your responses in parts (b) and (c), what do you expect $x_t$ to approach as $t$ gets large? Explain your reasoning.
g) Let $x_0 = 2$, $r = 2.5$. Use Matlab to plot $x_t$ for $0 \le t \le 50$. Make a rough sketch of the plot (or paste your Matlab plot) in the space below, and specify what $x_t$ approaches as $t$ gets large.

1. Consider the difference equation xt+1 = rxt(1 - xt).
a) Show that the equilibria are x = 0 and x = (r-1)/(r).
b) Use the theorem from class to determine the values of r for which x = 0 is a stable equilibrium.
c) Use the theorem from class to determine the values of r for which x = (r-1)/(r) is a stable equilibrium.
d) Suppose that x0 = 2, r = 0.8. Based on your responses in parts (b) and (c), what do you expect xt to approach as t gets large? Explain your reasoning.
e) Let x0 = 2, r = 0.8, use Matlab to plot xt for 0 ≤ t ≤ 50. Make a rough sketch of the plot (or paste your plot in the space below), and specify what xt approaches as t gets large.
f) Suppose that x0 = 2, r = 2.5. Based on your responses in parts (b) and (c), what do you expect xt to approach as t gets large? Explain your reasoning.
g) Let x0 = 2, r = 2.5. Use Matlab to plot xt for 0 ≤ t ≤ 50. Make a rough sketch of the plot (or paste your Matlab plot) in the space below, and specify what xt approaches as t gets large.

Added by Sarah H.

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