In the previous problem, you drew horizontal and vertical lines all the way to the axes so that you could (1) read off the new value of the state variable (from the vertical $x_{n+1}$ axis), and then (2) start off using that new value in the next step (by matching the new value with the horizontal $x_n$ axis).
We can take a shortcut by observing that the horizontal line from step one (at vertical position $x_1$) and the vertical line from step two (at horizontal position $x_1$) cross exactly at the diagonal line. (Online, you can see light gray lines that extend each line and show the intersection.) Exploiting this fact, we can, after moving vertically to the graph of $f$ to find the output value, draw a horizontal line to the diagonal rather than to the vertical axis. The point on the diagonal gives our next horizontal position, so we can simply move vertically to the graph of $f$ again to find the next value of $x$.

This procedure works because moving horizontally to the diagonal makes your horizontal position match your vertical position, which was the output value of the previous step.

Verify that this procedure works for calculating $x_1$, $x_2$, and $x_3$, starting with the initial condition $x_0=1$. You should get the same answers as you did in the previous problem. In fact, each time you move horizontally to the diagonal, you should reach the same vertical line that you used in the previous problem.

If all went well, you should get the same estimates for the values of $x$:

$x_1 =$

, $x_2 = $

, and $x_3=$

.

##### Hint

Online, drag the

*step* slider to 1, which will let you set the initial condition $x_0$ (black point). Then, change

*step* to 2, which will freeze the initial condition point and reveal a blue point which you can move to calculate $f$ evaluated at the initial condition. Move the blue point so it is straight above the initial condition on the graph of $f$. The point's vertical position becomes an estimate of $x_1=f(x_0)$.

Change *step* to 3 to translate the value of $x_1$ to the horizontal position. The previous blue point freezes and a new black point appears, which you can move horizontally to the diagonal. The coordinates of this black point will then be $(x_1,x_1)$. Both its vertical and horizontal position represent $x_1$.

Change *step* to 4 to estimate $x_2=f(x_1)$. The black point freezes and a new blue point appears. If you move the blue point vertically from the black point to the graph of $f$, its second coordinate becomes $x_2=f(x_1)$.

Change *step* to 5 and then 6 to repeat this process to calculate $x_3=f(x_2)$.

Remember: to estimate the next iteration, move horizontally to the diagonal (using a black point) then move vertically to the graph of $f$ (using a blue point).

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