Science:Math Exam Resources/Courses/MATH103/April 2014/Question 09 (b)/Solution 1

Cobweb

The purpose of a cobweb plot is to determine graphically the sequence of points $x_{0},F(x_{0}),F(F(x_{0})),\ldots$ determined by iterating $F$ (for some initial condition $x_{0}$). Starting at $x_{0}$, our first goal is to determine $F(x_{0})$. To do so, begin by tracing a vertical line from $(x_{0},0)$ to $(x_{0},F(x_{0}))$.

Once we have found $(x_{0},F(x_{0}))$ (see the first hint), we want to determine where $F(x_{0})$ lies on the $x$-axis (so that we can repeat this procedure). To do so, we begin by finding $(F(x_{0}),F(x_{0}))$. Since the $x$- and $y$-coordinates of this point agree with each other, it lies on the line $y=x$. Thus, we sketch the line $y=x$ and determine the point on this line with height $F(x_{0})$.

To find the point on the line $y=F(x)$ of height $F(x_{0})$, take the point $(x_{0},F(x_{0}))$ found in Hint 1 and draw a horizontal line towards the line $y=x$. The intersection of these two lines is $(F(x_{0}),F(x_{0}))$.

Once we have found $(F(x_{0}),F(x_{0}))$ (see Hint 3), we can get determine the point $F(x_{0})$ on the $x$-axis simply by drawing a vertical line from this point to the $x$-axis. We can now repeat this procedure (replacing $x_{0}$ by $F(x_{0})$, $F(x_{0})$ by $F(F(x_{0}))$, etc.).