Science:Math Exam Resources/Courses/MATH103/April 2015/Question 06 (c)

MATH103 April 2015

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Question 06 (c)
Consider the function $F(x)=\lambda xe^{-x}$, where $\lambda >0$ is a parameter. Now consider the iterated map $x_{t+1}=F(x_{t})$ for $t=0,1,2,3,\cdots$. (c) Starting at $x_{0}$, use cobwebbing to draw $x_{t}$ up to $x_{5}$ in the two graphs below, which show $F(x)$ for two different values of $\lambda$. graphs

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Note that the dash line on each graph is the graph of $y=x$.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. If we use cobwebbing, we obtain the following graphs and the points $x_{1},\cdots ,x_{5}$: cobweb cobweb The process is as follows. By drawing a vertical line from $(x_{0},0)$, we obtain the point $(x_{0},F(x_{0}))$. Then, drawing a horizontal line from the point, we find the intersection point with $y=x$ graph, which is $(x_{1},x_{1})=(F(x_{0}),F(x_{0}))$. Projection of this point on $x-$axis gives the point $(x_{1},0)$. Repeating this process, we get $x_{1},\cdots ,x_{5}$.