Math 214
Homework for Section 2.10
Spring 2016
This problem set is out of 55 points. It is due on Thursday, March 24. It is optional–
it will only count if it raises your overall homework grade.
Problem 1. (7 points) Show using only the definitions that any contraction on Rn is continuous.
Problem 2. (5 points) Sketch a graph of a differentiable function F : R → R whose
derivative is continuous everywhere and such that 0 < F 0 (x) < 1 for all x ∈ R, but where
the graph of y = F (x) never intersects the graph of y = x. (Note that you are not being
asked to find a formula for F . Just sketch a graph.) Explain why this shows that the r < 1
condition is necessary in the statement of the contraction mapping theorem.
−2 1
1
1
2
2
~x +
.
Problem 3. Let F : R → R be given by F (~x) = 10
1 −2
0
1. (5 points) Solve F (~x) = ~x to find the correct fixed point of F in R2 and check that the
fixed point is unique.
2. (5 points) Show that F is a contraction on R2 . There are several ways to do this.
(Hint: Determine the eigenvalues of the given matrix, M , and show that the largest
eigenvalue, in absolute value, provides an c such that ||M x|| ≤ c||x|| for all x ∈ R2 .
How does this help?)
3. (5 points) Let x0 = 0. Use a computational tool such as Maple or Matlab to compute
iterates x1 , x2 , ..., x5 and verify experimentally that the iterates are converging to the
fixed point.
Problem 4. (10 points) Consider the transformation T : R3 → R3 which converts cylindrical coordinates to cartesian coordinates. Express T in terms of coordinate functions, find
DT and det(DT ) and use this information and the inverse function theorem to determine
on which parts of R3 the conversion is invertible.
Problem 5. Complete Problems 2.10.2 (5 points) and 2.10.3 (8 points) in the book
Problem 6. (5 points) Find an iterated function system generator online and draw a pretty
iterated function system fractal and turn in a picture of it.
1