Takehome Midterm
Math 323, Spring 2011
These problems must be written up neatly (preferably typed), and are due on Thursday,
March 17.
Rules: This is a midterm. You must solve the problems entirely on your own, and you
should not discuss the problems with anyone but me.
1. Most calculus books discuss Newton’s method, a procedure for finding roots of functions.
Given a differentiable function f : R → R, define a new function N : R → R by
N (x) = x −
f (x)
.
f 0 (x)
Then orbits of N have a tendency to converge to roots of f .
(a) Let r be a root of f . Assuming that f is twice differentiable and f 0 (r) 6= 0, prove
that r is an attracting fixed point for N .
(b) For f (x) = x3 − x − 1, compute the first six points in the orbit of 2 under N .
Does the orbit converge to a root of f ?
(c) For f (x) = x3 − 2x + 2, prove that N has an attracting 2-cycle.
2. Let f (x) = cx(1 − x), and let c be the smallest positive value for which 1/2 is periodic
under f with period 6.
(a) Determine the value of c. Your answer must be correct to six decimal places.
(b) Find the six points in the attracting cycle, and indicate which points map to
which. Give your answers to four decimal places.
(c) The points of the six-cycle partition the interval [0, 1] into seven subintervals
I, J, K, L, M, N, O. Draw the transition graph for f corresponding to this partition.
(d) Based on this transition graph, which periods exist in f ? Given a sample itinerary
for each possible period.
3. Let f : R2 → R2 be the function f (x, y) = (0.4 x2 + y, 0.5(x + 1)y).
(a) Find all of the fixed points of f .
(b) Use the Jacobian to classify each fixed point as a sink, a source, or a saddle.
4. Let f : [0, 2] → [0, 2] be the map
(
x + 1 if 0 ≤ x ≤ 1
f (x) =
4 − 2x if 1 < x ≤ 2.
(a) Find a point x ∈ [0, 2] whose orbit under f has Lyapunov number 2.
√
(b) Find a point x ∈ [0, 2] whose orbit under f has Lyapunov number 2.
√
(c) Find a point x ∈ [0, 2] whose orbit under f has Lyapunov number 5 8.
Note: Your answers in parts (a), (b), and (c) should be expressed as fractions.
5. Consider the functions f , g, h, and k whose graphs are shown below:
3
5
4
4
2
3
2
3
2
2
1
0
1
2
fHxL
3
0
1
1
1
0
3
0
1
2
3
gHxL
4
5
0
0
1
2
hHxL
3
4
0
0
1
2
3
kHxL
(a) Exactly two of these functions are conjugate. Which ones? Describe the evidence
that you used to reach this conclusion.
(b) Find a continuous bijection C so that C ◦ A = B ◦ C, where A and B are the two
conjugate functions you found in part (a).