Question

Use the cobweb diagram to show that the map f(x) =e^-x has a unique fixed point. Also determine the stability character of the fixed point from the diagram.

Answer #1

Show that the function has a unique fixed point in
the interval .
g(x)=(3x+19)^1/3 [0,infinite]

Show, if f : X → X is continuous, X is compact, and f does not
have a fixed point, then there is an E > 0 such that d(x, f(x))
≥ E for all x ∈ X.

Show E[f(X)g(X)]≥E[f(X)]E[g(X)] for f,g bounded,
nondecreasing.

Use the three-point center-difference formula to approximate f ′
(0), where f(x) = e x , for (a) h=0.1; (b) h=0.01; (c) h=0.001.

Let f(x, y) = (2y-x^2)(y-2x^2) a. Show that f(x, y) has a
stationary point at (0, 0) and calculate the discriminant at this
point. b. Show that along any line through the origin, f(x, y) has
a local minimum at (0, 0)

Show, if f : X → X is continuous, X is compact, and f does not
have a fixed point, then there is an > 0 such that d(x, f(x)) ≥
for all x ∈ X.

Prove that, if f(x) has a limit as x->a, then the limit is
unique.
In other words, if limf(x) = L1 and lim f(x) = L2, then
L1=L2

Find the fixed point of e-e^(x) to within 0.1 by
using xed point iteration.

Use intermediate theorem to show that theer is a root of
f(x)=-e^x+3-2x in the interval (0, 1)

consider solving x=cos(x) using fixed point iteration to find p,
the fixed point. Show that the convergence is linear for any
starting value p0 sufficiently close to p.

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