C3 THEORY OF COMPUTATIONAL DYNAMICS
EXAMPLE SHEET 1
Stars indicate level of difficulty.
1.
For each of the following functions, calculate the derivative:
i)
x2
ii)
x logx
iii)
λx(1 - x)
iv)
sinx
v)
|x|
vi)
|x |
(where |x| = x if x ≥ 0 and |x| = -x if x < 0)
2.
For each of the functions in Q1, compute the second and third derivatives, and hence write
down the Taylor expansion about an arbitrary point x 0 . Discuss for which x0 this expansion
exists.
3.
Consider the function f(x) = xα, for α > 0. Discuss to what order the Taylor expansion of f at x = 0
3
5
exists. Write down the expansion for α = 2 and α = 2.
4.** Let f be the function
f(x)
=
( )
exp − x1
2
for x ≠ 0, and f(0) = 0. Compute the Taylor expansion of f at x = 0, and show that it does not
converge to f.
5.
Let f be the logistic map f(x) = λx(1 - x). Compute the derivative of f°f and f°f°f. Here ° indicates
composition, thus f°f(x) = f(f(x)) and so on.
6.
Compute the second derivative of fºf where f(x) = λx(1 - x).
7.
Let f : R 2 → R 2 be the Hénon map f(x,y) = (1 - ax2 + y, b x) where a and b are parameters.
Compute the derivative of f.
8.* Let g : R2 → R be the function
g(x,y)
=
xy
x +y2
2
for (x,y) ≠ (0,0), and g(0,0) = 0. Show that both partial derivatives of g at (0,0) are 0 (think, don’t
calculate!). By considering the line x = y, or otherwise, show that g is not differentiable at (0,0).
Is it differentiable at other points?
9.
Suppose f : R n → R m and g : R r → R n are differentiable. By expanding f(g(x + h), or otherwise,
show that D xf°g = Dg(x)f.D xg.
C3 Exercise Sheet 1
2
10. With f and g as in Q9, write out the ith co-ordinate of f(g(x + h) to first order in h (in terms of the
partial derivatives of f and g).
11. Let f : R2 → R2 be the Hénon map f(x,y) = (1 - ax2 + y, bx). Compute the derivative of f°f.
12. Show that if all the partial derivatives of a function f : Rn → R exist and are continuous at a point
x∈Rn then f is differentiable at x. Contrast this with the example in Q8 above.
13. Let f : R2 → R 2 be the Hénon map f(x,y) = (1 - ax 2 + y, bx). Compute the second derivative of f,
and hence write down the Taylor expansion of f about the point (0,0) to second order. What is
the expansion to order r for r > 2 (think, don’t calculate)? Now compute the expansion about an
arbitrary point (x,y).
14. Let F : R 3 → R 3 be the Lorenz vector field F(x,y,z) = (σ (y - x), rx - y - xz, xy - bz). Compute the
second derivative of F, and hence write down the Taylor expansion of F about an arbitrary point
(x,y,z) to second order. What is the expansion to order r for r > 2 (think, don’t calculate)?
15. Suppose x(t)∈R satisfies the differential equation
dx(t )
dt
Evaluate
=
d 4 x (t )
d 4t
F(x(t))
in terms of the derivatives of F at x(t).
16.* Repeat Q15, but now with x(t)∈R n. You may prefer to write everything out in terms of coordinates and partial derivatives.
17. Consider the modified Euler method
xt+h
=
1
xt + hF(xt + 2hF(xt))
What order accuracy does this give?
18. Let
dx
dt
=
λx
Write down an exact formula for xt+h in terms of x t, and compare this to the approximate
solutions given by Euler, modified Euler (Q17) and 2nd and 4th order Runge-Kutta schemes.
19. Write out explicit formulae for the Euler, modified Euler (Q17) and 2nd order Runge-Kutta
schemes, when used to solve
..
.
x + α(x2 - 1)x + x
=
β cos ωt
20. Obtain an explicit error estimate for the Euler method: i.e. compute a constant K (in terms of F
and its derivatives) such that
| xt+h - xt - hF(xt) |
for all sufficiently small h.
≤
Kh2