MA1024B-23S2 Tutorial 08 Page 1 of 2
Numerical Methods
1. Consider the equation f (x) = x3 + 7x − 2 = 0.
(a) Use algebraic manipulation to show that each of the following has a fixed point
at p precisely when f (p) = 0.
i. x = g1 (x) = x − x3 − 7x + 2
ii. x = g2 (x) = 17 (2 − x3 )
iii. x = g3 (x) = x − ex (x3 + 7x − 2)
3
iv. x = g4 (x) = x − x 3x +7x−2
2 +7
(b) If possible, apply the fixed-pint iteration method on each of the functions g
defined in part (a) to determine a solution accurate to within 10−2 on [0,1]. Use
the initial approximation as p0 = 0.5.
(c) Which function do you think gives the best approximation?
2. Use the fixed-point iteration method to find an approximation to the fixed-point that
is accurate to within 10−2 for g(x) = π2 +0.5[sin (x)+cos (x)] on [0, 2π]. Estimate the
number of iterations required to achieve the given accuracy. Compare the theoretical
estimate to the number actually needed.
3. Let A be a given positive constant and g(x) = 2x − Ax2 .
(a) Show that if fixed-point iteration converges to a non-zero limit, then the limit is
p = A1 , so the inverse of a number can be found using only multiplications and
subtractions.
1
(b) Find an interval about A for which fixed-point iteration converges, provided p0
is in that interval.
4. (a) Show that the sequence defined by,
1 1
xn = xn−1 + ;n≥1
2 xn−1
√ √
converges to 2 whenever x0 > 2.
√ 2 √ √
(b) Use the fact
√ that 0 < (x 0 − 2) whenever x 0 ̸
= 2 to show that if 0 < x0 < 2,
then x1 > 2.
(c) Use√the results of parts (a) and (b) to show that the sequence xn convergence
to 2 whenever x0 > 0.
Multivariate Calulus & PDE
1. Determine the 1st-degree polynomial approximation L(x, y) and the 2nd-degree
polynomial approximation Q(x, y) for the following functions near the given points:
(a) f (x, y) = xey + y 2 near the point (1, 0)
MA1024B-23S2 Tutorial 08 Page 2 of 2
(b) f (x, y) = ex sin(y) near the point (0, 0)
2. Compute the gradient vector and the Hessian matrix of the function
f (x, y) = sin(xy) + x2 y
at a point a = (0, 1). Then find the second-degree polynomial approximation of f
at a.
3. Calculate the following integrals:
R 2 R y2
(a) 0 0 (x2 + y) dx dy
R π/2 R sin x
(b) 0 (x + y) dy dx
RR 20 y
(c) R x e dydx, where R = [1, 3] × [0, 2].
4. Let f be a function with continuous second partial derivatives on a rectangular
domain R with vertices (x1 , y1 ), (x1 , y2 ), (x2 , y2 ) and (x2 , y1 ), where x1 < x2 and
y1 < y2 . Use the fundamental theorem of calculus to show that
∂ 2f
ZZ
dA = f (x1 , y1 ) − f (x2 , y1 ) + f (x2 , y2 ) − f (x1 , y2 )
R ∂y ∂x
5. Change the order of integration to show that
Z xZ v Z x
f (u) du dv = (x − u)f (u) du ; ∀x > 0
0 0 0
Hence, show that
Z xZ vZ u Z x
1
f (w) dw du dv = (x − w)2 f (w) dw ; ∀x > 0
0 0 0 2 0
6. Show that ∞
arctan(πx) − arctan(x)
Z
π
dx = ln π
0 x 2
by first expressing the integral as an iterated integral.
