臺灣大學數學系
八十五學年度碩士班甄試入學考試試題 分析(應用數學組)
[回上頁]
1.
(a)
Given a function defined on and three distinct points , , find a
polynomial of degree 3 such that for and
. (b)
Derive the Simpson rule for numerical integration on interval :
where . (c)
Prove that the Simpson rule has an error bounded above by .
2.
(a)
Find the rectangular parallelepiped of greatest volume inscribed in the ellipsoid:
. (b)
Describe and prove the Lagrange's method of undetermined multiplier for the extremal problem with constrain, namely, find the extreme value of a function
under the constraint .
3.
Consider damped forced motion of a mass on a spring, govern by
(1)
where is the mass of the spring, is a damping constant, is the spring modulus,
and and ω are the amplitude and frequency of forcing respectively.
(a)
Write the general solution in the form of , where is the homogeneous solution by considering in (1), and is the particular solution. For the , be sure to discuss various cases in choosing the constants , , and .
(b)
What is the periodic and amplitude of the particular solution ? (c)
How do we choose ω so that the amplitude of the particular solution is a maximum?
4.
Solve the following linear system of ordinary differential equations:
(a)
with initial condition .
(b)
with initial condition and .
5.
The operator norm of a matrix is defined by
where and are the 2-norms of and , respectively. Show that satisfies
(Hint: and is symmetric and nonnegative definite.)
