SCIE 001 MATHEMATICS ASSIGNMENT 6 (Due Nov. 14, 2012)
There are two parts to this assignment. The first part is on WeBWorK and is due at 11:59 pm on Fri. Nov.
9. The second part consists of the questions on this page, and is due before 10:00 am on Wed. Nov. 14. For
these questions, you are expected to provide full solutions with complete arguments and justifications. You
will be graded primarily on the correctness, clarity and elegance of your solutions. Your answers must be
typeset or very neatly written. They must be stapled, with your name and student number at the top of each
page.
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14
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1
1. Note that
=
. Prove that there are infinitely many pairs of distinct positive numbers a and
2
4
b such that aa = bb .
2. Let f (x) = ln x, 0 < x < ∞.
(a) Find the Taylor polynomials T1 (x), T2 (x) and T3 (x) of f , all at a = 1. (T1 (x) = L(x) is the linear
approximation, T2 (x) = Q(x) is the quadratic approximation, and T3 (x) could be called the cubic
approximation.)
(b) Let R1 (x) = f (x) − T1 (x) be the error (or “remainder”) in the linear approximation, and let R2 (x) =
f (x)−T2 (x) be the error in the quadratic approximation. Make the following table with five columns,
and then fill it in with numbers accurate to the 8th decimal place.
x
f (2) (1)
2! (x
− 1)2
f (3) (1)
3! (x
R1 (x)
− 1)3
R2 (x)
1.10000000
0.90000000
1.01000000
0.99000000
Based on your numbers in the table, discuss how you could estimate the accuracy of a linear or
quadratic approximation.
3. Recall the Lennard-Jones potential
V (r) = ε
r 12
eq
r
−2
r 6 eq
r
,
0 < r < ∞,
where ε and req are both positive constants. Let
U (r) = V (r) + ε,
which just shifts the graph of the original function up by ε units (we still call it the Lennard-Jones
potential).
(a) Find the quadratic approximation Q(r) (or T2 (r)) of U at req .
(b) Find the angular frequency of (classical) small oscillations in a diatomic H2 molecule if the LennardJones potential U (r) is used to model the interaction energy of the two hydrogen atoms, with ε =
7.22 × 10−19 J and req = 74.0 pm, and then the Lennard-Jones potential is approximated by a the
potential for a spring of equilibrium length req , for r near req .
(c) Using the same “tiny spring” approximation of the bond, find the lowest (quantum) vibrational
energy level possible, and the wavelength of electromagnetic radiation absorbed when the energy is
excited from the lowest to the next-lowest energy state.