MATH 1321-004 Homework #11
due 2013-APR-15, 11:35 a.m.
1
Assignments
by (2), (3), and the following
All numbers below refer to the textbook:
Calculus Concepts and Contexts, J. Stewart, 4th Ed.
a + ib = c + id ⇔ a = c, and b = d.
(5)
(II) should also give your some insights on the relations of Taylor series of ex , sin x, and cos x.
This problem exemplifies
• §13.2: 3, 7, 11, 17, 19, 21, 39, 43
• §13.3: 1, 5, 9, 11, 15, 17, 21, 23, 25, 32, 33, 34
• Extra credit
What you have learnt helps you learn more.
– §13.2: 37, 47, 48,
– §13.3: 19, 29, 35, 36,
3
– The problem in the next section.
Acknowledgment
Since the extra credit problem in Section 2 is triggered
Boldfaced problems weigh 2 points each and all other
by separate questions from Andrew Hurlbut and Curtis
problems 1 point each. A random subset of problems will
Houston, I am giving each of them one-point extra credit
be graded for a total point of 15.
for asking the questions.
2
Buy one, get three free
4
Directions
The purpose of this problem is to help you remember
double-angle formulae by connecting them to complex
numbers and Taylor series. The bottom line is that you
can derive double-angle formulas so long as you remember the Euler’s formula (2).
The imaginary unit i satisfies
Vector calculus is the punchline of this course and it requires a solid understanding of materials covered in previous chapters. So I am taking a much slower pace for
this chapter to make sure that you understand it well.
In case you missed a lecture, you should be able to
figure out most of the problems by reading the examples
i2 = −1
(1) in the book.
Additional 25% credits will be given to you if you
and it allows the real numbers R to be extended to comtypeset your solutions in LATEX. You can also get partial
plex numbers C. A milestone of complex analysis is Euextra credit for typesetting solutions of some problems.
ler’s formula discovered around 1740:
From those of you who uses LATEX, I will select a
ix
∀x ∈ R,
e = cos x + i sin x.
(2) winner and base the released solution on her/his LATEX
source. The criteria will be
(I) Prove
(i) correctness,
(a + ib)(c + id) = ac − bd + i(ad + bc).
(3)
(ii) number of extra credit problems finished,
(II) Prove (2) using (1) and Taylor series of ex , sin x,
and cos x in Table 1 on page 613 of the textbook. (iii) the logical flow of explaining the steps.
(III) Prove the double angle formulae
2
2
2
cos(2θ) = cos θ − sin θ = 1 − 2 sin (θ),
sin(2θ) = 2 sin θ cos θ.
2 tan θ
.
tan(2θ) =
1 − tan2 θ
The homework winner will get another 25% extra
credit.
(4a)
Note: Please send me your latex source (.tex) via
(4b)
email if you typeset your solution in LATEX and would
(4c) like to participate in the competition for the 25% extra
credit.
1