A:
1 An Introduction to Complex Numbers
1.1 Introduction
We are familiar with quadratic functions of the form
f(a:) = a332 + ta: + c,
where a, b, and C are Legalw DAMS— . Such functions, as we
saw in high school math, may be represented by pizzade in
the
'vwvvvvv—wwvvvvvv'v‘r'v'vvvvvvvvvvvvvvvvvvvvvv
1
Now we know from hypothesis (ii).that 6A and so,
rearranging we have that
We) - «Wm-'46:») /
Now to (ﬁnally!) prove L’Hopital’s rule!
Proof of L’Hépz'tal’s Rule:
Case 1: Let a: e I so that :17 > a.
By h
5.3 Integration by Trigonometric Substitution
Previously, we learned how to integrate by using the method of
substitution. This method allowed us to change a d M -
l 00 an integral into one that we can more 8% deal
with. Let’s review this brieﬂy with an e
5.2 Integrating Products of Trig Functions
Consider the following integral:
/00s(:r)si112(23) dx = S (1050:)
a 1 )
M—bw :-co.s(x
This is easy; we can just use chLQFLrLﬂ rgif; iﬂ
rw or 5mm with u = . Since u’
appears exactly, reverse power rule gives us:
1
An Introduction to Complex Numbers
1.1
Introduction
We are familiar with quadratic functions of the form
f (x) = ax2 + bx + c,
where a, b, and c are r
n
. Such functions, as we
saw in high school math, may be represented by p
the r
p
. Then, the r
in
of
Practice Term Test 1
Math*1210, Winter 2015
Friday, February 6 at 5:30 pm
Last Name:
First Name:
Instructions for right now:
Ensure that all elds are clearly lled out on the rst and second pages of the test.
There are eight pages total, including this p
functions instead.
Le} x“— it
Wed'
3L3
(go £394:
= @5130?) -— smk"'(.-+)
: (cask(ﬂ)1 -Gmkﬂny
: (ud4)?” — (E Siam);
= C0520) 'Sst‘n-LH)
=4
. 1.
. C S + + t z
x.°_.,.,£mlwi3w/
asthma/sf
:— j 5 KS /_,
65
Example 3. With the result of the previous emmple,
We come up with six new formulas:
w
I?»
. QEQFCSHIBED : 7:—
d _l
0 ——-(arcoos(:r) :
dx "——-—-l _ x1
d
——"l
o dJars an(_a:) I + x1
dr < > ——'—'——-—r._
. -— z
d3: arocse w) b4 x14
o E;(arcseo(;c) : M )8"
0 —(arocot(:c) : _:_.l_.
d3: l +><1
Thes
We know that for f to be a function, the curve representing must
pass a V er—h‘ cal
line 1; es+
For the inverse function to exist, f must not only pass a vertical
line test, but also a hm line test. If f possesses this
special property, we call f a 0 he —
Part A: Quick Answer — 15 marks
Read each question carefully and please ensure that you ﬁll in your ﬁnal answers in the boxes
provided at the right side of the page. Each correct answer is worth 1 mark.
A1. If 2 = 4 — 62' then ﬁnd 2*.
A2. What is the re