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Math 100 UBC/UBCO full course notes 20+ pages

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Course
MATH 100 (MATH)

Institution
University Of British Columbia (UBC )

From start to finish MATH 100 notes with practice problems and examples 20+ pages full of notes and graphs and examples. ORGANIZED.

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Uploaded on May 14, 2023
Number of pages 22
Written in 2022/2023
Type Class notes
Professor(s) Wayne broughton
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MATH108

101

, ↑ AF(z) 1 % % with wasne.D

TUESDAY JC 10 Introduction
-
to limits

-> pre-cpic review:

1. Given that x=2 is of
proot x3-x-2x+12. factor the polynomial completely and findall i ts roots
of
-

>Che(R: 23 22
- -
8(2) 12 +
X-2 is a f (( +0r

8
= -

4 -
16 12
+ -
x3 -
x" -
8X 12 +
(X
= -
2)(X +x -
6)

(x -
2) (x + 3) (X -
2) The roots are x 1
=

andX = -
3

2.x z
-

x2 +1 -

- rte1)

4(x- ( -
1)- xe
(X
- -
-

2
x -
1

>12 -
2x 1+
-
y2 -
1
=

-
2X

xz - 1 xz -
1

3. Find the equation of
t he line that
p asses through the points (2,5) and 1-1, 2) in the plane
xx

5-1
Findslope
=
->

2 -
1 -
1)

following:y-Y' m(X X')
↓
= -

-> y 3/2X 6
=
+

y
- -
3 3/2(X 2)
= -

3/2X
=
-

3 5 +

1 3/2x 2
=
= +

4. Findthe exact value of COS(-π /6)
7 L
68
2
v 45
=1880:300 using special triangle
=

2452
1

v/2 38
13 1

THURSDAY, JAN 12:PRECALL & INTROTO CALC

1

1: X X 8
=
REFRESHON BASIC GRAPHS

shifted sin (X) graph
M

L
Shifted COS(X) grOPM Y:COS(X) Domain:x0 or

1.Sin (X)
(0,0)u(0,p)

EcosixOY ↳ <
1 >

*
-

2

iπ
-
- 1

1:109eX:10n 1

y: 109 - XM Y:109 X
> 7
[0,0 1 -
1 Exponential fxn: y.e*
1 ⑲
Xx,0 ④
↓
&
i
U:104b(V)

V =bu

·finan
piecewise fXn
>

9
Y :(X/,0 ·
(x) X,X>,0

D:( -8,0) Range:1R -

X, ifX <0
↳(0,0
Y: arc an(X)
+

#
I tan(x) 20 (X)
= +

,-> calculus:single variable -> differential (aboutr ates change) the
of main concept is the derivative

·unifiying theme:limits

Limits:Chapter 2 Introducing limits
~If itisitwill
-> -> cause the answer

·consider y f(x) xx 1 x 1 = what is f164)?
:=I 3 i s the
what domain t he
of flx)? x, andx*I
- = -
=
=

x5 -
5x 3 -
1

<(0,1) UC2, 0) - know how to write/readinterval notation

o The graph of this fan seems to keep going
through x 1
=

(where its under), o f(x) near
lets 100k x 1
=

+ (0.9):80.9-1=1.48698.... -> a bit bigger f(1.1) 1.51203=
or even closer to 1 f(0.99):1.490743/f (1.01):1.501245 (f(0.9999):1.499988
29-1
f(1.00001):1.50003// As x gets closer to 1 the fan gets closer to 1.5

· s eems that f(x) is
It getting closer to 1.5 as s ets
x closer to 1:this is the behavior f (x) nedr
of 1
x=

mathematically) we can make f(x) as close
This turns out to be true (can be proved
->
to 1.5 as we want close) if we use
carbitrarily any X value "close

enough"to 1 limit notation

We say the
*
o f f(x):x-1
limit is 1.5 as x approaches 1 im f(x):1.5
x 1
"3-1
->

x
or we can write it out as f(x)-1.5

CS X -
1

7/3 25
-> Now consider f(x) for near
x x 64 f(64)
we know that is or so what
doesMyfex) mean
-
=

of (63.9):2.33287 1 + (63.99999):2.3333287 (f(64.00001):2.333379

25 =
tyMyf(x):
It looks like f(x) is getting
arbitrarily
close to as x is getting close to 64 2

5 253 same
(imf(x) f(64): question
answer, different
so 2
= =

TUESDAY JAN #3
17:LECTURE

LIMITS:CIP1.2 -> Instantaneous velocity

-> If I travel 8 0km/h
at for 1 hr, then I have gone 80km. or i travel 160km in 2 hours, or in one minitravel 800 8/6km
=

over time period:distance travelled
>velocity
time taken

0 What does m ean
it to be going 80
at km/h at one point in time (an instant)?

I
travel a distance in time so v =

Ca "
-> We lookata time interval around instant:
that >t

we can define aus velocity:a travelled on thatinterval = change in position
length of time change in time

As
* interval
that gets shorter & shorter, the due velocity
a pproaches a limit
value -> t he
That limiti s the instantaneous velocity
-

CLP-1:1.3 - EXPLORING LIMITS

taking enough"to the (x+a)
0ximf(x)=L:means we can make f(x) as close to (as we like ("arbitrarily
close"), by X "close a
value of

-.
-
AMPLE:
Dimsincx), since is defined
not at 10bc is

sincxl
-> x X sin(0.1) 0.998334.....
=

3 i n trig fans is in radians
in calculus, we assume x
0.10.448334 8.2

Whatifwe use -0.1: same:0.998334

0.01:0.999983 -
8.82:0.99999....
I It180ks like
MMsincx)
y
X
= 1 This turns out to be true!

2(1m(x+2): in
we plug
cant x 2 Yes we could (and is the rightanswer), but limit
w hat
t hats not
-
=

means

↳ 1.999:(1.999) +2 5.996001:It100ks like =

imz (x+ 2) is the same as 22 +

2 6 =

-mes Dlim doesnt
exist

EG#3) Le f(x) +
sin),
=

for x is
=O- what
limo sin(i)? H(t)

I
SAS x =
0, gets larger (+ or -- so sink) oscillates faster faster btwn-1 and 1

so
him sin(i) DNE:sin(A) does not "settle down"close to one value justkeeps
it jumping again
away
1

4) "Differentvalues on the sizes
left andright -> H(t):
Let

Goit 1 if
+ co

t >I
-
t
8

H:heavyside txn

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