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Lecture 1

1000 Lecture 1: All math 1

24 Pages
65 Views
Summer 2017

Department
Economics
Course Code
1000
Professor
simo
Lecture
1

This preview shows pages 1-3. Sign up to view the full 24 pages of the document.
www.nafham.com
if y=logax then x=ay
xlog
b
x=b
logaa=1
logaax
( )
=x
alog
a
x=x
logaMN
( )
=logaM+logaN
logaMp
( )
=PlogaM
logaM
N
=logaMlogaN
a
M
N
=a
n
M
Sx=xa+bi
( )
[ ]
xabi
( )
[ ]
=x22ax +a2+b2
( )
cos2x+sin2x=1
sin 2
( )
=2sin
( )
cos
( )
cos 2
( )
=cos2sin2
cos 2
( )
=12sin2
cos 2
( )
=2cos
21
sin 2
= ± 1cos
2
cos 2
= ± 1+cos
2
tan 2
( )
=2tan
1tan2
tan 2
=1cos
sin =sin
1+cos
logax=ln x
ln a
sin +sin =2 sin +
2cos
2
sin sin =2 cos +
2sin
2
cos +cos =2 cos +
2cos
2
cos cos =2 sin +
2sin
2
sin +
( )
=sin cos +cos sin sin
( )
=sin cos cos sin
cos +
( )
=cos cos sin sincos
( )
=cos cos +sin sin
ax
2
+bx +c=0
x=b±b
2
4ac
2a
Page 1
Math Reference Trigonometry & Analysis James Lamberg
cos π
2+x
= sin x
cos π
2x
=sin x
cos π ± x
( )
= cos x
cos 3π
2+x
=sin x
cos 3π
2x
= sin x
sin π
2±x
=cos x
sin π + x
( )
= sin x
sin π − x
( )
=sin x
sin 3π
2±x
= cos x
tan π
2+x
= cot x
tan π
2x
=cot x
tan π+x
( )
=tan x
tan π−x
( )
= tan x
tan 3π
2+x
= cot x
tan 3π
2x
=cot x
for every
element of
there exists
such that
therefore
Q since
¬ not
and
or
d derive
integrate
x=a evaluate with x =a
proportional to
p precedes
f follows
congruent to
union
intersection
subset
superset
proper subset
propor superset
C (fancy) compliment
implies
double implication
~ negation or ¬
( )
Q.E.D: Quod Erat Demonstandum
"that which was to be proved"
tan
( )
=tan tan
1+tan tan
tan +
( )
=tan +tan
1tan tan
Mid =x
2
x
1
2,y
2
y
1
2
Dist =x
2
x
1
( )
2
+y
2
y
1
( )
2
A = 1
2r2, sector area
www.nafham.com
c2=a2+b22ab cos
b2=a2+c22ac cos
a2=b2+c22bc cos
sin
a=sin
b=sin
c
K=1
2bc sin
y=Asin x -
( )
+K, > 0
Amplitude = A=Mm
2
Period = 2π
Phase Shift =
Frequency = 1
period =2π
Critical Points = period
4
Unit Circle
cos, sin
( )
0,0°=1,0
( )
π
6,30
°=3
2,1
2
π
4,45
°=2
2,2
2
π
3,60
°=1
2,3
2
π
2,90
°=0,1
( )
2π
3,120°= 1
2,3
2
3π
4,135°=−2
2,2
2
5π
6,150°=−3
2,1
2
π,180
°= −1,0
( )
7π
6,210
°=−3
2,1
2
5π
4,225
°=−2
2,2
2
4π
3,240
°= 1
2,3
2
3π
4,270
°=0, 1
( )
5π
3,300
°=1
2,3
2
11π
6,330
°=3
2,1
2
Page 2
\Math Reference Trigonometry & Analysis James Lamberg
www.nafham.com
ab =ab
a2=a
a
b=a
b
an=an
if a<b&b<c, then a<c
if a<b&c<d, then a+c<b+d
if a<b then a+k<b+k
if a<b&k>0, then ak <bk
if a
<b&k<0, then ak >bk
a+ba+b
¯aaa
ak iff ¯k ak
ak iff a-k of a k
xo
limsin x
x=1
xo
lim1cos x
x=0
xc
lim
f x
( )
f c
( )
x
c
d
dx x
n
=nx
n1
d
dx C=0
d
dx tanx=sec
2
x
d
dx sin x=cos x
d
dx cos x=¯sin x
d
dx cot x=¯csc
2
x
d
dx sec x=sec xtan x
f t
( )
=1
2gt
2
+v
0
t+s
0
v t
( )
=f't
( )
a t
( )
=v't
( )
=f"t
( )
d
dx csc x=¯csc xcot x
d
dx f x
( )
g x
( )
( )
=f x
( )
g'x
( )
+g x
( )
f'x
( )
d
dx u=u'u
u,u0
Profit = Revenue -Cost=SoldPrice -Cost
Critical # when f'x
( )
=0 or f' x
( )
DNE
f x
( )
increasing if f'x
( )
>0
f x
( )
decreasing if f'x
( )
<0
MVT, f'c
( )
=f b
( )
f a
( )
baon a,b
[ ]
IPs if f"x
( )
=0 and f"x
( )
changes sign
ydy =f'x
( )
dx
f x
( )
concave down on a,b
( )
if f'x
( )
is DEC x in a,b
[ ]
, f"x
( )
<0
f x
( )
concave up on a,b
( )
if f'x
( )
is INC x in a,b
[ ]
, f"x
( )
>0
d
dx f x
( )
dx
( )
=f x
( )
k
( )
dx =kx +C
x
n
( )
dx =xn+1
n+1+C
TrapRule
f x
( )
dx ba
2n
a
b
f a
( )
+f b
( )
+2f x
1
( )
+f x
2
( )
+... +f x
n
( )
( )
( )
Euler' s Method
Start @ x,y f' x,y
( )
→ ∆x→ ∆y = xf' x,y
( )
Use y for change in next y
kf'x
( )
( )
dx =kf x
( )
+C
f'x
( )
±g'x
( )
( )
dx =f x
( )
±g x
( )
0
( )
dx =C
sec2x
( )
( )
dx =tan x
( )
+C
csc2x
( )
( )
dx =Ccot x
( )
sec x
( )
tan x
( )
( )
dx =sec x
( )
+C
csc x
( )
cot x
( )
( )
dx =Ccsc x
( )
c
i=1
n
=cn
i
i=1
n
=n n +1
( )
2
i
2
i=1
n
=n n +1
( )
2n+1
( )
6
i3
i=1
n
=n2n+1
( )
2
4
cos x
( )
( )
dx =sin x
( )
+C
sin x
( )
( )
dx =Ccos x
( )
x→∞
lim f ci
( )
i=1
n
⋅ ∆x,xiacixi,x=ba
n
f x
( )
dx
a
a
=0
f x
( )
b
a
dx = − f x
( )
a
b
dx
f x
( )
a
b
dx =f x
( )
a
c
dx +f x
( )
c
b
dx
kf x
( )
a
b
dx =k f x
( )
a
b
dx
f x
( )
±g x
( )
a
b
dx =f x
( )
a
b
dx ±g x
( )
a
b
dx
f x
( )
a
b
dx =F b
( )
F a
( )
,F'x
( )
=f x
( )
F'x
( )
=d
dx f t
( )
a
x
=f x
( )
If f is even, f x
( )
-a
a
dx =2 f x
( )
0
a
dx
Average Value of a
Function : 1
b-af x
( )
dx
a
b
If f is odd, f x
( )
-a
a
dx =0
d
dx
f x
( )
g x
( )
=g x
( )
f'x
( )
f x
( )
g'x
( )
g x
( )
( )
2
Math Reference Calculus James Lamberg
Page 1

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Description
Math Reference Trigonometry Analysis JamesLamberg y x if y = log x a then x = a loa(MN )= logaM+log Na loga )= x aN = 1 2 2 p aN cos x + sin x = 1 log M = log M log N loga( ) Plog M a 2tan a N a a xlob = b tan2 )= 2 si(2 )= 2sin( )cos( ) 1tan a lag = x log a = 1 M M log x = ln x aN = n a S = [ (a +bi][ x (abi )]= x 2ax+ a(+b 2) a ln a x 1 cos 1 cos sin 1 + cos sin = tan 2 = sin = 1 + cos cos = 2 2 2 2 2 2 2 2 cos(2 )= 2cos 1 co(2 )= cos sin cos(2 )=12sin + + cos cos = 2sin 2 sin 2 sin sin = 2cos 2 sin 2 + + cos + cos = 2cos cos sin + sin = 2sin cos 2 2 2 2 2 sin( + )= sin cos +cos sin sin( )= sin cos cos sin ax +bx+c = 0 2 cos( )= cos cos +sin sin co( + )= cos cos sin sin x = b b 4ac 2a cos + x = sinx tan + x = cot x therefore 2 2 sin 2 x = cosx Q since cos x = sinx tan x = cotx sin(+ x =) sin x not 2 2 and sin( x ) = sinx cos( x =) cos x tan(+ x ) = tanx or 3 3 tan( x ) = tan x sin 2 x = cos x cos 2 + x = sinx tan 3 + x = cotx p precedes C (fancy) compliment 3 2 cos x = sin x f follows implies 2 3 congruent to double implication subset tan x = cot x 2 union ~ negation (r ) superset Q.E.D: Quod Erat Demonstandum intersection proper subset d derive that which was to be proved for every propor superset x x y y element of integrate Mid = 2 1, 2 1 tan +tan 2 2 there exists x=aevaluate with x=a tan( + )=1tan tan 2 2 proportional to Dist = (2 1) ( y2y 1) such that tan( )= tan tan 1 2 1+tan tan A = r , sector area 2 Page 1 www.nafham.com .
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