ページ3:

ReviewDerivative
The Derivative of
a Function
function too)
def f(x) = lim f(x+h)-fog)
620
h
the difference quotient for f at x:
f(x+h)-f(x)
h
description Differentiable @ a point x
called
a
Differentiable @ a point
functim is
Differentiable @
every point of Domain
a differentiable function
then is continuous at x=L.
(differentiable),
(可微分必連續)
Theorem If a function has
a derivative at x=L
proof To show f is continuous,
(at x=C)
we need to show 1. f(c) exist.
2. I'm fox) east.
XAL
> we know for sure!
3. lin fox)=f(c).
we
know: 1. im (x-c) =
I'm f(x)-f(c)
2. XAL X-L
50 xin (x-1) x lim
X-91
fox)-f(c)
XX-L
I'm [(x-1) f(x)-(c)]
= I'm [foxy-fis)]
=
=
Rules of Basic Dif.
1. (c)=0
3. (cu)= c du
exists
= 0xm=0
n-1
> lim [fes- fes] = lim fexs - ft) = 0
4. 1 (u±v) = du = d.
*
5. 11 (uv) = du v + udu = u'u+uv' (symmetry)
6.
(4) = uv-uv
7. n is negative interger and x#0, de (x²)=nx²+
J
RED.
✓74-sub.
Chain Rule:
=
141
atx
at u(x)
rewrite: dx Ju.
atx.
器
TRIG Dif. (symmetry here) proving.
d
1, đc sinx Fest
①def
physics: SHM
(SMX)
已
(cosk)
+
3. dx tanx = Seix
4.
Secx = tanxsexx
2. A
Cosx = -sinx.
5.
ttxe
cscx
6. CSC x= - cotxesex
sin tacos
sin
cos
tan
平方(相采
Cot
tan
Sec
CSL
Sec
☑☑
cot
CSC
上乘
對稱加負號
sinx tui xa
tan'x + 1 = sec²x
cot'x +/ = <sc²x

ページ4:

Other Dif Rules
+ b (x) = +
Q:
?
2
=
d
=d[en[an]
= e
en (a)
•en(a*) & [en(a*)]
e
.
a. In (a)
dx.
[x In[a]]
a² = a². In (a)
The road to MUT:
We need to know max and min — relative/absolute
(local)
e.g.
相對極大vs.絕對板大......
The first derivative theorem for Local Extreme Values (FOT)
-max
= (a,b)`
interior point c of an interval fis defined,
If f has local or at
and f' is defined at x=C,
then filmin
proof without works
Local max
slope o
slope so
(never positive)
(never negative
f(c) exisex+c+
lim f(x)-f)
= lim
f(x)-f(c)
!= 0 = f(4) = lim
fox)-fo
X-L
x-+-
X-L
x-
Rolle's Theorem (RT).
Suppose that y = f(x) is continuous on [a,b]
and differentiable in (a,b).
If f (a) = f(b)=0
y= f(x)
MUT: if f is cont. on [a,b]
and dif. in (a,b)
there exist in (a,b)
S.t.
f(b)-f(a)
b-a
wwwm
=
slope of secant.
=fes.
f(s) = slope = M2
slope=m
fes
a
d(x).
Let
d (x) = f(x)-g(x).
We know
g(x) = fras + f(b)-f(a)
b-a
)-f(a) (x-a).
→ d(x) = foc) - f(a)- (x-a):
'd' = f'-g'
f(b)-fla).
b-9
b-a
d'(x) = f(x)-
f(b)-f(a).
From RT,
= x=c
=(a,b)
we know for d(x)
s.t. d'(c)=0.
d' (c) = f(c) - b-a
f(63-f(a) =0.
f(b)-f(9)
f(c)=
b-a
QED.
then there exist at least one c
s.t. f(c) = 0.
proof
€(a,b)
we can find extreme values on [9,6]
at 1. end points
C
b
0 = min (f) = f(x) ≤ max(f) = 0 → c is every point in
2. f'co
=
→one or tho
of extreme
value
3. f' X exist. * (againse hypothesis)
(a,b)
not on xa
or x=b
FDT
on x=c, f(c) = extreme
f(c)=0

ページ5:

Inverse Trig Functions
The Arc Sine
y=sin"'x
(sin x)=?
Let y = sin x < x=sing
(sink) =
->>
SITE dx = sin x + c.
Domain: -1≤x≤|
Range: -1/2 =4≤7/2
→
do
dx = | = cosy
√1-X2
-
dx = x (sin x) = cosy =
'
√1-x2
(-14×41)
(-1<*<1)
note that (sinx)'= s/nx = cscx + sin x.
Sinx
②sin'(-x)=-sin'(x) → odd function.
cos'(x) + cos(x) = π | (cos"x)=?
②The Arc Cosine
y
y=cos"
identities:
Domain: -1≤x≤1
②
sin" (x) + cos(x) = .
Range: Osyst
genme me proof
| = Lx = = siny Lxx
-π
x
sin'x
tan
4=742
y=tan'x
Domain:-<x<
Range: -7/cy</2
ax tan x =?
Y = tan x x = tany
dx
Y=-72
= Sexy
d
y=secx
•
sec'y
=
3x+
ST+ x2 dx = tan' x + C
-34
"
Tu
y=secx
Domain:1x/>1
Range: Osyst, y4/2
Secy (-1) Secy>
√x21
not
def.
Cotx
x=cosy
☑
(45) = -
(1x11)
= -
√√√1-X2
(12x1<1)
·St)ch=105x+C
y=cotx
y=π
a/2
Domain: -00<
Range: 0<y<.
×
(cot x) =>
←
y=cot"> <x=coty → =|= -css'y ex dy = - (y == sinty
=
2x+15
1+x2
S(-1)=='x+C
(cat)=-
->
1+x²
(sec⭑x)>?
->
Sec-x=y
x=sexy / tuny secy .
tany+1=sec³y
-
not def.
dx
tany secy ±√x²/.x
:fal
=
xxx fx-1
12. // TT (IN>1) <>
M

ページ6:

(secx) = x/J
(sec" (x1)
=
J
·(-1).
(chain rule)
√ TX JATT = secx + C
= Sec
sec |x|+C
(x'>1)