Summary AP Calculus AB Study Guide

AB Calculus AP Study Guide
UNIT 1
continuity not continuous
& defined
f(x) =
① fal
L
=
undefined
② limf(x) value ② im
f(x) +
Uma-f(x)
=

X- a X-at
③ lim + (x)
=
f(a) ③ limf(x) + f(a)
* ->a X +a



SqueezeTheorem
If n(x) = f(x) =
g(x) and
limhx L
lmg
= =

,

then limf(x) =
L

am
HM
X - C =
1
X- 6



timx =
1
Jhorizonta
asymptote
a
timf(x) =
@J vertical
asymptote

UNIT 2
·

differentiation implies continuity but continuity does
# imply differentiation no sharp pt
_
① must have local linearity him f(x)
lm f(x)
=


+
X- a - a



& must be continuous him f(x) =
+(a)
*-a




③ cannot have a vertical tangent or place where slope und.

IROC derivative
Ex H f(x) f(x)
= = -
=




f (a) x) f(a)
I differencetient
1 a
= -




X -
a


AROC =
Slope (x+ h) f(x) =
= Yz Y
, -
-




U
X2 X , -




Intermediate value theorem (IVT) -




If f(x) is continuous
and +(a) #f(p) and c is in btwn fcal and f (b) , then +(c) =
K

D find f(a) and f(D) <
make sure f(x) is continuous

② is < btwn fla) and f(b) => are fca) and (b) different ?
③ find( values
& make sure c is in domain - answer format : +(a) <
K < f(b)

, differentiation rules
productRule
[] = 0 c = constant /




& (f(x)g(x)] = f(x)g(x) g(x) + (x)
+




* [KX] = K K = constant
Quotient Rule

& [x] =


-g(x
nx"nnteger =
& (k (x)] 1 8'(x)
-
=
+
.




& [f(x) = g(x) = f(x) =
g(x)


Transcendental derivatives

y sin(X)
=

y
= COS) y tan(X)
=
y =
sec(X
y
=
coS(X) y =
sin(X)
-




y cot (X)
=
y = -
es- (x)
=
sec(X) y =
sec(tan(*
) = ex y = eX
y y
cs(X) cS(X) co+ (x) y en(X) y=
' =
y
= -




y


Graphing Derivatives
& find local maximums a minimums (where t 0 =




② find recognizable shapes and use them to find a
↳ eX
: A
⑤ the more severe the slope ,
the bigger thenumberl will be



Absolute value
f(x) (x =

-
-


4) X
-

n= j
X= 4
↓
4) ;
S
-

(X -

X24
f(x) =

-

( -X 4) ; -

x 14
+ -
-


x+4 1 x -

4
f(x) = -
1 f(x) =
1
-
17 1
UNIT 3
·

differentiability must be continuous