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AP Calculus AB Study Guide
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A 36 pages year-long study guide of AP Calculus AB course.
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Unit : L nit 4 :im its + Unit 2 : L nit 3: Differential
Continuity Differentiation Intergrat Equations
1 M 1
2EC) V99L JE :
ABC ABC Procedures
& + -
11) Equations/theorems
ABC Terms
Example
, v :
v
T -
Say : "Limit of f(x) ,
& methods of evaluate Limits :
as X approach c is L + Table (Ex 1)
Graph (Ex2)
Notation: Limf(x)L
+
+ Algebraic (Ex 3 ,
4 ,
5)
facior' conjugates tris identities
Table :
Ex : WhatDoes y approach as
X get closer 10-1 ?
x + 5x + 6 = + 3
a) f(x) = b) f(x) =
↑ #
x + 2
&
X f(x) X f(x)
-
2 .
1 0 .
9 Y
-
2 .
1 -
9 Y
-
1 .
01 0 99 .
~
-
1 .
01 -
99 ~
-
2 001
.
0 .
999 ~ 1 -
2 . 001 -
999 -
-
1000
-
L N/A -
L N/A M
M
-
1 .
995 1 .
001 1 -
1 .
995 1001 10001
1 1
-
1 99 .
1 .
01 -
1 99.
101
-
1 5 1
. 1 Y -
1 5 11 Y
. .
x 5x 6 Lim X + 3
Lim + +
= 1 = DNA
2 Xc 2 2
2
-
Xc -
+ X +
x
,graphi
f(x) Lim f(x) = L
- -
X
-
- a
>
-
a I
⑱ f(a) = M
-
-
I
Y = a
Lim f(x) = DNE
x -
a
-
-
Ex 2 :
- -
I I .
I
I
I ↑
Lim f(x) = 1 Lim f(x) = DNE Lim f(x) I 3
X =
3 X - 6 X S 0-
Lim f(x) = DNE
X ,
0
Lim f(x) = 1 5 . Live f(x) = 3 Lim f(x) =
- 3
X >
Ot
x
-
4 X(
Algebraic :
Strategy : w = = b = real# factor
Lim -(t) c f (a) Coningate
Xia
Direct sub = 0/0= Intermedian Trig Identities
form
, factor :
Excim
~
Ex x If
99.9
: Lim -
1
I
X13
X -
3 S
Lim 1 Lim X 1
3)(x 3)
x
X
-
(x
-
Lim Lim +
-
I
x -
1 x -
1
X13
=
X13 x3 -
1 (x -
1)(x2 + x + 1)
X -
3
·
Lim
Lim x + 3 I
=
X13
= 3 + 3 = 6
&
I 1
Conjugate :
Ex :
~
Lim x + 1 -
11 -
1 8 If
- =
X = 8
X 8 G
Lim X + 1 1 Lim X + 1 1
( in In
- -
-
X = 8 X = 8
X X
( x + 11)(x + 1 + 1) Lim X Lim 1
= =
0
= X x + 1 +
x + 1 + 1 -
1
x ,
x(x + 1 1)
+
X > O
(x + 1 + 1)
1 1 1
I = I
(0 + 1 + 1) (x + 1) 2
Tris Identities :
Ex : O
1 cos
1 O If
==
Lim -
cos
-
X =
0 T
2 Sir O O
Cos
Lim 1 -
cos O Lim 1 -
cos &
X =
0 = X ·
o
2 sin
"
O (1 + cos@)(1-cosO)
-cost 1 + =
sin 1 -cos"E
~
= Lim 1
I
X O
(1 + cos@)
1 1 1
I = I
1 + cosO 1 + 1 2
, I
Limit Properties :
Lim
x - C
f(x) = g(x) =
x
Lim
- C
f(x) ! Lim
x - C
g(x)
Assumeinc a
= 1 + M
Lim f (x) · g(x) = Lim f(x) · Lim g(x) Limkf(x) = k Lim f(x)
x - C x - C x - C x - C x - C
I L .
M = kL
T
Lim(f(x))
(Lim f(x)
Lim
cf(x) =
imI
X ·
L x - C x - C
= -
M
Lim M
g(x) I L
S
X > C
Composite func
: im f(g(x) =
f(ximal
>
If : im g(x) = M exist + continue at M
Ex : Lim
s(h(x)) = ?
X 3 -
1
1 1
Lim
O 3 3
Lim
· (n(x)) = 9(h(x)) = 3
1-
-
2 2
x3 -
n(x)c j
9 h
1 1
Lim
Lim
9(h(x))
L
-
3 - 2 -
l
-
1
1 2
·
3
& 2
-
3 - 2 -
l
-
1
1 2 3
&
x . -1
I
- =
n(x)c - 2S(n(x)) = 3
⑭ 2 8 O 2
Lim
g(n(x))
- -
= 3
-
S ⑧ -
S X 3 -
1
~ ~
Limits x Piecwise function :
I
+ + 4x))
Ex : Let +(x) = firD
Lim
f(x)
12. x 2
3x 1
.
+ x
↑ (x
↓
Lim
. 2
-
3x + 1 = 7
~
im f(x) = DNE
/
Lim
2
+ x + 4 = S
x .
-
2
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