x a
8. Continuity at a if lim f  x   f  a   lim f  x 
x a
9
7
10
y
6
TYPES OF DISCONTINUITY
7
11
y
4
5
4
4
3
-6
3
2
-5
-4
-3
-2
-1
2
1
1
2
3
1
x
4
5
-1
1
Point or removable
2
3
Jump
5
2
y
1
1
-1
-2
2
x
3
-2
-1
x
1
2
-1
-3
-4
-5
x
4
12
y
3
2
1
6
5
-1
x a
-2
Infinite
Oscillating
Basic Rules
 constant 
d
d
d
1.
c  0  Constant  2.
c  f ( x)  c
f ( x) 

dx
dx
dx
 multiple 
d n
d
d
d  Sum & 
3.
x  nx n 1  power  4.
u  v   u  v 

dx
dx
dx
dx  Difference 
d
d
v
uu
v
d
d
d
d u 
dx
dx  Quotient 
5.
(uv )  v
uu
v  Product  6.

 
2
dx
dx
dx
dx  v 
v
1
7.  f 1   x  
Inver
se


f   f 1  x  
d
d
d
8.
f  g  x  
f  u  g  x  where u  g  x   Quotie nt 
dx
dx
dx
WHERE THE DERIVATIVE DOES NOT EXIST
26
27
y
7
28
y
6
2
4
1
2
x
1
2
Corner
-1
Discontinuity
1
2
3
4
-1
x
5
Cusp
y
1
-1
1
3
-1
29
y
2
5
1
-2
WWW.MATHGOTSERVED.COM
Exponential and Logarithm ic Functions
d
1 du
d
1 du
20.
ln u 
21.
log a u 
dx
u dx
dx
u ln a dx
d u
du
d u
du
22.
e  eu
23.
a  a u ln a
dx
dx
dx
dx
d
1 du
d
u du
24.
u 
25.
u 
dx
dx
u dx
2 u dx
2
Linearization: L x  f  a  f  a x  a
x
-1
x
1
1
-2
Vertical tangent
ARITHM ETIC OF INFINITY
2. n     
1.      2. n     
  
  
3.   0  
4.0     
3. 0    Ind.


1.  /   Ind. 


1.     Ind.
2. n /   0




2. n    
3. / n  


3.   n  
4. n / 0  
  
  



5.
n
/
0


4. n  n   0    1

 



6.
n
/
0


1


5. n   n  0  
7.
0
/
0
=
Ind.



1.  1  sin     1 
1.    
2.  n   

 Trig
3.  0  Ind
4. 0   0 
2.  1  cos     1 
 power
5. n   
6. 1  Ind 

7. 0 0  Ind.

1.     
LIMITS(1)
sin x
sin x
sin 2 x
1. lim
1
2. lim
 0 3. lim
0
x0
x 
x0
x
x
x
sin  ax 
sin  ax 
 bx   a
cos x  1
4. lim
 0 5. lim
 lim
 lim
x0
x

0
x

0
x

0
x
sin  bx 
sin  ax  b
 bx 
6.lim f  x   L  exists  If and only if lim f  x   lim f  x   L
x a
xa
xa
7. f  x  is cont a if lim f  x   f  a 
Trigonom etric F unctions
d
du
d
du
8.
sin u  cos u
9.
cos u   sin u
dx
dx
dx
dx
d
du
d
du
2
2
10.
tan u  sec u
11.
cot u   csc u
dx
dx
dx
dx
d
du
d
du
12.
sec u  sec u tan u
13.
csc u   csc u cot u
dx
dx
dx
dx
Inverse Trigonom etric F unctions
d
1
du
d
 1 du
14.
sin  1 u 
15.
cos  1 u 
2 dx
dx
dx
1 u
1  u 2 dx
d
1 du
d
 1 du
1
1
16.
tan u 
17.
cot u 
dx
1  u 2 dx
dx
1  u 2 dx
d
1
du
d
1
du
18.
sec  1 u 
19.
csc  1 u 
2
2
dx
dx
u u  1 dx
u u  1 dx
30.Mean Value Theorem: If f is cont on  a, b on and diff on  a, b 
f  b  f  a
 exist a c   a, b  s.t. f   c  
ba
f b  f  a
31.Rolle's Theorem : MVT where f   c  
0
ba
32. IntermediateValueTheorem. : If f is cont on  a, b and d   f  a  , f  b  
then there is a c   a, b st f  c   d
Definition of Derivative
f  x  h  f  x
33. f   x   lim
h 0
h
f  x  f a
34. f   a   lim
x a
xa
DERITAVITIVES(2)
LIMIT LAWS
35. f   x   Average rate of change
= Slope of Secant line

f  b  f  a 
ba
11. LRAMn = w f  x1   f  x2  ..  f  xn1   or w1 f  x1   w2 f  x2  ..  wn1 f  xn1  1. xn dx  xn1  C 2. 1 dx  ln x  C 3. 1 dx  ln ax  b  C 4. ekx dx  ekx  C

x
 ax  b

n 1
a
k
12. RRAMn = w f  x2   f  x3   f  xn   or w1 f  x2   w2 f  x2  ..  wn1 f  xn 
au
cos kx
sin kx
x
5.
a
dx


C
6.
sin
kx
dx



C
7.
cos
kx
dx


C



  x x   x x 
 x  x 
ln a
k
k
13. MRAMn = w f  1 2   f  2 3   f  n1 n  or
2
8.
sec
x
tan
x
dx

sec
x

C
9.
sec
x
dx

tan
x

C
10. csc x cot x dx   csc x  C


2
2
2








 x2  x3 
 xn1  xn 
 x1  x2 
w1 f 
 w2 f  2  wn1 f  2 
1 b
 2 




15. Average Value of f av f   fave 
f  x dx
a
b a
b

a
Note: w
and applies only for equal sub intervals
n
b
x
w
1
16. FTC I:  f   x  dx  f  b   f  a  17 FTC II i   f  t  dt  F  x 
14. T=
 y1 2y2 2yn1  yn  or  w1  y1  y2   w2  y2  y3  ...
n
a
a
2
2
APPROXIMATING AREA

20. Area =
  f  x
xright
xleft
21. Vol of rev =

xright
xleft

top
 f  x down  dx or Area =


ytop
ydown
 f  y   f  y   dy
left
right 


 f  x  a   f  x   a dx Vol Cross sect.= A x dx

top
down


 
2
2
b
a
ii 
d
d
f  t  dt  f  x  iii )
f  t  dt  f  g  x    g   x   h  g  x    h   x 
dx a
dx hx 
x
g x
18.Integration by parts:  vdu  uv   vdu
19.Integration by substitution:
use LIPET to select u
 f  g  x   g   x    f  u  du
INTEGRALS (3)
ANTIDIFFERENTION (INTEGRATION) RULES
1. Derivative of f at
a:
2. Critical Number
c
Verbal Description
Symbolic
The instantaneous rate of change
of the function at a or the slope
of the tangent line at a
f a 
 lim
h 0
df
dx
Graphical
y
f  a  h  f a 
xa
a
h
c   a, b  where
A number c in an open (a, b)
interval where the derivative is
zero or does not exist
x
f  c   0
f   c   0 or f   c  DNE
f   c  DNE
y
y
-1
x
1
-2
2
y
6
5
4
1
1
-2
7
2
2
3
-1
Corner
2
x
1
1
2
-1
1
Discontinuity
y
2
-1
1
-1
4. Concavity Test
f’: Is a point where f’ changes from increasing to
decreasing or decreasing to increasing
f’’: Is a point where f’’ changes from positive
to negative or negative to positive
Motion definitions and Equations
s t   x b   x  a  
v t   s t 
8. Velocity: A Vector quantity that
represents the rate of change of position
represents the rate of change of velocity
Reciprocal
1
csc x
1
cos x 
sec x
1
tan x 
cot x
1
sin x
1
sec x 
cos x
1
cot x 
tan x
sin x 
csc x 
0
sin x
0
cos x
1
tan x
0
csc x
Und.
1
2
3
1
2
sec x
1
2
cot x
Und.
2
2
3
3
3
2
2
1
2
3
2
2
2
2
1
2
2
3
1
2
2
3
2
2
1
3
3
Und.
 
min

f  c
x
 
x
-2
max

f  c
c
 

y
y
f   x   0  f  x  is CU
b) If f   c   0 on I
Concave Up
 f  x  is CD on I
Concave Down
x
x
y
f ' s from  to  or  to 
f  x ' s from + to  or  to
Point of Inflection
c
x
f   c  ' s from + to -
 f  c is a POI
 v t 
b
a
x
f   x   0  f  x  is CU
f ' s from CU to CD or CD to CU
d  t   x b  x a   v t  dt
7. Distance: A scalar quantity that
represents total movement regardless of
sign
b
a
Speed  v  t 
11. Given initial position s  a   C the final position is
given by s  b   s  a    s   t  dt
b
a
is given by
Sine Curve
Cosine Curve
y
y
1
1
-2π
-3π/2
-π
-π/2
π/2
π
3π/2
2π
x
-2π
-3π/2
-π
-π/2
π/2
π
3π/2
2π
x
-1
-1
y
y
y
1
0
f  c
c
 f  x  is CU on I
Quotient
Pythogorean
sin x sin 2 x  cos 2 x  1
tan x 
cos x
tan 2 x  1  sec 2 x
cos x
cot x 
cot 2 x  1  csc2 x
sin x
4

1
y
b)
9. Speed: A scalar quantity that
represents the rate of covering distance
  30    45    60    90 
6
a) If f   c   0 on I
a  t   v   t   s   t 
10. Acceleration: A vector quantity that
 
 f '  c  is a max
f: Is a point where the concavity of f changes
6. Displacement: A Vector quantity that
represents the net change in position
f  c
b) If f   c   ' s from +to 
a) If the second derivative is positive on
an interval I then the function is
Concave Up on I
b) If the second derivative is negative
on an interval I the function is
Councave down on I
5. Point of
Inflection at c
a)
2
Vertical tangent
y
 f '  c  is a min
x
LIMITS (1)
3. First Derivative
Test
5
-1
x
1
Cusp
a) If f   c   ' s from  to 
4
1
2
a) If the first derivative changes from
negative to positive at c then the
function has a relative minimum at c
b) If the first derivative changes from
positive to negative at c then the
function has a relative maximum at c
3
y
VOCABULARY (4)
Term
y
x
x
x
Linear y  x
x
quadratic y  x 2
y
Cubic y  x 3
Radical y 
y
y
y
x
1
Und.
0
x
x
x
Logarithmic y  ln x
Exponential y  e x Absolute value y  x
AP Calculus Survival Kit
x
circular y  9  x 2