Complete Course Packet

Calculus AB
Derivatives & Integral Rules
Properties:
d
 c  f   c  f 
 f  g   f   g 
c  0
dx
d
f  x  dx  f  x 
 c  f  x  dx  c f  x  dx
dx 
  f  x   g  x  dx   f  x  dx   g  x  dx


Derivative Rules:
Product:
 f  g  
f  g  g  f 
Quotient:
 f  g  f   f  g 
  
g2
g
Chain Rule (Composition of functions):
dy dy du
d
du
d

 , or
f  u    f   u   , or
f  g  x   f   g  x   g   x 

dx du dx
dx
dx
dx
Integration: Use Algebra, Trigonometry, or u-substitution to get the integral into one of the
patterns below.
Derivatives & Integrals:
Chain rule
Integral(Antiderivative)
u n 1
d n
du
n
u
du

c
u  n  u n 1

n 1
dx
dx


 
d
du
 sin u   cos u
dx
dx
 cos u du  sin u  c
d
du
 cos u    sin u
dx
dx
 sin u du   cos u  c
d
du
 tan u   sec2 u
dx
dx
 sec
d
du
 cot u    csc2 u
dx
dx
 csc
d
du
 sec u   sec u tan u
dx
dx
 sec u tan u du  sec u  c
d
du
 csc u    csc u cot u
dx
dx
 csc u cot u du   csc u  c
 
2
2
u du  tan u  c
u du   cot u  c
d u
du
e  eu
dx
dx
 e du  e
d
1 du
ln u  

dx
u dx

u
u
c
du
 ln u  c
u
d
1 du
log a u  

dx
u ln a dx
 
d u
du
a  au ln a
dx
dx












u
 a du 
d
1 du
sin 1 u 
dx
1  u 2 dx
au
c
ln a

u
 sin 1    c
a
a u
a
2
du
2
2
d
1 du
cos 1 u  
dx
1  u 2 dx
d
1 du
tan 1 u 
dx
1  u 2 dx
du
1
u
 tan 1    c
2
u
a
a
d
1 du
cot 1 u  
dx
1  u 2 dx
d
1
du
sec1 u 
2
dx
u u  1 dx
u
du
u a
2
2

1 1 u
sec
c
a
a
d
1
du
csc1 u  
2
dx
u u  1 dx
 tan u du  ln sec u  c
 cot u du  ln sin u  c
 sec u du  ln sec u  tan u  c
`
 csc u du  ln csc u  cot u  c
Calculus AB often used formulas
1) Calculator in Radian Mode
2) Continuity and Differentiability:

g  x , x  a
then
f  x  

h  x  , x  a
?

g  x  , x  a
f  x  

h  x  , x  a
?
Continuity: lim g  x   h  a   lim h  x 
x a
x a
?
Differentiability: Continuity plus lim g   x   lim h  x 
x a
b
3) Area   height dx ,
x a
b
Volume   Area dx
a
a
4) Relative Max/Min of k  x   k   x   0 or und (critical points)
Max  k  changes from  to 
Min  k  changes from  to 
5) Absolute Max/Min (max/min) of k  x  @ critical points or endpoints
6) Points of inflection of k  x   k   x   0 or und and k  changes sign
b
7) Average value: f ave 
1
f  x  dx
b  a a
Application of average value: vave
8) Position-Velocity-Acceleration:
b
s b  s  a 
1

v  t  dt 

ba a
ba
v  t   s  t 

a  t   v  t   s  t  
x
9) Initial condition formula:  x0 , y0  , y  x   y0   y  t  dt
x0
t
Application  t0 , s0  , s  t   s0   v  x  dx
t0
 v t  dt  s t   c
 a t  dt  v t   c
10) Distance- Displacement:
b
Dis tan ce   v  t  dt  area
a
b
Displacement   v  t  dt  net signed area
a
11) Fundamental Theorem of Calculus
b
 f   x  dx  f  b   f  a 
a
d
dx
f  x
 h  t  dt  h  f  x    f   x 
a
12) Given a graph of a function f,
x
and g  x    f  t  dt , then
a
g   x   f  x  make sign line here
g   x   f   x  make sign line here
dV
dr
, given
and V   r 2 h
dt
dt
Find connection between r and h and substitute ( in cones look for similar triangles).
Differentiate with respect to t. Don't forget the chain rule.
V   r 2 h, h  2r
13) Related rates: find
Example: V   r 2  2r   2 r 3
dV
dr 

 2  3r 2 
dt
dt 

dy
 x 2  y  1
dx
Solve: find y - separate variables, integrate both sides, only one c is necessary, use initial
condition. If 2 branches, use the branch the contains the point given (initial value)
14) Slope fields: Slope field given-
Trigonometry Functions
sin  
Reciprocal functions
1
csc
Functions in terms of sin & cos tan  
sec x 
cos  
sin 
cos 
1
sec
tan  
cot  
1
cot 
cos 
sin 
csc x 
1
sin x
1
cos x
Cofunctions: sine – cosine, etc




sin      cos  , cos      sin 
2

2

Negative functions
sin      sin  (odd function)




csc      sec  , sec      csc 
2

2

cos     cos  (even function)
tan      tan  (odd function)




tan      cot  , cot      tan 
2

2

Pythagorean Identities
sin2   cos2   1
divide by sin2   1  cot 2   csc2 
divide by cos2   tan 2  1  sec2 
Cosine: Sum / Difference
 Cosine double angle
cos      cos  cos  sin  sin   cos 2  cos      cos2   sin 2 
Cosine
Double angle  Power reduction  Half angle

1  cos 2
1  cos 2
substitute for cos2   cos 2  1  2sin 2   sin 2  
 sin   
2
2
1  cos 2
substitute for sin 2   cos 2  2cos2   1  cos2  

2
1  cos 2
cos   
2
Sine: Sum / Difference
Sine double angle

sin 2  sin      2sin  cos 
sin(   )  sin  cos   sin  cos  
2
2
1.5
1
0.5
3
2 2 1
1 2
2
1.5
1
fx = sinx
1
3
2
gx = cosx
2
2 1
2
0.5
0
0
-0.5
-1
  
6 4 3
1

2
2
3
4
5
6
  
6 4 3
1
0
-0.5
-1
-1.5
-1.5
-2
-2
"  " is a special # and   3.142
0

2
2
3
4
5
6
Exponential and Logarithmic Functions
Exponential and logarithmic with the same base are inverse functions (reflect over the y  x
line)
Exponential function: f  x   a x , Domain : x , Range : y  0
lim a x  0, lim a x  
x 
x 
Properties of Exponential functions:
4.5
an  am  an  am
a n  a m  a nm
a   a
a b   a
n m
an 
n
3.5
3
nm
n m k
fx = ex
kn km
b
2.5
2
1
an
a a
limxf is +
4
1.5
0,1
1
1
n
limx-f = 0
a0  1
-3
0.5
-2
-1
1
2
3
4
Logarithmic function: f  x   log n a, Domain : x  0, Range : y 
lim logn a  , limlogn a  
x 0
x
Properties of Logarithmic functions:
y  log a x  x  a y
2.5
log a  nm   log a n  log a m
n
log a    log a n  log a m
m
log a n m  m log a n
log n m 
log a m
log a n
limxf is +
2
1.5
gx = lnx
1
0.5
1
2
3
1, 0
-0.5
log10 n  log n
-1
log e n  ln n
-1.5
log n 1  0; ln e  1; log10  1
limx0 f = -
-2
Note: log x is understood to be log10 x and ln x is log e x
“e” is special base and e 2.718
4
5
6
7