AP CALCULUS BC
Final Notes
Trigonometric Formulas
1.
2.
3.
4.
5.
6.
7.
8.
9.
sin   cos   1
1  tan 2   sec 2 
1  cot 2   csc 2 
sin(  )   sin 
cos(  )  cos
tan(  )   tan 
sin( A  B)  sin A cos B  sin B cos A
sin( A  B)  sin A cos B  sin B cos A
cos( A  B)  cos A cos B  sin A sin B
2
2
10. cos( A  B)  cos A cos B  sin A sin B
11.
sin 2  2 sin  cos
12. cos 2  cos
2
  sin 2   2 cos 2   1  1  2 sin 2 
sin 
1

cos  cot 
cos 
1

14. cot  
sin  tan 
1
15. sec  
cos 
1
16. csc  
sin 
13. tan  
17. cos(
18. sin(

2

2
  )  sin 
  )  cos 
Differentiation Formulas
1.
2.
3.
4.
5.
6.
7.
8.
9.
d n
( x )  nx n 1
dx
d
( fg )  fg   gf 
dx
d f
gf   f g 
( )
dx g
g2
d
f ( g ( x))  f ( g ( x)) g ( x)
dx
d
(sin x)  cos x
dx
d
(cos x)   sin x
dx
d
(tan x)  sec 2 x
dx
d
(cot x)   csc 2 x
dx
d
(sec x)  sec x tan x
dx
10.
11.
12.
13.
14.
15.
16.
17.
d
(csc x)   csc x cot x
dx
d x
(e )  e x
dx
d x
(a )  a x ln a
dx
d
1
(ln x ) 
dx
x
d
1
( Arc sin x) 
dx
1 x2
d
1
( Arc tan x) 
dx
1 x2
d
1
( Arc sec x) 
dx
| x | x2 1
dy dy du


Chain Rule
dx dx dx
Integration Formulas
1.
 a dx  ax  C
2.
n
 x dx 
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
1
 x dx  ln x  C
 e dx  e  C
x
x
ax
 a dx  ln a  C
 ln x dx  x ln x  x  C
x
 sin x dx   cos x  C
 cos x dx  sin x  C
 tan x dx  ln sec x  C or  ln cos x  C
 cot x dx  ln sin x  C
 sec x dx  ln sec x  tan x  C
 csc x dx  ln csc x  cot x  C   ln csc x  cot x  C
 sec x d x  tan x  C
 sec x tan x dx  sec x  C
 csc x dx   cot x  C
 csc x cot x dx   csc x  C
 tan x dx  tan x  x  C
2
2
2
18.
a
19.

20.
x n 1
 C , n  1
n 1
x
dx
1
x
 Arc tan    C
2
a
x
a
dx
 x
 Arc sin    C
a
a2  x2
2
dx
x2  a2

x
1
1
a
Arc sec  C  Arc cos  C
a
a
a
x
1.
Formulas and Theorems
Limits and Continuity:
A function y  f (x) is continuous at x  a if
i).
f(a) exists
ii).
lim f ( x) exists
iii).
xa
lim  f (a)
xa
Otherwise, f is discontinuous at x = 1.
The limit lim f  x  exists if and only if both corresponding one-sided limits exist and are equal –
x a
that is,
lim f  x   L  lim f  x   L  lim f  x 
x a
2.
3.
x a
x a
Even and Odd Functions
1.
A function y  f (x) is even if f ( x)  f ( x) for every x in the function’s domain.
Every even function is symmetric about the y-axis.
2.
A function y  f (x) is odd if f ( x)   f ( x) for every x in the function’s domain.
Every odd function is symmetric about the origin.
Periodicity
A function f (x ) is periodic with period p ( p  0) if f ( x  p )  f ( x) for every value of
x.
Note: The period of the function y  A sin( Bx  C ) or y  A cos( Bx  C ) is
The amplitude is
4.
A . The period of y  tan x is  .
Intermediate-Value Theorem
A function y  f (x) that is continuous on a closed interval
between f (a ) and f (b) .
Note: If f is continuous on
5.
2
.
B
a, b takes on every value
a, b and
f (a ) and f (b ) differ in sign, then the equation
f ( x)  0 has at least one solution in the open interval (a, b) .
Limits of Rational Functions as x  
f ( x)
lim
 0 if the degree of f ( x)  the degree of g ( x)
i).
x   g ( x)
x 2  2x
Example: lim
0
x   x3  3
ii).
iii).
f ( x)
is infinite if the degrees of f ( x)  the degree of g ( x)
x   g ( x )
x3  2x
Example: lim

x   x2  8
f ( x)
lim
is finite if the degree of f ( x)  the degree of g ( x)
x   g ( x )
lim
2 x 2  3x  2
2
Example: lim

5
x   10 x  5 x 2
6.
Horizontal and Vertical Asymptotes
1.
A line y  b is a horizontal asymptote of the graph y  f (x) if either
2.
7.
 0 0  and x1, y1  are points on the graph of
Average and Instantaneous Rate of Change
i).
Average Rate of Change: If x , y
ii).
8.
lim f ( x)  b or lim f ( x)  b .
x
x  
A line x  a is a vertical asymptote of the graph y  f (x) if either
lim f ( x)   or lim   .
x  a
x  a-
y  f (x) , then the average rate of change of y with respect to x over the interval
x0 , x1  is f ( x1 )  f ( x0 )  y1  y0  y .
x1  x0
x1  x0 x
Instantaneous Rate of Change: If x0 , y 0  is a point on the graph of y  f (x) , then
the instantaneous rate of change of y with respect to x at x 0 is f ( x0 ) .
Definition of Derivative
f ( x)  lim
h0
9.
The latter definition of the derivative is the instantaneous rate of change of f  x  with respect to
x at x = a.
Geometrically, the derivative of a function at a point is the slope of the tangent line to the graph of
the function at that point.
The Number e as a limit
i).
ii).
10.
11.
a, b and differentiable on a, b such that f (a) 
is at least one number c in the open interval a, b such that f (c)  0 .
Mean Value Theorem
If f is continuous on
14.
a, b such that
f (b) , then there
a, b and differentiable on a, b , then there is at least one number
c
f (b)  f (a )
 f (c) .
ba
Extreme-Value Theorem
If f is continuous on a closed interval
on
13.
n
 1
lim 1    e
n   n 
1
 n
lim 1   n  e
n  0 1 
Rolle’s Theorem
If f is continuous on
in
12.
f ( x  h)  f ( x )
f  x   f a 
of f '  a   lim
x a
x a
h
a, b.
a, b,
then f (x ) has both a maximum and minimum
To find the maximum and minimum values of a function y  f (x) , locate
1.
the points where f (x ) is zero or where f (x ) fails to exist.
2.
the end points, if any, on the domain of f (x ) .
Note: These are the only candidates for the value of x where f (x ) may have a maximum or a
minimum.
Let f be differentiable for a  x  b and continuous for a a  x  b ,
1.
2.
x in a, b , then f is increasing on a, b .
If f ( x )  0 for every x in a, b , then f is decreasing on a, b .
If f ( x )  0 for every