Quiz yourself on definitions and theorems for AP Calculus AB

1. AP Calculus AB Definition
and Theorems
BINGO

2. lim ( ) lim ( )
x a x a
f x f x 
 

We say the ______ exists if:

3. lim ( ) lim ( )
x a x a
f x f x 
 

We say the limit exists if:

4. if f is continuous on [a, b] , then there are x1 , x2 such
that and1 2,a x x b 
1 2) ( ) and ( ) ( ),for all [( , ]f x f x ff x ax x b  
What theorem is this?

5. if f is continuous on [a, b] , then there are x1 , x2 such
that and1 2,a x x b 
1 2) ( ) and ( ) ( ),for all [( , ]f x f x ff x ax x b  
Extreme Value Theorem

6. )lim ( ) exists, but ( ) does not or (lim ( )
x a x a
f x f a f x f a
 

_________ occurs at x = a if

7. )lim ( ) exists, but ( ) does not or (lim ( )
x a x a
f x f a f x f a
 

A removable discontinuity occurs at x = a if

8. Suppose g (f (x)) = x and f is differentiable.
If '( ( )) 0, thenf g a 
1
'( )
( ( ))
g a
f g a


What theorem is this?

9. Suppose g (f (x)) = x and f is differentiable.
If '( ( )) 0, thenf g a 
1
'( )
( ( ))
g a
f g a


Inverse Function Theorem

10. ( ( )) ( ( ))f g x g f x x 
g is the _________ of f is defined if

11. ( ( )) ( ( ))f g x g f x x 
g is the inverse of f is defined if

12. ( ) ( )f b f a
b a


the _________ of f on [a, b] is defined to be

13. ( ) ( )f b f a
b a


the average rate of change of f on [a, b] is defined to be

14. 1
( )
b
a
f x dx
b a 
the _________ of f on [a, b] is defined to be

15. 1
( )
b
a
f x dx
b a 
the average value of f on [a, b] is defined to be

16. lim ( ) exists
( ) exists
lim ( ) = ( )
x a
x a
f x
f a
f x f a


A function is ______ if:

17. lim ( ) exists
( ) exists
lim ( ) = ( )
x a
x a
f x
f a
f x f a


A function is continuous if:

18. '( ) 0f x 
A function is ______ if:

19. '( ) 0f x 
A function is increasing if:

20. '( ) 0f x 
A function is ______ if:

21. '( ) 0f x 
A function is decreasing if:

22. ''( ) 0f x 
A function is ______ if:

23. ''( ) 0f x 
A function is concave down if:

24. if f is continuous on [a, b] there is a
(
1
( ) )
b
a
f c f x dx
b a

 
What theorem is this?

25. if f is continuous on [a, b] there is a
(
1
( ) )
b
a
f c f x dx
b a

 
Mean Value Theorem for Integrals

26. lim ( ) DNE
x a
f x

_________ occurs at x = a if

27. lim ( ) DNE
x a
f x

A nonremovable discontinuity occurs at x = a
if

28. ( ) ( ) ( )
b
a
f x dx F b F a 
if a function f is continuous and '( ) ( )F x f x
Theorem
What theorem is this?

29. ( ) ( ) ( )
b
a
f x dx F b F a 
if a function f is continuous and '( ) ( )F x f x
Fundamental Theorem of Calculus I

30. 0
( ) (
l
)
im
h
f x h f x
h
 
_________ is defined to be

31. 0
( ) (
l
)
im
h
f x h f x
h
 
The derivative of f is defined to be

32. " changes signf
_________ occurs at a point at which

33. " changes signf
An inflection point of f occurs at a point
at which

34. ' 0 orf DNE
_________ occurs at a point at which

35. ' 0 orf DNE
A critical point of f occurs at a point at
which

36. ' changes from + tof 
________occurs at a point at which

37. ' changes from + tof 
A relative max of f occurs at a point at
which

38. ' changes from tof  
_________ occurs at a point at which

39. ' changes from tof  
A relative minimum of f occurs at a
point at which

40. ( ) '( )( )y f a f a x a  
_________ to f at x = a has an equation

41. ( ) '( )( )y f a f a x a  
A tangent line to f at x = a has an
equation

42. ( ) ( )
x
a
d
f t dt f x
dx
 
 
 

if a function f is continuous
Theorem
What theorem is this?

43. ( ) ( )
x
a
d
f t dt f x
dx
 
 
 

if a function f is continuous
Theorem
The Fundamental Theorem of Calculus II

44. an _____________ of f is a function F such that
'( ) ( )F x f x

45. an antiderivative of f is a function F such that
'( ) ( )F x f x

46. the _____________ is defined to be
1
lim
n
n
k
b a b
f a k
a
n n

     
   
   

 
 


47. the integral of f from a to b is defined to be
1
lim
n
n
k
b a b
f a k
a
n n

     
   
   

 
 


48. if f is continuous on [a, b] and differentiable on (a, b)
there is a
( ) ( )
( , ) such that '( )
f b f a
a b f c
b a
c

 

What theorem is this?

49. if f is continuous on [a, b] and differentiable on (a, b)
there is a
( ) ( )
( , ) such that '( )
f b f a
a b f c
b a
c

 

The Mean Value Theorem for Derivatives

50. if f is continuous on [a, b] , for every M between
( ) and ( )f a f b
]the ure his a [ , s c t (hat )ac b f c M 
Theorem
What theorem is this?

51. if f is continuous on [a, b] , for every M between
( ) and ( )f a f b
]the ure his a [ , s c t (hat )ac b f c M 
Theorem
Intermediate Value Theorem

52. "( ) 0f x 
A function is ______ if:

53. "( ) 0f x 
A function is concave up if:

54. lim ( )
x a
f x

 
_________ occurs at x = a where a is

55. lim ( )
x a
f x

 
A vertical asymptote occurs at x = a
where a is

56. lim ( ) or lim ( )
x x
f x f x
 
_________ occurs if
is not + or - infinity

57. lim ( ) or lim ( )
x x
f x f x
 
A horizontal asymptote occurs if
is not + or - infinity