2020 AP CALCULUS AB FORMULA LIST

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Definition of the derivative:

   

0

( ) lim

h

f x h f x

fx h







     

lim

xa

f x f a

fa xa







(Alternative form)

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Definition of continuity: f is continuous at c if and only if

1) f (c) is defined;

2)

lim ( ) exists;

xc

fx



3)

lim ( ) ( ).

xc

f x f c



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Mean Value Theorem: If f is continuous on [a, b] and differentiable on (a, b), then there

exists a number c on (a, b) such that

   

( ) .

f b f a

fc ba





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Intermediate Value Theorem: If f is continuous on [a, b] and k is any number between f (a)

and f (b), then there is at least one number c between a and b such that f (c) = k.

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1

2

11

2

nn du

d d d

dx

x nx u

dx dx dx x x

u





   

 

 

             

 

       

 

2

f x g x f x f x g x

dd

f x g x f x g x g x f x

dx dx g x gx

df g x f g x g x

dx







  







 

 

[sin ] cos [cos ] sin

d du d du

u u u u

dx dx dx dx

  

22

[tan ] sec [cot ] csc

d du d du

u u u u

dx dx dx dx

  

[sec ] sec tan [csc ] csc cot

d du d du

u u u u u u

dx dx dx dx

  

11

[ln ] [log ] ln

a

d du d du

uu

dx u dx dx u a dx



[ ] [ ] ln

u u u u

d du d du

e e a a a

dx dx dx dx



22

11

[arcsin ] [arccos ]

11

d du d du

uu

dx dx dx dx

uu

  



22

11

[arctan ] [arccot ]

11

d du d du

uu

dx u dx dx u dx

  



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Definition of a definite integral:

 

       

011

lim lim

nn

xn

k k k k

kk

b

af x dx f x x f x x

  



     





AP Calculus AB Formula List, Cheat Sheet of Calculus

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  ^ ^ ^  ^ 

 ^ 

 ^ ^  ^ ^ ^ ^ 

Definition of a definite integral:          

 cos^ u du^ ^ sin^ u^ ^ C^ sin^ u du^  ^ cos u^  C

 ^ 

 f^ ^ ^ g^ ^ x^ ^ ^ g^ ^ x dx ^^ ^ f^ ^ g^ ^ x^  C

Let f be defined at c. If f ^   c 0 or if f is undefined at c , then c is a critical number of f.

containing c. If f is differentiable on the interval, except possibly at c , then f   c

1) If f  x changes from negative to positive at c , then  c , f   c is a relative

2) If f  x changes from positive to negative at c , then  c , f   c is a relative

1) If f  ^ c  0 and f ^   c  0 , then  c , f   c is a relative minimum.

2) If f ^   c  0 and f ^   c  0 , then  c , f   c is a relative maximum

1) If f ^  x   0 for all x in I , then the graph of f is concave upward in I.

2) If f   x   0 for all x in I , then the graph of f is concave downward in I.

A function f has an inflection point at  c , f   c 

1) if f    c 0 or f   c does not exist and

2) if f  changes sign from positive to negative or negative to positive at x  c

OR if f  x changes from increasing to decreasing or decreasing to increasing at x = c.

First Fundamental Theorem of Calculus:      

Second Fundamental Theorem of Calculus:    

Chain Rule Version:  