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2020 AP CALCULUS BC FORMULA LIST
Definition of the derivative :
0
( ) lim h
f x h f x f x h
lim x a
f x f a f a x a
(Alternative form)
Definition of continuity : f is continuous at c if and only if
f ( c ) is defined;
lim ( ) exists; x c
f x
- lim ( ) ( ). x c
f x f c
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 ( ) c^ f^ ^ b ^^ f^ ^ a . b a
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.
1 2
n n
du d d (^) dx d x nx u dx dx (^) u dx x x
^ ^ ^ ^
^
^ ^ ^ ^ ^ ^
2
d d^ f^ x^ g^ x^ f^ x^ f^ x^ g^ x f x g x f x g x g x f x dx dx g x (^) g x
d f g x f g x g x dx
(^) ^
^
[sin ] cos [cos ] sin
d du d du u u u u dx dx dx dx
2 2 [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
[ln ] [log ] ln
a
d du d du u u dx u dx dx u a dx
[ ] [ ] ln
d (^) u u du d (^) u u du e e a a a dx dx dx dx
2 2
[arcsin ] [arccos ] 1 1
d du d du u u dx (^) u dx dx (^) u dx
2 2
[arctan ] [arc cot ] 1 1
d du d du u u dx u dx dx u dx
Definition of a definite integral:
0 1 1
lim lim
n n
x k^ k^ n k^ k k k
b
a
f x dx f x x f x x
(^) (^)
cos^ u du^ ^ sin^ u^ ^ C^ sin^ u du^ ^ cos u^ C 1 , 1 1
n n x x dx C n n
du ln u C u
^
ln
u u u u a e du e C a du C a
f^ ^ ^ g^ ^ x^ ^ ^ g^ ^ x dx ^^ ^ f^ ^ g^ ^ x^ C
Definition of a Critical Number:
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.
First Derivative Test:
Let c be a critical number of a function f that is continuous on an open interval I
containing c. If f is differentiable on the interval, except possibly at c , then f c
can be classified:
- If f x changes from negative to positive at c , then (^) c , f (^) c is a relative
minimum of f.
- If f x changes from positive to negative at c , then (^) c , f (^) c is a relative
maximum of f.
Second Derivative Test:
Let f be a function such that the second derivative of f exists on an open interval containing c.
If f ^ c 0 and f ^ c 0 , then (^) c , f (^) c is a relative minimum.
If f ^ c 0 and f ^ c 0 , then (^) c , f (^) c is a relative maximum
Definition of Concavity:
Let f be differentiable on an open interval I. The graph of f is concave upward on I if (^) f is increasing on the interval and
concave downward on I if f is decreasing on the interval.
Test for Concavity:
Let f be a function whose second derivative exists on an open interval I.
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.
Definition of an Inflection Point:
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:
b
a
f x dx f b f a
final initial + change
initial final change
b
a b
a
f b f a f x dx
f a f b f x dx
Second Fundamental Theorem of Calculus:
x
a
d f t dt f x dx
Chain Rule Version:
g x
a
d f t dt f g x g x dx
Integration by Parts:
Indefinite: udv uv vdu Definite:
b (^) b
a a
udv uv^ vdu
Order for choosing “u”
L – Logarithms
I – Inverse Trig Functions
P – Polynomials
E – Exponential Functions
T – Trig Functions
Area Under a Curve
or
or
_____________________________________________________________________
Volume
Disk Method (no hole)
A. Horizontal Axis of Rotation:
B. Vertical Axis of Rotation:
Washer Method (with hole: whole-hole)
A. Horizontal Axis of Rotation:
B. Vertical Axis of Rotation:
Cross Sections
A. Cross sections are perpendicular to x-axis
B. Cross sections are perpendicular to y-axis
Length of a Curve
or
b
a
A f x g x dx
^
b
a
A upper lower dx
d
c
A f y g y dy
d
c
A right left dy
b 2
a
V upper lower dx
d 2
c
V right left dy
2 2
b
a (^) whole hole
V upper lower upper lower dx
^
2 2
d
c (^) whole hole
V right left right left dy
^
b
a
V A x dx
d
c
V A y dy
2
1
b
a
dy L dx dx
2
1
d
c
dx L dy dy
Definition of a Taylor polynomial:
If f has n derivatives at c , then the polynomial
2 3 ... 2! 3!!
n
n f c f c f c n P x f c f c x c x c x c x c n
is called the nth Taylor polynomial for f at c****.
Test Series Converges Diverges Comment
Nth Term Test for
Divergence
This test cannot be
used to show
convergence.
Geometric Series If converges:
P-Series
Alternate Series Converges if :
terms are decreasing
Remainder:
Ratio No conclusion if:
1
n n
a
lim (^) n 0 n
a
1
n
n
ar
r 1 r 1
1
1
a S r
1
p n n
p 1 p 1
1
1
n n n
a
^
an 0
lim (^) n 0 n
a
Rn an (^) 1
1
n n
a
1 lim 1
n n n
a
a
1 lim 1
n n n
a
a
1 lim 1
n n n
a
a
