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Source: adapted from notes by Nancy Stephenson, presented by Joe Milliet at TCU AP Calculus Institute, July 2005
AP CALCULUS FORMULA LIST
Definition of e: lim 1^1 n e (^) n→∞ n = ^ +
Absolute Value: 0 0 x x^ if^ x x if x
= ^ ≥
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( ) (^ )^ (^ )^ ( ) (^ )^ (^ )
( ) (^ )^ (^ )
( ) (^ )^ (^ )
Definition of the Derivative: ' lim ' lim
' lim
' lim
x h
h
x a
f x f^ x^ x^ f^ x^ f x f^ x^ h^ f^ x x h f a f^ a^ h^ f^ a derivative at x a h f x f^ x^ f^ a alternate form x a
∆ →∞ →∞
→∞
→
∆ ∆
= +^ −^ = +^ −
= +^ − =
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( ) ( ) ( ) ( )
Definition of Continuity: is continuous at iff: (1) is defined (2) lim exists (3) lim
x c x c
f c f c f x f x f c
→ →^ =
Average Rate of Change of f (^) ( x (^) ) on (^) [ a b, (^) ] = f^ (^ bb)^ −−af^ (^ b)
[ ] ( ) ( ) ( ) ( ) ( )
Rolle's Theorem: If is continuous on , and differentiable on , and if , then there exists a number on , such that ' 0.
f a b a b f a f b c a b f c
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[ ] (^ ) ( ) ( ) (^ )^ (^ )
: Rolle's Theorem is a special case of The Mean Valu
Mean Value Theorem: If is continuous on , and differentiable on , , then there exists a number on , such that '.
Note
f a b a b c a b f c f^ b^ f^ a b a
( ) ( ) ( ) (^ )^ (^ )^ (^ )^ (^ )
e Theorem If f a = f b then f ' c = f^ ab^ −−a^ f b^ = f^ ab^ −−af^ a =0.
Source: adapted from notes by Nancy Stephenson, presented by Joe Milliet at TCU AP Calculus Institute, July 2005
[ ] ( ) ( ) ( )
Intermediate Value Theorem: If is continuous on , and is any number between and , then there is at least one number between and such that.
f a b k f a f b c a b f c =k
( )
Definition of a Critical Number: Let f be defined at c. If f ' c =0 or f ' is is undefined at c, then c is a critical number of f.
( )
First Derivative Test: Let be a critical number of the function that is continuous on an open interval containing. If is differentiable on , except possibly at , then can be
c f I c f I c f c ( ) ( ) ( ) ( ) ( ) ( )
classified as follows. 1 If ' changes from negative to positive at , then is a relative minimum of. 2 If ' changes from positive to negative at , then is a relative maximum of.
f x c f c f f x c f c f
( )
( ) ( ) ( ) ( ) ( ) ( )
Second Derivative Test: Let be a function such that ' 0 and the second derivative exists on an open interval containing. 1 If " 0, then is a relative minimum. 2 If " 0, then i
f f c c f c f c f c f c
< s a relative maximum.
Definition of Concavity: Let be differentiable on an open interval. The graph of is concave upward on if ' is increasing on the interval, and concave downward on , if ' is dec
f I f I f I f reasing on the interval.
( ) ( ) ( ) ( )
Test for Concavity: Let be a function whose second derivative exists on an open interval. 1 If " 0 for all in , then the graph of is concave upward on. 2 If " 0 for all
f I f x x I f I f x x
< in I , then the graph of f is concave downward on I.
( ( )) ( ) ( ) ( ) ( )
Definition of an Inflection Point: A function has an inflection point at , if 1 " 0 or " does not exist, and if 2 changes concavity at.
f c f c f c f c f x c
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Exponential Growth: dy dt = ky y t ( ) =Cekt
Source: adapted from notes by Nancy Stephenson, presented by Joe Milliet at TCU AP Calculus Institute, July 2005
TRIGONOMETRIC IDENTITIES
Pythagorean Identities:
sin 2 x + cos 2 x = 1 tan 2 x + 1 = sec 2 x 1 + cot 2 x =csc^2 x
Sum & Difference Identities
( ) ( ) ( )
sin sin cos cos sin cos cos cos sin sin tan tan^ tan 1 tan tan
A B A B A B A B A B A B
A B A^ B
A B
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Double Angle Identities
2 2 2 2 2 2
2
sin 2 2sin cos cos sin cos 2 1 2 sin cos 1 cos 2^ sin^1 cos 2 2 cos 1 2 2 tan 2 2 tan 1 tan
x x x x x x x x x^ x x x x x x
= ^ − = +^ = −
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Half Angle Identities
sin 1 cos^ cos 1 cos^ tan^1 cos^ sin^1 cos 2 2 2 2 2 1 cos 1 cos sin
x x x x x x x x x x x
= ± −^ = ± +^ = ± −^ = = −
Source: adapted from notes by Nancy Stephenson, presented by Joe Milliet at TCU AP Calculus Institute, July 2005
CALCULUS BC ONLY
Integration by Parts: (^) ∫ u dv = uv −∫ v du
( ) [ ] ( ) 2
Arc Length of a Function: For a function with a continuous derivative on , : : (^) a^ b 1 '
f x a b Arc Length is s = (^) ∫ + f x dx
( ) [ ] ( ) ( ) 2
Area of a Surface of Revolution: For a function with a continuous derivative on , : : (^2) a^ b 1 '
f x a b Surface Area is S = π (^) ∫ r x + f x dx
( ( )^ ( )) ( ( ) ( )) ( ( ) ( ))
( )
2 2
Parametric Equations and the Motion of an Object: Position Vector , Velocity Vector ' , ' Acceleration Vector " , "
Speed (or, magnitude of the velocity vector):
Distan
x t y t x t y t x t y t
v t dx^ dy dt dt
= ^ ^ +^
2 2 ce traveled from to is: : . Slope 1'st derivati
b a t a t b s dx^ dy dt dt dt Note The distance traveled by an object along a parametric curve is the same as the arc length of a parametric curve
= = = ^ ^ +^
∫
( ) ( ( ) ( ))
( ( ) ( ))
2 2
ve of curve at , is:
Second derivative of curve at , is:
C x t y t dy^ dy dt dx dx dt d dy C x t y t d^ y^ d^ dy^ dt^ dx dx dx dx dx dt
^
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( ) ( )
( ) ( )
L'Hôpital's Rule: lim lim ' x c x c '
f x f x → (^) g x = → g x
