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AP Calculus
Formula Sheet
Michael Florip
Derivatives & Limits
Definition of Derivative
Derivative Rules
(,;711x) 1
'° cos X ( c.o,, -x )'-=. -S , n X
(to.nx)'-= c;ec-2-x
( c.ot x )' ~ - c1c"';><.
(~e_c_ XJ1 = ',el 'X 1n'1)<
(C~( x.)'= _ (.~C 1- C.o + X
Inverse Trig Derivatives
Limits
lt14 ~X.:c,() i.-,«>
UM x,o'° )n X = ... 114
(In i<)' <: ~
(\ n u )'= -b_ · J I (o~ x ')l =. i . _L ~ .JO. 'I X \na
( \o,,LL )' ~ ~. LI I. _!___ \A \n°'
d c~f '( l4) ~ cit -DO< ,c< ~
( \A.)' 4 I a = a · \l ·ln o..
Derivative of Absolute Value
-r6)-= H f'(k):! *
©Michael Florip 2021
S sin x dp. -c.o<,)( +C
r Co$ X <Ax = ~~ (\ X + C
f fan x ~ ~ " - I o \ co'> x \ -+ c.
Integrals
s C, s C )( ,!.x "- -I~ I C.SlX-1-Cd\-X l t C
S,; e c. x J. )( = \ a \ s e.c. x t f-an -x l -t C.
Other Integrals
j ¼ Jx ~ \(\ \ l( I + C
) e,1' J~ : ex + C
Michael Flori p 2021
Trig Rules & Identities
_eythagorean Identities
s iol0-t c<>s~e == \
tani e --r}.:: se (., 2g
\-+c.e f le ~ csc.20
Double Angle Formulas
ce~(!..e)= cos 2& - 5jo2.e
~ 2 U)~2e - (
::: l - 2sio'2.e
tan(w} 2-tone l-inn'"b
Half Angle Identities
si~1a. I (1-cos(ze~
cos19 e ½ ( \-1-C.OJ (ze~
t~n2e = \- cos(2e) \ ~ Ce4i (?.e)
Sum & Difference Formulas
5tn(A ±B)= ~inA C,f)jB-t C9f A ~ir1 a cos(A±S)= cosAc.osB =t ~in/t 4io B
i l~1B'\= +a~ A ± -\r.t1B ()f\ ~ '/ l + -fun A t-n.B
Product to Sum Formulas
~in A ~io B =-~ [ CoS(,H~) - c.os( A~ B]
co~ Acor8==½ l ~s(II-B)-1-C•5(Ati)]
~io Ac.~ Ba 4 [~in (11+~) •Hlo (1t -s~
Co5A sin B= ~ Gi11 (,n 8)-siD (A-B~
Sum to Product Formulas
©Michael Flori p 2021
Volume & Parametric Formulas
Volume of a Circular Disk
Around y-axis
V \l I t ~i,~ o-\" ,o-htion Volume of a Cylindrical Shell 4- l°'-"'(). C. ~ 0
V = 2 < j x . V(_ll) J.x
\J: 211 :I y-t{y) 'Y
Volume of a Cross Section
Arc Length Formula
Parametric Formulas
(! (.0.1\n ot it\ttrz,Gt,t i1~elf e-r. ce~ t "r enJ.paitrl-5
©Michael Florip 2021
Converge-Diverge Tests
P-Series Test
i ._!_ xr
it p > l, conver3es
if f ~ I, Ji\le r,es
N-th Term Test
z:~" ;f liM a J:0 dh/tr~tS
n➔«' () / J
It li/\1 a == 0 intof\ chi Si ve (\➔ ti) ft I
Ratio Test
hM j QM~ = L n~otJ a o
.. f L ) t, J i\le.r j (L ~
jf L::-l, illlaftcl1.4sive..
Geometric Test
£ 0.11, er ~ U • fn -I
i-t \ c( < \ ., c.o fl "e r~ e ~ il lr\2.l, di\Jtr~tS
Alternate Series
Root Test
Z(aAJ
~~~\ar ~ l !! la~\= L
i~ L '>\ , c!Ne.r-~e~ it L=-l, in,onc.\\Asive.
r f L '- \ , Col\ v e. ri e. 5
Integral Test
Lt On
7~ Un d.x lt co.n bt TC\te.j<~ -+e.a, c.o "\te r~ e.s. ,ff.ie.r wi 5e.
Direct ComQarison Test d ;"e r«je s
00 t)O
23 flo ~ hn f\= \ n.~1
f ,d,": a,,~ b(l ~I\J an~ bl\~ 0
if' t l\ 0 di"e r~et, fl= I
~eA\ ~ b,, J j\l e,r~C~ n.:1
©Michael Florip 2021
Converge-Diverge Tests (Continued)
Checking for Absolute/Conditional Convergence
eked~ i~
L, \ 0.n \ t on"trje. S
/ flo
--~,{ c.~e.c.k it
ctn (.QA\Jtr1es
I " ye-.",,
Interval of Convergence
r-----~ 'L,an COl\'lef ~e~
c.on~.fiottmh y
CD Use. ra:\i o +eif an~ se+ to < l [) ~olve ~"~ '>(
® C.',,ied,. eod roiAts wi"'1 e,OAV~/ Ji'IU'jfl hsts
6 a" <Ah~l~tely con Vt! r:_~eS
©Michael Florip 2021
Sequences and Series
Maclaurin Series (Taylor Series where center= 0).
'1 '1 x' x~ co~ 'X '! l - £ 1-~ - - -t - + ...
2t ii! ,, '8~ .
~x = I + X-+
Taylor Series
tor f( x) c.~tr¼e re.J c,.f ~-=-C ✓ ()0 <~) Y'
t(x) ~ LJ f (c) · (x-c) ~o rd
Alternating Series Remainder
i~ ie..-;ec; sq¼i,f.e s o.HerM'A+iA~ Serres te.st,
-¥.,ten yov. to.A \,He ~ t;c;t ter~ fltr il\c.\lA!tj .\-o e shMtAte ~ teMtAiAkr.
\ ~ l .£ llt~in ~ Lagrange Error Bound Formula
I M Ix_ c I ,,., R n = rcMQindcr(err .. 0
R" ~ - C n-+') t )( " ':) i~f.l\ x-val~e 0. ':. c.e.nte.<' e f rol111o~i•I t\ ~ tl~ru. J y-l Y"oµi A I
/_ )n Zn ~l X
(1") l
Infinite Sum (TelescoP-ing Series).
j.f t;e.riE!', i"> ·h.\e~<.q,;n, 1
S' "M =- (A I -t 0. n-t I
Infinite Sum (Geometric Series).
o.., l ~w~= 1 <I i-r j ,r
M.: M&ti~IAM ~ ~f.4.n X a~! C ~1 l,,l,.;1~ o.t \tee, C\C it ~~IA~l ""'k ~ nt\ tle,i\l(rive ot t(t)
©Michael Florip 2021
Other Things to Know
Derivative of Inverse Functions
ComP-ound Interest (Continuously)_
ComP-ound Interest (comP-ounded n times P-er t).
A -Y ( l t % )(\t
Euler's Method
M\Ait be. ,i .. e.n:
ii :: f.{x, r) j~flf ~;ze ~ j ( X, I Yo) X 11t I ::: X11 +~
/rrfl =-Y11 -th ·1(><1t,Y~ ExP-onential Growth
1 = A e kt A~ IM.ff al k~ ce~illAt vo.\u.e. t-:. tjMe.
Logistic Growth
M
Polar Area
Intermediate Value Theorem (IVT).
1f f(i) it, c.eAttt\\A.O(A~ ln [A,j],
t(o.) < K < f(\c,);!A °'< C < b
f(c)~ K o. c \>
©Michael Florip 2021
