Note : This comprehensive formula sheet covers all major topics in AP Calculus BC.
[Limits & Continuity]
[Derivatives]
[Integration]
[Series]
[Parametric/Polar]
[Differential Equations]
lim ā” x ā 0 sin ā” x x = 1 \lim_{x \to 0} \frac{\sin x}{x} = 1 lim x ā 0 ā x s i n x ā = 1 lim ā” x ā 0 1 ā cos ā” x x = 0 \lim_{x \to 0} \frac{1-\cos x}{x} = 0 lim x ā 0 ā x 1 ā c o s x ā = 0 lim ā” x ā ā ( 1 + 1 x ) x = e \lim_{x \to \infty} (1 + \frac{1}{x})^x = e lim x ā ā ā ( 1 + x 1 ā ) x = e
Power Rule: d d x [ x n ] = n x n ā 1 \frac{d}{dx}[x^n] = nx^{n-1} d x d ā [ x n ] = n x n ā 1
Product Rule: d d x [ f ( x ) g ( x ) ] = f ā² ( x ) g ( x ) + f ( x ) g ā² ( x ) \frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x) d x d ā [ f ( x ) g ( x )] = f ā² ( x ) g ( x ) + f ( x ) g ā² ( x )
Quotient Rule: d d x [ f ( x ) g ( x ) ] = f ā² ( x ) g ( x ) ā f ( x ) g ā² ( x ) [ g ( x ) ] 2 \frac{d}{dx}[\frac{f(x)}{g(x)}] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} d x d ā [ g ( x ) f ( x ) ā ] = [ g ( x ) ] 2 f ā² ( x ) g ( x ) ā f ( x ) g ā² ( x ) ā
Chain Rule: d d x [ f ( g ( x ) ) ] = f ā² ( g ( x ) ) g ā² ( x ) \frac{d}{dx}[f(g(x))] = f'(g(x))g'(x) d x d ā [ f ( g ( x ))] = f ā² ( g ( x )) g ā² ( x )
Function Derivative sin ā” x \sin x sin x cos ā” x \cos x cos x cos ā” x \cos x cos x ā sin ā” x -\sin x ā sin x tan ā” x \tan x tan x sec ā” 2 x \sec^2 x sec 2 x e x e^x e x e x e^x e x ln ā” x \ln x ln x 1 x \frac{1}{x} x 1 ā a x a^x a x a x ln ā” a a^x \ln a a x ln a
ā« x n d x = x n + 1 n + 1 + C \int x^n dx = \frac{x^{n+1}}{n+1} + C ā« x n d x = n + 1 x n + 1 ā + C n ā ā 1 n \neq -1 n ī = ā 1 ā« 1 x d x = ln ā” ā£ x ā£ + C \int \frac{1}{x} dx = \ln|x| + C ā« x 1 ā d x = ln ā£ x ā£ + C ā« e x d x = e x + C \int e^x dx = e^x + C ā« e x d x = e x + C ā« sin ā” x d x = ā cos ā” x + C \int \sin x dx = -\cos x + C ā« sin x d x = ā cos x + C ā« cos ā” x d x = sin ā” x + C \int \cos x dx = \sin x + C ā« cos x d x = sin x + C
U-Substitution
ā« f ( g ( x ) ) g ā² ( x ) d x = ā« f ( u ) d u \int f(g(x))g'(x)dx = \int f(u)du ā« f ( g ( x )) g ā² ( x ) d x = ā« f ( u ) d u
Integration by Parts
ā« u ā d v = u v ā ā« v ā d u \int u\,dv = uv - \int v\,du ā« u d v = uv ā ā« v d u
Partial Fractions
Decompose rational functions
Ratio Test
lim ā” n ā ā ā£ a n + 1 a n ā£ \lim_{n \to \infty}|\frac{a_{n+1}}{a_n}| lim n ā ā ā ā£ a n ā a n + 1 ā ā ā£ < 1: Converges
1: Diverges
= 1: Inconclusive
Root Test
lim ā” n ā ā ā£ a n ā£ n \lim_{n \to \infty}\sqrt[n]{|a_n|} lim n ā ā ā n ā£ a n ā ā£ ā
p-Series
ā 1 n p \sum \frac{1}{n^p} ā n p 1 ā
f ( x ) = ā n = 0 ā f ( n ) ( a ) n ! ( x ā a ) n f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n f ( x ) = ā n = 0 ā ā n ! f ( n ) ( a ) ā ( x ā a ) n
e x = ā n = 0 ā x n n ! e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} e x = ā n = 0 ā ā n ! x n ā sin ā” x = ā n = 0 ā ( ā 1 ) n x 2 n + 1 ( 2 n + 1 ) ! \sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} sin x = ā n = 0 ā ā ( 2 n + 1 )! ( ā 1 ) n x 2 n + 1 ā cos ā” x = ā n = 0 ā ( ā 1 ) n x 2 n ( 2 n ) ! \cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} cos x = ā n = 0 ā ā ( 2 n )! ( ā 1 ) n x 2 n ā 1 1 ā x = ā n = 0 ā x n \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n 1 ā x 1 ā = ā n = 0 ā ā x n
d y d x = d y d t d x d t \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} d x d y ā = d t d x ā d t d y ā ā
x = r cos Īø
y = r sin Īø
r 2 = x 2 + y 2 r^2 = x^2 + y^2 r 2 = x 2 + y 2 Area: A = 1 2 ā« Ī± Ī² r 2 d Īø A = \frac{1}{2}\int_Ī±^Ī² r^2 dĪø A = 2 1 ā ā« Ī± Ī² ā r 2 d Īø
Arc Length: L = ā« Ī± Ī² r 2 + ( d r d Īø ) 2 d Īø L = \int_Ī±^Ī² \sqrt{r^2 + (\frac{dr}{dĪø})^2}dĪø L = ā« Ī± Ī² ā r 2 + ( d Īø d r ā ) 2 ā d Īø
Separable
d y d x = f ( x ) g ( y ) \frac{dy}{dx} = f(x)g(y) d x d y ā = f ( x ) g ( y ) Separate and integrate
Linear
d y d x + P ( x ) y = Q ( x ) \frac{dy}{dx} + P(x)y = Q(x) d x d y ā + P ( x ) y = Q ( x ) Use integrating factor
Growth/Decay
d y d t = k y \frac{dy}{dt} = ky d t d y ā = k y Solution: y = C e k t y = Ce^{kt} y = C e k t
Logistic
d y d t = k y ( 1 ā y M ) \frac{dy}{dt} = ky(1-\frac{y}{M}) d t d y ā = k y ( 1 ā M y ā ) Solution: y = M 1 + C e ā k t y = \frac{M}{1 + Ce^{-kt}} y = 1 + C e ā k t M ā
y n + 1 = y n + f ( x n , y n ) Ī x y_{n+1} = y_n + f(x_n,y_n)\Delta x y n + 1 ā = y n ā + f ( x n ā , y n ā ) Ī x
Angle sin cos tan 0Ā° 0 1 0 30Ā° 1/2 ā3/2 1/ā3 45Ā° 1/ā2 1/ā2 1 60Ā° ā3/2 1/2 ā3 90Ā° 1 0 undef
e ā 2.71828
Ļ ā 3.14159
ln(e) = 1
e^(ln x) = x
š” Pro Tip: This formula sheet is provided for reference - understanding how to use these formulas is more important than memorizing them!
