AP Calculus Formula Sheet — AB & BC Complete Reference A complete AP Calculus reference covering both AB and BC topics. The AP exam doesn't provide a formula sheet, so this is everything you should know cold for free-response and multiple-choice sections. BC-only topics are marked.
Limits & Continuity Limit Definition
lim x → a f ( x ) = L \lim_{x \to a} f(x) = L x → a lim f ( x ) = L Continuity at a
lim x → a f ( x ) = f ( a ) \lim_{x \to a} f(x) = f(a) x → a lim f ( x ) = f ( a ) Intermediate Value Theorem
f continuous on [ a , b ] ⟹ takes every value between f ( a ) and f ( b ) f \text{ continuous on } [a, b] \implies \text{takes every value between } f(a) \text{ and } f(b) f continuous on [ a , b ] ⟹ takes every value between f ( a ) and f ( b ) Special Trig Limit
lim x → 0 sin x x = 1 \lim_{x \to 0} \frac{\sin x}{x} = 1 x → 0 lim x sin x = 1 L'Hôpital's Rule
lim x → a f g = lim x → a f ′ g ′ ( 0 0 or ∞ ∞ ) \lim_{x \to a} \frac{f}{g} = \lim_{x \to a} \frac{f'}{g'} \quad\left(\tfrac{0}{0} \text{ or } \tfrac{\infty}{\infty}\right) x → a lim g f = x → a lim g ′ f ′ ( 0 0 or ∞ ∞ )
Derivatives Definition (Difference Quotient)
f ′ ( x ) = lim h → 0 f ( x + h ) − f ( x ) h f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} f ′ ( x ) = h → 0 lim h f ( x + h ) − f ( x ) Power Rule
d d x [ x n ] = n x n − 1 \frac{d}{dx}[x^n] = n x^{n-1} d x d [ x n ] = n x n − 1 Product Rule
( f g ) ′ = f ′ g + f g ′ (fg)' = f' g + f g' ( f g ) ′ = f ′ g + f g ′ Quotient Rule
( f g ) ′ = f ′ g − f g ′ g 2 \left(\tfrac{f}{g}\right)' = \frac{f' g - f g'}{g^2} ( g f ) ′ = g 2 f ′ g − f g ′ Chain Rule
[ f ( g ( x ) ) ] ′ = f ′ ( g ( x ) ) g ′ ( x ) [f(g(x))]' = f'(g(x)) g'(x) [ f ( g ( x )) ] ′ = f ′ ( g ( x )) g ′ ( x ) Implicit Differentiation
d d x [ F ( x , y ) ] = 0 , solve for d y d x \frac{d}{dx}[F(x, y)] = 0,\ \text{solve for } \tfrac{dy}{dx} d x d [ F ( x , y )] = 0 , solve for d x d y Derivative Formulas (Functions) Sine, Cosine
( sin x ) ′ = cos x , ( cos x ) ′ = − sin x (\sin x)' = \cos x,\ (\cos x)' = -\sin x ( sin x ) ′ = cos x , ( cos x ) ′ = − sin x Tangent
( tan x ) ′ = sec 2 x (\tan x)' = \sec^2 x ( tan x ) ′ = sec 2 x Sec, Csc, Cot
( sec x ) ′ = sec x tan x , ( csc x ) ′ = − csc x cot x , ( cot x ) ′ = − csc 2 x (\sec x)' = \sec x \tan x,\ (\csc x)' = -\csc x \cot x,\ (\cot x)' = -\csc^2 x ( sec x ) ′ = sec x tan x , ( csc x ) ′ = − csc x cot x , ( cot x ) ′ = − csc 2 x Exponential
( e x ) ′ = e x , ( a x ) ′ = a x ln a (e^x)' = e^x,\ (a^x)' = a^x \ln a ( e x ) ′ = e x , ( a x ) ′ = a x ln a Logarithm
( ln x ) ′ = 1 x , ( log a x ) ′ = 1 x ln a (\ln x)' = \tfrac{1}{x},\ (\log_a x)' = \tfrac{1}{x \ln a} ( ln x ) ′ = x 1 , ( log a x ) ′ = x l n a 1 Arcsine
( sin − 1 x ) ′ = 1 1 − x 2 (\sin^{-1} x)' = \tfrac{1}{\sqrt{1 - x^2}} ( sin − 1 x ) ′ = 1 − x 2 1 Arctangent
( tan − 1 x ) ′ = 1 1 + x 2 (\tan^{-1} x)' = \tfrac{1}{1 + x^2} ( tan − 1 x ) ′ = 1 + x 2 1 Applications of Derivatives Mean Value Theorem
f ′ ( c ) = f ( b ) − f ( a ) b − a f'(c) = \frac{f(b) - f(a)}{b - a} f ′ ( c ) = b − a f ( b ) − f ( a ) Critical Point
f ′ ( c ) = 0 or undefined f'(c) = 0 \text{ or undefined} f ′ ( c ) = 0 or undefined Second Derivative Test
f ′ ′ ( c ) > 0 ⇒ local min ; f ′ ′ ( c ) < 0 ⇒ local max f''(c) > 0 \Rightarrow \text{local min};\ f''(c) < 0 \Rightarrow \text{local max} f ′′ ( c ) > 0 ⇒ local min ; f ′′ ( c ) < 0 ⇒ local max Concavity
f ′ ′ ( x ) > 0 : concave up ; f ′ ′ ( x ) < 0 : concave down f''(x) > 0: \text{ concave up};\ f''(x) < 0: \text{ concave down} f ′′ ( x ) > 0 : concave up ; f ′′ ( x ) < 0 : concave down Linear Approximation
L ( x ) = f ( a ) + f ′ ( a ) ( x − a ) L(x) = f(a) + f'(a)(x - a) L ( x ) = f ( a ) + f ′ ( a ) ( x − a ) Integrals (Antiderivatives) Power Rule
∫ x n d x = x n + 1 n + 1 + C , n ≠ − 1 \int x^n\,dx = \frac{x^{n+1}}{n+1} + C,\ n \ne -1 ∫ x n d x = n + 1 x n + 1 + C , n = − 1 Reciprocal
∫ 1 x d x = ln ∣ x ∣ + C \int \frac{1}{x}\,dx = \ln|x| + C ∫ x 1 d x = ln ∣ x ∣ + C Exponential
∫ e x d x = e x + C \int e^x\,dx = e^x + C ∫ e x d x = e x + C Trig
∫ sin x d x = − cos x + C , ∫ cos x d x = sin x + C \int \sin x\,dx = -\cos x + C,\ \int \cos x\,dx = \sin x + C ∫ sin x d x = − cos x + C , ∫ cos x d x = sin x + C u-Substitution
∫ f ( g ( x ) ) g ′ ( x ) d x = ∫ f ( u ) d u \int f(g(x)) g'(x)\,dx = \int f(u)\,du ∫ f ( g ( x )) g ′ ( x ) d x = ∫ f ( u ) d u Integration by Parts (BC)
∫ u d v = u v − ∫ v d u \int u\,dv = uv - \int v\,du ∫ u d v = uv − ∫ v d u Fundamental Theorem & Applications FTC Part 1
d d x ∫ a x f ( t ) d t = f ( x ) \frac{d}{dx}\int_a^x f(t)\,dt = f(x) d x d ∫ a x f ( t ) d t = f ( x ) FTC Part 2
∫ a b f ′ ( x ) d x = f ( b ) − f ( a ) \int_a^b f'(x)\,dx = f(b) - f(a) ∫ a b f ′ ( x ) d x = f ( b ) − f ( a ) Average Value
f avg = 1 b − a ∫ a b f ( x ) d x f_\text{avg} = \frac{1}{b - a}\int_a^b f(x)\,dx f avg = b − a 1 ∫ a b f ( x ) d x Area Between Curves
A = ∫ a b [ f ( x ) − g ( x ) ] d x A = \int_a^b [f(x) - g(x)]\,dx A = ∫ a b [ f ( x ) − g ( x )] d x Volume (Disk Method)
V = π ∫ a b [ f ( x ) ] 2 d x V = \pi \int_a^b [f(x)]^2\,dx V = π ∫ a b [ f ( x ) ] 2 d x Volume (Washer)
V = π ∫ a b [ R ( x ) 2 − r ( x ) 2 ] d x V = \pi \int_a^b [R(x)^2 - r(x)^2]\,dx V = π ∫ a b [ R ( x ) 2 − r ( x ) 2 ] d x Volume (Shell, BC)
V = 2 π ∫ a b x f ( x ) d x V = 2\pi \int_a^b x f(x)\,dx V = 2 π ∫ a b x f ( x ) d x Arc Length
L = ∫ a b 1 + ( f ′ ( x ) ) 2 d x L = \int_a^b \sqrt{1 + (f'(x))^2}\,dx L = ∫ a b 1 + ( f ′ ( x ) ) 2 d x Parametric, Polar & Vector (BC Only) Parametric Derivative
d y d x = d y / d t d x / d t \frac{dy}{dx} = \frac{dy/dt}{dx/dt} d x d y = d x / d t d y / d t Parametric Arc Length
L = ∫ a b ( x ′ ( t ) ) 2 + ( y ′ ( t ) ) 2 d t L = \int_a^b \sqrt{(x'(t))^2 + (y'(t))^2}\,dt L = ∫ a b ( x ′ ( t ) ) 2 + ( y ′ ( t ) ) 2 d t Polar Area
A = 1 2 ∫ α β r 2 d θ A = \tfrac{1}{2}\int_\alpha^\beta r^2\,d\theta A = 2 1 ∫ α β r 2 d θ Polar Slope
d y d x = r ′ sin θ + r cos θ r ′ cos θ − r sin θ \frac{dy}{dx} = \frac{r' \sin\theta + r \cos\theta}{r' \cos\theta - r \sin\theta} d x d y = r ′ cos θ − r sin θ r ′ sin θ + r cos θ Vector Position
r ⃗ ( t ) = ⟨ x ( t ) , y ( t ) ⟩ \vec{r}(t) = \langle x(t), y(t) \rangle r ( t ) = ⟨ x ( t ) , y ( t )⟩ Speed
∣ v ⃗ ( t ) ∣ = ( x ′ ( t ) ) 2 + ( y ′ ( t ) ) 2 |\vec{v}(t)| = \sqrt{(x'(t))^2 + (y'(t))^2} ∣ v ( t ) ∣ = ( x ′ ( t ) ) 2 + ( y ′ ( t ) ) 2 Series (BC Only) Geometric Series
∑ n = 0 ∞ a r n = a 1 − r , ∣ r ∣ < 1 \sum_{n=0}^{\infty} a r^n = \tfrac{a}{1 - r},\ |r| < 1 n = 0 ∑ ∞ a r n = 1 − r a , ∣ r ∣ < 1 p-Series Test
∑ 1 n p converges ⟺ p > 1 \sum \tfrac{1}{n^p} \text{ converges} \iff p > 1 ∑ n p 1 converges ⟺ p > 1 Ratio Test
L = lim ∣ a n + 1 a n ∣ ; L < 1 ⇒ converges L = \lim \left|\tfrac{a_{n+1}}{a_n}\right|;\ L < 1 \Rightarrow \text{converges} L = lim a n a n + 1 ; L < 1 ⇒ converges Taylor Series
f ( x ) = ∑ n = 0 ∞ f ( n ) ( a ) n ! ( x − a ) n f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^n f ( x ) = n = 0 ∑ ∞ n ! f ( n ) ( a ) ( x − a ) n Maclaurin: e^x
e x = ∑ n = 0 ∞ x n n ! e^x = \sum_{n=0}^{\infty} \tfrac{x^n}{n!} e x = n = 0 ∑ ∞ n ! x n Maclaurin: sin x
sin x = ∑ n = 0 ∞ ( − 1 ) n x 2 n + 1 ( 2 n + 1 ) ! \sin x = \sum_{n=0}^{\infty} \tfrac{(-1)^n x^{2n+1}}{(2n+1)!} sin x = n = 0 ∑ ∞ ( 2 n + 1 )! ( − 1 ) n x 2 n + 1 Maclaurin: cos x
cos x = ∑ n = 0 ∞ ( − 1 ) n x 2 n ( 2 n ) ! \cos x = \sum_{n=0}^{\infty} \tfrac{(-1)^n x^{2n}}{(2n)!} cos x = n = 0 ∑ ∞ ( 2 n )! ( − 1 ) n x 2 n Lagrange Remainder
∣ R n ( x ) ∣ ≤ M ∣ x − a ∣ n + 1 ( n + 1 ) ! |R_n(x)| \le \tfrac{M |x - a|^{n+1}}{(n+1)!} ∣ R n ( x ) ∣ ≤ ( n + 1 )! M ∣ x − a ∣ n + 1
