AP Calculus BC Formula List

Sep 20, 2026

Uploaded by Finn

Comprehensive AP Calculus BC formula sheet compiling key results for derivatives and limits, continuity, and the mean value and intermediate value theorems. Includes rules for differentiation such as the chain rule, product/quotient-related derivative forms, and antiderivative tools like integration by parts and common integral templates. Covers definite integrals, the fundamental theorems of calculus, optimization tests, concavity and inflection point criteria, and applications including L’Hospital’s Rule, average/instantaneous rates, and particle motion relationships.

LanguageEnglish

CategoryExam

FormatPDF

Pages5

Reading Time13 min

AccessFree to download

Derivative and Limit Foundations

Continuity, MVT, and IVT

Differentiation Rules and Trig/Log Derivatives

Integral Foundations and Applications

Definite Integrals and Fundamental Theorems

Integration by Parts and Common Integral Forms

L’Hospital’s Rule

Optimization, Concavity, and Motion

Critical Numbers and Derivative Tests

Concavity and Inflection Points

Particle Motion and Rate In/Out

2020 AP CALCULUS BC FORMULA LIST Definition of the derivative: 􀁣 f 􀀋x 􀀎 h􀀌􀀐 f 􀀋x􀀌 􀁣 f 􀀋x􀀌􀀐 f 􀀋a􀀌 f (x) 􀀠 lhim􀁯0 h f 􀀋a􀀌 􀀠 lxim􀁯a x 􀀐 a (Alternative form) Definition of continuity: f is continuous at c if and only if 1) f (c) is defined; 2) limx􀁯c f (x) exists; 3) limx􀁯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􀀌b~~ ~~􀀐~~ ~~fa~~ ~~􀀋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. ddx xn 􀀠 nxn􀀐1 ~~ ~~ddx u~~ ~~ 􀀠 d2u~~dx~~u ddx f 􀀋x 􀀌g 􀀋x 􀀌 􀀠 f 􀀋x 􀀌g 􀁣 􀀋x 􀀌􀀎 g 􀀋x 􀀌 f 􀁣 􀀋x 􀀌 ddx 􀀋 f 􀀋 g 􀀋x 􀀌􀀌􀀌 􀀠 f 􀁣 􀀋 g 􀀋x 􀀌􀀌􀂘 g 􀁣 􀀋x 􀀌 ddx [sin u] 􀀠 cosu ddux ddx [tan u] 􀀠 sec2 u ddux d du [sec u] 􀀠 secu tanudx dx d 1 du [ln u] 􀀠 dx u dx ddx [eu ] 􀀠 eu ddux d 1 du [arcsin u] 􀀠 dx 1 􀀐 u dx d 1 du [arctan u] 􀀠 2 dx 1 􀀎 u dx d 􀂪 1 􀂺 1 dx 􀂬􀂫 x 􀂼􀂻 􀀠 􀀐 x2 d 􀂪 f 􀀋x 􀀌􀂺 g 􀀋x 􀀌 f 􀁣 􀀋x 􀀌􀀐 f 􀀋x 􀀌g 􀁣 􀀋x 􀀌 ~~ ~~ 􀂫 ~~ ~~ 􀂻 􀀠 2 dx 􀂬g 􀀋x 􀀌 􀂼 􀀋 g 􀀋x 􀀌􀀌 ddx [cos u] 􀀠 􀀐 sin u ddux ddx [cot u] 􀀠 􀀐 csc2 u ddux d du [csc u] 􀀠 􀀐 cscu cot udx dx d 1 du [log u] 􀀠 dx a ulna dx ddx [au ] 􀀠 au ln a ddux d 1 du [arccosu] 􀀠 􀀐 dx 1 􀀐 u dx d 1 du dx [arc cot u] 􀀠 􀀐 1 􀀎 u 2 dx Definition of a definite integral: 􀂳ba f n 􀀋x 􀀌dx 􀀠 l􀀧ixm􀁯0 1 f n 􀀋xk 􀀌􀂘 􀀋􀀧xk 􀀌 􀀠 lnim􀁯􀁦 􀂦 f k 􀀠 1 􀀋xk 􀀌􀂘 􀀋􀀧xk 􀀌 􀂳 cosu du 􀀠 sin u 􀀎 C 􀂳 xn dx 􀀠 xnn~~ ~~􀀎􀀎11 􀀎 C, n 􀁺 􀀐1 􀂳 eu du 􀀠 eu 􀀎 C 􀂳 f 􀁣 􀀋 g 􀀋x 􀀌􀀌􀂘 g 􀁣 􀀋x 􀀌dx 􀀠 f 􀀋 g 􀀋x 􀀌􀀌 􀀎 C 􀂳 sin u du 􀀠 􀀐 cosu 􀀎 C 􀂳 1u du 􀀠 ln u 􀀎 C 􀂳 au du 􀀠 lanua 􀀎 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: 1) If f 􀁣 􀀋x􀀌 changes from negative to positive at c, then 􀀋 c, f 􀀋c 􀀌􀀌 is a relative minimum off. 2) If f 􀁣 􀀋x􀀌 changes from positive to negative at c, then 􀀋 c, f 􀀋c 􀀌􀀌 is a relative maximum off. Second Derivative Test: Let f be a function such that the second derivative of f exists on an open interval containing c. 1) If f 􀁣 􀀋c􀀌 􀀠 0 and f 􀁣􀁣 􀀋c􀀌 􀀡 0 , then 􀀋 c, f 􀀋c 􀀌􀀌 is a relative minimum. 2) 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: 􀂳ba f 􀁣 􀀋x􀀌dx 􀀠 f 􀀋b􀀌􀀐 f 􀀋a􀀌 f 􀀋b 􀀌 􀀠 f 􀀋a 􀀌 􀀎 􀂳 ba f 􀁣 􀀋x 􀀌dx f 􀀋a 􀀌 􀀠 f 􀀋b 􀀌 􀀐 􀂳ba f 􀁣 􀀋x 􀀌dx final 􀀠 initial + change initial 􀀠 final 􀀐 change Second Fundamental Theorem of Calculus: ddx 􀂳 xa f 􀀋t 􀀌dt 􀀠 f 􀀋x􀀌 Chain Rule Version: ~~ ~~ddx 􀂳 ag 􀀋x􀀌 f 􀀋t 􀀌dt 􀀠 f 􀀋 g 􀀋x􀀌􀀌􀂘 g 􀁣 􀀋x􀀌 Average rate of change of f (x) on [a, b]: _____________________________________________________________________________________________ Average value of f (x) on [a, b]: fAVE 􀀠 b~~ ~~~~ ~~a 􀂳ba f (x)dx Particle Motion If an object moves along a straight line with position function s

FAQs

What conditions define continuity at a point c?

Continuity at c requires that f(c) is defined, the limit limx→c f(x) exists, and limx→c f(x) equals f(c).

How do the Mean Value Theorem and Intermediate Value Theorem differ?

The Mean Value Theorem guarantees a point c where f'(c) equals the average rate of change on (a, b) under continuity and differentiability. The Intermediate Value Theorem guarantees that for any value k between f(a) and f(b), there exists c in [a, b] with f(c)=k under continuity.

Which tests help determine local extrema and concavity?

First and second derivative tests classify relative minima and maxima using sign changes in f'(x) or f''(x). Concavity and inflection points are determined using the sign of f''(x) and whether f''(x) changes sign (or the function transitions between increasing/decreasing).

Sep 20, 2026

Uploaded by Finn