MATH · Mathematics

AP Calculus Formula Sheet: Essential AB and BC Formulas

A structured AP Calculus AB and BC study sheet covering limits, derivatives, integrals, differential equations, applications, parametric and polar calculus, and infinite series.

Published by Insomnia Studyroom · 2026-09-16 · Editorial standards · Report a correction

Reviewed: September 16, 2026 · Aligned to the 2026–27 AP Calculus AB and BC course framework

Before you use this formula sheet

This is an independently written study reference, not an official College Board exam handout. AP Calculus is not listed among the subjects receiving a general printed reference sheet for the 2027 AP Exams. Learn when each relationship applies rather than expecting this page to be available during the test.

AB + BC labels content shared by both courses. BC only identifies the additional BC curriculum. Always check hypotheses such as continuity, differentiability, convergence, interval, sign, and units.

1. Limits and continuity (AB + BC)

\[ \lim(f\pm g)=\lim f\pm\lim g,\qquad \lim(fg)=(\lim f)(\lim g),\qquad \lim\frac fg=\frac{\lim f}{\lim g}\ \text{if }\lim g\ne0 \]

\[ f\text{ is continuous at }a \iff f(a)\text{ exists},\ \lim_{x\to a}f(x)\text{ exists, and } \lim_{x\to a}f(x)=f(a) \]

\[ \lim_{x\to0}\frac{\sin x}{x}=1,\qquad \lim_{x\to0}\frac{1-\cos x}{x}=0 \]

Trigonometric limit formulas require radians. A two-sided limit exists only when the one-sided limits agree.

2. Derivative definitions and rules (AB + BC)

\[ f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}h =\lim_{x\to a}\frac{f(x)-f(a)}{x-a} \]

Core differentiation rules
Rule	Formula
Power	\(\dfrac d{dx}x^n=nx^{n-1}\)
Product	\((fg)'=f'g+fg'\)
Quotient	\(\left(\dfrac fg\right)'=\dfrac{f'g-fg'}{g^2}\)
Chain	\(\dfrac d{dx}f(g(x))=f'(g(x))g'(x)\)
Inverse	\((f^{-1})'(a)=\dfrac1{f'(f^{-1}(a))}\)

\[ y-f(a)=f'(a)(x-a),\qquad L(x)=f(a)+f'(a)(x-a) \]

3. Common derivatives (AB + BC)

Exponential, logarithmic, trigonometric, and inverse-trigonometric derivatives
Function	Derivative	Function	Derivative
\(e^x\)	\(e^x\)	\(a^x\)	\(a^x\ln a\)
\(\ln\|x\|\)	\(1/x\)	\(\log_a x\)	\(1/(x\ln a)\)
\(\sin x\)	\(\cos x\)	\(\cos x\)	\(-\sin x\)
\(\tan x\)	\(\sec^2x\)	\(\cot x\)	\(-\csc^2x\)
\(\sec x\)	\(\sec x\tan x\)	\(\csc x\)	\(-\csc x\cot x\)
\(\arcsin x\)	\(1/\sqrt{1-x^2}\)	\(\arccos x\)	\(-1/\sqrt{1-x^2}\)
\(\arctan x\)	\(1/(1+x^2)\)	\(u(x)^{v(x)}\)	Use logarithmic differentiation

For composite functions, multiply by the inner derivative: \(\dfrac d{dx}e^u=e^uu'\) and \(\dfrac d{dx}\ln|u|=u'/u\).

4. Function analysis and major theorems (AB + BC)

\(f'>0\): increasing; \(f'<0\): decreasing.
\(f''>0\): concave up; \(f''<0\): concave down.
A critical number lies in the domain and satisfies \(f'=0\) or \(f'\) does not exist.
An inflection point requires an actual change in concavity.
First derivative test: \(+\to-\) gives a local maximum; \(-\to+\) gives a local minimum.
If \(f'(c)=0\), then \(f''(c)>0\) gives a local minimum and \(f''(c)<0\) gives a local maximum.

\[ \text{Mean Value Theorem:}\qquad f'(c)=\frac{f(b)-f(a)}{b-a} \]

The MVT requires continuity on \([a,b]\) and differentiability on \((a,b)\). The Extreme Value Theorem requires continuity on a closed interval. L'Hôpital's Rule applies only after verifying an indeterminate form such as \(0/0\) or \(\infty/\infty\):

\[ \lim\frac{f(x)}{g(x)}=\lim\frac{f'(x)}{g'(x)} \]

5. Integration and the Fundamental Theorem (AB + BC)

\[ \int_a^bf(x)\,dx =\lim_{n\to\infty}\sum_{i=1}^nf(x_i^*)\Delta x, \qquad \Delta x=\frac{b-a}{n} \]

\[ \frac d{dx}\int_a^xf(t)\,dt=f(x),\qquad \int_a^bf(x)\,dx=F(b)-F(a)\ \text{when }F'=f \]

\[ \frac d{dx}\int_{g(x)}^{h(x)}f(t)\,dt =f(h(x))h'(x)-f(g(x))g'(x) \]

\[ f(b)=f(a)+\int_a^bf'(x)\,dx,\qquad f_{\mathrm{avg}}=\frac1{b-a}\int_a^bf(x)\,dx \]

A definite integral is signed accumulation. Geometric area may require splitting where the integrand changes sign.

6. Common antiderivatives (AB + BC)

Frequently used antiderivatives
Integrand	Antiderivative
\(x^n,\ n\ne-1\)	\(\dfrac{x^{n+1}}{n+1}+C\)
\(1/x\)	\(\ln\|x\|+C\)
\(e^x\)	\(e^x+C\)
\(a^x\)	\(a^x/\ln a+C\)
\(\cos x\)	\(\sin x+C\)
\(\sin x\)	\(-\cos x+C\)
\(\sec^2x\)	\(\tan x+C\)
\(\csc^2x\)	\(-\cot x+C\)
\(\sec x\tan x\)	\(\sec x+C\)
\(\csc x\cot x\)	\(-\csc x+C\)
\(1/(1+x^2)\)	\(\arctan x+C\)
\(1/\sqrt{1-x^2}\)	\(\arcsin x+C\)

\[ \int f(g(x))g'(x)\,dx=\int f(u)\,du \]

7. Differential equations and motion

AB + BC:

\[ \frac{dy}{dx}=f(x)g(y) \Longrightarrow \int\frac1{g(y)}\,dy=\int f(x)\,dx \]

\[ \frac{dy}{dt}=ky\Longrightarrow y=y_0e^{kt} \]

\[ v=s',\qquad a=v'=s'',\qquad \text{displacement}=\int_a^bv(t)\,dt,\qquad \text{distance}=\int_a^b|v(t)|\,dt \]

BC only:

\[ y_{n+1}=y_n+f(x_n,y_n)\Delta x \]

\[ \frac{dP}{dt}=kP\left(1-\frac PK\right),\qquad P(t)=\frac K{1+Ae^{-kt}} \]

8. Area, volume, and arc length

AB + BC:

\[ A=\int_a^b(\text{top}-\text{bottom})\,dx \quad\text{or}\quad A=\int_c^d(\text{right}-\text{left})\,dy \]

\[ V=\int_a^bA(x)\,dx,\qquad V=\pi\int_a^b\left(R(x)^2-r(x)^2\right)\,dx \]

Find intersections before setting bounds, and construct the cross-sectional area from the stated geometry.

BC only:

\[ \text{Arc length}=\int_a^b\sqrt{1+(f'(x))^2}\,dx \]

9. Additional integration techniques (BC only)

\[ \int u\,dv=uv-\int v\,du \]

For rational functions, divide first when necessary, factor the denominator, and use partial fractions when appropriate.

\[ \int_a^\infty f(x)\,dx =\lim_{b\to\infty}\int_a^bf(x)\,dx \]

An improper integral converges only when its defining limit exists and is finite. Split at every infinite discontinuity inside the interval.

10. Parametric, vector-valued, and polar calculus (BC only)

\[ \frac{dy}{dx}=\frac{dy/dt}{dx/dt},\qquad \frac{d^2y}{dx^2} =\frac{\dfrac d{dt}\left(\dfrac{dy}{dx}\right)}{dx/dt} \]

\[ \vec r=\langle x,y\rangle,\quad \vec v=\langle x',y'\rangle,\quad \vec a=\langle x'',y''\rangle,\quad \text{speed}=\sqrt{(x')^2+(y')^2} \]

\[ \text{distance}=\int_a^b\sqrt{(x'(t))^2+(y'(t))^2}\,dt \]

\[ x=r\cos\theta,\qquad y=r\sin\theta,\qquad \frac{dy}{dx} =\frac{r'\sin\theta+r\cos\theta}{r'\cos\theta-r\sin\theta} \]

\[ A_{\mathrm{polar}}=\frac12\int_\alpha^\beta r(\theta)^2\,d\theta \]

11. Infinite sequences and series (BC only)

\[ \sum_{n=0}^{\infty}ar^n=\frac a{1-r}\quad(|r|<1), \qquad \sum_{n=1}^{\infty}\frac1{n^p}\text{ converges iff }p>1 \]

Nth-term test: \(\lim a_n\ne0\) proves divergence; \(\lim a_n=0\) is inconclusive.
Comparison tests: use known positive-term benchmark series.
Integral test: require a positive, continuous, decreasing function.
Alternating series: convergence follows when \(b_n\) decreases to zero.
Ratio test: \(\rho=\lim|a_{n+1}/a_n|\); converge if \(\rho<1\), diverge if \(\rho>1\), inconclusive if \(\rho=1\).

\[ |R_N|\le b_{N+1},\qquad \int_{N+1}^{\infty}f(x)\,dx\le R_N\le\int_N^{\infty}f(x)\,dx \]

For a power series, find the radius and test both endpoints separately.

12. Taylor and Maclaurin series (BC only)

\[ f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n,\qquad |R_N(x)|\le\frac{M|x-a|^{N+1}}{(N+1)!} \]

Common Maclaurin series
Function	Series	Convergence
\(e^x\)	\(\displaystyle\sum_{n=0}^{\infty}\frac{x^n}{n!}\)	All real \(x\)
\(\sin x\)	\(\displaystyle\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{(2n+1)!}\)	All real \(x\)
\(\cos x\)	\(\displaystyle\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{(2n)!}\)	All real \(x\)
\(1/(1-x)\)	\(\displaystyle\sum_{n=0}^{\infty}x^n\)	\(\|x\|<1\)

For the Lagrange bound, \(M\) must bound \(|f^{(N+1)}|\) on an interval containing the center \(a\) and the approximation point \(x\).

Final AP-style checklist

Name the requested quantity: value, rate, net change, distance, area, volume, or approximation.
Identify whether the information is analytical, graphical, numerical, or verbal.
Check every theorem's hypotheses before applying it.
Write the mathematical setup before entering calculator values.
Include units and interpret negative signs in context.
Check the answer against the graph, endpoints, scale, and domain.

The best way to remember the sheet is through connections: a derivative is a limit, slope, and rate; a definite integral is a Riemann-sum limit, signed area, and accumulation; and the Fundamental Theorem explains why differentiation and integration are inverse processes.

To work beyond memorizing formulas, study the definition of the derivative and antiderivatives and indefinite integrals. The AP Calculus course outline connects the remaining topics in this sheet.