Calculus Reference Sheet
Every formula you need for the course, on one printable page — limits, derivatives, integrals, the Fundamental Theorem, the application formulas, and the BC series toolkit.
A companion to keep beside you while you work, aligned to the College Board AP Calculus AB/BC framework. The rules below are the ones worth knowing cold; the worked examples and full explanations live inside each module. Use the print button to save a clean copy.
Limit Laws
Where a function is headed as \(x\to c\). When each piece has a limit, limits distribute across the operations.
The basic laws
Let \(\displaystyle\lim_{x\to c} f(x)=L\) and \(\displaystyle\lim_{x\to c} g(x)=M\), with \(k\) constant.
- Sum / difference: \(\displaystyle\lim_{x\to c}\bigl[f(x)\pm g(x)\bigr] = L \pm M\)
- Constant multiple: \(\displaystyle\lim_{x\to c}\,k\,f(x) = kL\)
- Product: \(\displaystyle\lim_{x\to c}\bigl[f(x)\,g(x)\bigr] = L\,M\)
- Quotient: \(\displaystyle\lim_{x\to c}\frac{f(x)}{g(x)} = \frac{L}{M}\), provided \(M\neq 0\)
- Power: \(\displaystyle\lim_{x\to c}\bigl[f(x)\bigr]^{n} = L^{n}\)
Key tools & facts
- Existence: \(\displaystyle\lim_{x\to c} f(x)\) exists \(\iff\) \(\displaystyle\lim_{x\to c^-} f(x) = \lim_{x\to c^+} f(x)\).
- Squeeze Theorem: if \(g(x)\le f(x)\le h(x)\) near \(c\) and \(\displaystyle\lim_{x\to c} g = \lim_{x\to c} h = L\), then \(\displaystyle\lim_{x\to c} f = L\).
- Two special limits: \[ \lim_{x\to 0}\frac{\sin x}{x}=1, \qquad \lim_{x\to 0}\frac{1-\cos x}{x}=0 \]
- L'Hôpital's Rule: for \(\tfrac{0}{0}\) or \(\tfrac{\infty}{\infty}\), \(\displaystyle\lim\frac{f}{g}=\lim\frac{f'}{g'}\).
Continuity at \(c\): \(f\) is continuous when \(f(c)\) is defined, \(\displaystyle\lim_{x\to c} f(x)\) exists, and the two are equal.
Derivative Rules
The instantaneous rate of change, \(f'(x)=\displaystyle\lim_{h\to0}\frac{f(x+h)-f(x)}{h}\), and the four rules that let you skip the limit.
The four rules
- Power Rule: \(\dfrac{d}{dx}\,x^{n} = n\,x^{n-1}\)
- Product Rule: \((fg)' = f'g + fg'\)
- Quotient Rule: \(\left(\dfrac{f}{g}\right)' = \dfrac{f'g - fg'}{g^{2}}\)
- Chain Rule: \(\dfrac{d}{dx}\,f\bigl(g(x)\bigr) = f'\bigl(g(x)\bigr)\,g'(x)\)
Also: constant multiple \((k f)' = k f'\), and sum \((f\pm g)' = f' \pm g'\).
| \(f(x)\) | \(f'(x)\) |
|---|---|
| \(\sin x\) | \(\cos x\) |
| \(\cos x\) | \(-\sin x\) |
| \(\tan x\) | \(\sec^{2} x\) |
| \(\sec x\) | \(\sec x \tan x\) |
| \(e^{x}\) | \(e^{x}\) |
| \(a^{x}\) | \(a^{x}\ln a\) |
| \(\ln x\) | \(\dfrac{1}{x}\) |
| \(\arcsin x\) | \(\dfrac{1}{\sqrt{1-x^{2}}}\) |
| \(\arctan x\) | \(\dfrac{1}{1+x^{2}}\) |
Antiderivatives & Integrals
Integration reverses differentiation. Every indefinite integral carries a constant \(+\,C\).
| \(\displaystyle\int f(x)\,dx\) | Result |
|---|---|
| \(\displaystyle\int x^{n}\,dx\) | \(\dfrac{x^{n+1}}{n+1}+C,\ n\neq -1\) |
| \(\displaystyle\int \frac{1}{x}\,dx\) | \(\ln|x|+C\) |
| \(\displaystyle\int e^{x}\,dx\) | \(e^{x}+C\) |
| \(\displaystyle\int a^{x}\,dx\) | \(\dfrac{a^{x}}{\ln a}+C\) |
| \(\displaystyle\int \cos x\,dx\) | \(\sin x+C\) |
| \(\displaystyle\int \sin x\,dx\) | \(-\cos x+C\) |
| \(\displaystyle\int \sec^{2} x\,dx\) | \(\tan x+C\) |
| \(\displaystyle\int \frac{dx}{1+x^{2}}\) | \(\arctan x+C\) |
| \(\displaystyle\int \frac{dx}{\sqrt{1-x^{2}}}\) | \(\arcsin x+C\) |
u-substitution
Reverses the chain rule. Choose \(u=g(x)\), so \(du=g'(x)\,dx\):
\[ \int f\bigl(g(x)\bigr)g'(x)\,dx = \int f(u)\,du \]
For a definite integral, change the limits to \(u\)-values: \(\displaystyle\int_a^b \to \int_{g(a)}^{g(b)}\).
Integration by parts BC
Reverses the product rule. Let \(u\) and \(dv\) be parts of the integrand:
\[ \int u\,dv = uv - \int v\,du \]
Choosing \(u\) — LIATE: prefer Log, Inverse trig, Algebraic, Trig, then Exponential for \(u\).
The Fundamental Theorem of Calculus
The bridge of the whole course: differentiation and integration are inverse operations.
Part 1 — the accumulation function
If \(f\) is continuous on \([a,b]\) and \(\displaystyle F(x)=\int_a^x f(t)\,dt\), then \(F\) is differentiable and
\[ \frac{d}{dx}\int_a^x f(t)\,dt = f(x) \]
Differentiating an integral with a variable upper limit hands back the integrand. With a chain, \(\dfrac{d}{dx}\displaystyle\int_a^{g(x)} f(t)\,dt = f\bigl(g(x)\bigr)g'(x)\).
Part 2 — evaluating a definite integral
If \(f\) is continuous on \([a,b]\) and \(F\) is any antiderivative of \(f\), then
\[ \int_a^b f(x)\,dx = F(b) - F(a) \]
Find an antiderivative, plug in the bounds, subtract. This is what turns an area problem into ordinary algebra.
Application Formulas
What the definite integral accumulates: area between curves, the volume of a solid of revolution, and the average value of a function.
Area between two curves
For \(f(x)\ge g(x)\) on \([a,b]\) (top minus bottom):
\[ A = \int_a^b \bigl[f(x) - g(x)\bigr]\,dx \]
Average value of a function
The mean height of \(f\) over \([a,b]\):
\[ f_{\text{avg}} = \frac{1}{b-a}\int_a^b f(x)\,dx \]
Volumes of revolution
Disk (region touches the axis), radius \(R(x)\):
\[ V = \pi\int_a^b \bigl[R(x)\bigr]^{2}\,dx \]
Washer (a gap to the axis), outer \(R\), inner \(r\):
\[ V = \pi\int_a^b \Bigl(\bigl[R(x)\bigr]^{2} - \bigl[r(x)\bigr]^{2}\Bigr)\,dx \]
Shell (cylindrical shells), radius \(x\), height \(h(x)\):
\[ V = 2\pi\int_a^b x\,h(x)\,dx \]
Series Convergence Tests BC
Does an infinite sum \(\displaystyle\sum a_n\) settle to a finite value? Pick the test that matches the shape of the terms.
Screening & benchmark series
- nth-Term Test (divergence): if \(\displaystyle\lim_{n\to\infty} a_n \neq 0\), the series diverges. (It can never prove convergence.)
- Geometric series: \(\displaystyle\sum_{n=0}^{\infty} ar^{n}\) converges \(\iff |r|<1\), to \(\dfrac{a}{1-r}\).
- p-series: \(\displaystyle\sum \frac{1}{n^{p}}\) converges \(\iff p>1\) (so the harmonic series \(p=1\) diverges).
Comparison & ratio tests
- Direct / Limit Comparison: compare \(a_n\) against a known \(b_n\) (often a \(p\)- or geometric series).
- Integral Test: for positive, decreasing \(f\), \(\displaystyle\sum f(n)\) and \(\displaystyle\int_1^{\infty} f(x)\,dx\) share the same fate.
- Ratio Test: with \(\displaystyle L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|\), converges if \(L<1\), diverges if \(L>1\), inconclusive if \(L=1\).
- Alternating Series Test: \(\displaystyle\sum (-1)^{n} b_n\) converges if \(b_n\) decreases to \(0\).
Key Maclaurin Series BC
A function rebuilt as an infinite polynomial centered at \(x=0\). These three are worth memorizing — each converges for all real \(x\).
\[ e^{x} = \sum_{n=0}^{\infty}\frac{x^{n}}{n!} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \cdots \]
\[ \sin x = \sum_{n=0}^{\infty}\frac{(-1)^{n}x^{2n+1}}{(2n+1)!} = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \cdots \]
\[ \cos x = \sum_{n=0}^{\infty}\frac{(-1)^{n}x^{2n}}{(2n)!} = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \cdots \]
The general Taylor series of \(f\) about \(x=a\) is \(\displaystyle\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^{n}\). A Maclaurin series is simply the case \(a=0\).