AP Calculus BC · formula sheet

AP Calculus BC: integration and series

Integration techniques, the convergence tests, the Taylor series worth knowing by heart, and the parametric, polar and differential-equation formulas.

27 formulas and facts in 5 sections

Graph of sin x together with its Taylor polynomials of degree 1, 3 and 5 about zero. — Each extra term of the Taylor series for sin x keeps the polynomial close to the curve over a wider interval.

Integration techniques

Integration by parts: $\int u\,dv = uv - \int v\,du$
Improper integral: $\int_a^{\infty} f(x)\,dx = \lim_{b\to\infty}\int_a^{b} f(x)\,dx$
It converges when this limit exists and is finite.
Partial fractions (distinct linear factors): $\frac{px+q}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}$
Average value of f on [a, b]: $f_{\text{avg}} = \frac{1}{b-a}\int_a^b f(x)\,dx$

Antiderivatives to know

Reciprocal: $\int \frac{1}{x}\,dx = \ln|x| + C$
Exponential: $\int e^{x}\,dx = e^{x} + C \qquad \int a^{x}\,dx = \frac{a^{x}}{\ln a} + C$
Trigonometric: $\int \sec^2 x\,dx = \tan x + C \qquad \int \sec x\tan x\,dx = \sec x + C$
Inverse trigonometric: $\int \frac{dx}{1+x^2} = \arctan x + C \qquad \int \frac{dx}{\sqrt{1-x^2}} = \arcsin x + C$

Convergence tests for series

nth-term test: $\lim_{n\to\infty} a_n \neq 0 \;\Longrightarrow\; \sum a_n \text{ diverges}$
If the limit is 0 the test says nothing.
Geometric series: $\sum_{n=0}^{\infty} ar^{n} = \frac{a}{1-r} \quad\text{for } |r| < 1$
It diverges when |r| ≥ 1.
p-series: $\sum_{n=1}^{\infty} \frac{1}{n^{p}} \text{ converges} \iff p > 1$
Ratio test: $L = \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|$
L < 1: converges absolutely. L > 1: diverges. L = 1: inconclusive.
Alternating series test: $\sum (-1)^{n} b_n \text{ converges if } b_n > 0,\; b_{n+1} \le b_n,\; \lim_{n\to\infty} b_n = 0$
Alternating series error bound: $|S - S_n| \le b_{n+1}$
Integral test: $\sum_{n=1}^{\infty} f(n) \text{ and } \int_1^{\infty} f(x)\,dx \text{ converge or diverge together}$
For f positive, continuous and decreasing on [1, ∞).
Limit comparison test: $\lim_{n\to\infty}\frac{a_n}{b_n} = c,\; 0 < c < \infty \;\Longrightarrow\; \text{both converge or both diverge}$
For series with positive terms.

Taylor and Maclaurin series

Taylor series about x = a: $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^{n}$
Exponential: $e^{x} = \sum_{n=0}^{\infty} \frac{x^{n}}{n!} \quad\text{for all } x$
Sine and cosine: $\sin x = \sum_{n=0}^{\infty} \frac{(-1)^{n}x^{2n+1}}{(2n+1)!} \qquad \cos x = \sum_{n=0}^{\infty} \frac{(-1)^{n}x^{2n}}{(2n)!}$
Geometric: $\frac{1}{1-x} = \sum_{n=0}^{\infty} x^{n} \quad\text{for } |x| < 1$
Natural logarithm: $\ln(1+x) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}x^{n}}{n} \quad\text{for } -1 < x \le 1$
Lagrange error bound: $|R_n(x)| \le \frac{M\,|x-a|^{n+1}}{(n+1)!}$
M is at least the largest value of |f⁽ⁿ⁺¹⁾| between a and x.

Parametric, polar and differential equations

Slope of a parametric curve: $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$
Arc length (parametric): $L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^{2} + \left(\frac{dy}{dt}\right)^{2}}\,dt$
Area in polar coordinates: $A = \frac{1}{2}\int_{\alpha}^{\beta} r^{2}\,d\theta$
Logistic growth: $\frac{dP}{dt} = kP\left(1 - \frac{P}{L}\right)$
L is the carrying capacity; growth is fastest when P = L/2.
Euler's method: $y_{n+1} = y_n + h\,f(x_n, y_n)$

A reference for revision. In an exam, use the formula sheet your exam board provides and the notation your course uses.