AP Calculus BC Formula Sheet Cheat Sheet

Key Facts

• The derivative definition is $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$, which gives the instantaneous rate of change at $x=a$.
• The Fundamental Theorem of Calculus states that if $F'(x)=f(x)$, then $\int_a^b f(x)\,dx=F(b)-F(a)$.
• Integration by parts is $\int u\,dv=uv-\int v\,du$, and it is useful when a product contains functions that simplify after differentiation.
• For parametric equations $x=x(t)$ and $y=y(t)$, the slope is $\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$ when $\frac{dx}{dt}\ne 0$.
• The polar area formula is $A=\frac{1}{2}\int_{\alpha}^{\beta} r^2\,d\theta$ for a region traced once by $r=f(\theta)$.
• A Taylor series centered at $x=c$ is $\sum_{n=0}^{\infty}\frac{f^{(n)}(c)}{n!}(x-c)^n$.
• The ratio test says a series $\sum a_n$ converges absolutely if $\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|<1$ and diverges if the limit is greater than $1$.
• Logistic growth has differential equation $\frac{dP}{dt}=kP\left(1-\frac{P}{L}\right)$, where $L$ is the carrying capacity.

Vocabulary

Derivative

A derivative measures the instantaneous rate of change of a function and is written as $f'(x)$ or $\frac{dy}{dx}$.

Definite Integral

A definite integral $\int_a^b f(x)\,dx$ represents signed area, net change, or accumulated quantity over the interval $[a,b]$.

Power Series

A power series is an infinite polynomial of the form $\sum_{n=0}^{\infty} a_n(x-c)^n$ centered at $x=c$.

Radius of Convergence

The radius of convergence $R$ is the distance from the center $c$ over which a power series converges.

Parametric Curve

A parametric curve defines position using $x=x(t)$ and $y=y(t)$ instead of writing $y$ directly as a function of $x$.

Polar Curve

A polar curve uses $r=f(\theta)$, where $r$ is distance from the pole and $\theta$ is the angle from the polar axis.