AP Calculus BC Important Formulas

Why so much random BS

Integration by Parts

\begin{aligned}

&\int u\,dv = u\,v \;-\;\int v\,du\\[6pt]

&\text{Hint: Choose }u\text{ (Logarithmic, Inverse, Algebraic, Trig, Exponential)}\\

&\quad\text{and }dv\text{ so that }du\text{ and }v\text{ simplify the integral.}

\end{aligned}

Partial Fractions

\begin{aligned}

&\frac{1}{(c x + d)\,(h x + k)}\\

&\quad= \frac{A}{c x + d}

+ \frac{B}{h x + k}

\\[8pt]

&\text{Multiply both sides by }(c x + d)(h x + k)\text{:}\\

&\quad1

= A\,(h x + k)

+ B\,(c x + d)

\\[6pt]

&\text{Equate coefficients:}\\

&\quad A\,h + B\,c = 0,\\

&\quad A\,k + B\,d = 1.

\\[6pt]

&\text{Hint: Clear denominators, then solve this linear system for }A,B.

\end{aligned}

Volume

\begin{aligned}

&\text{Disc Method: }

V=\pi\int_{a}^{b}\bigl[r(x)\bigr]^{2}\,dx\\

&\text{Washer Method: }

V=\pi\int_{a}^{b}\Bigl[R(x)^{2}-r(x)^{2}\Bigr]\,dx\\

&\text{Shell Method: }

V=2\pi\int_{a}^{b}r(x)\,h(x)\,dx\\

&\text{Cross‐Section Method: }

V=\int_{a}^{b}A(x)\,dx\\[6pt]

&\text{Hint: Use Disc/Washer when slicing perpendicular to the axis;}\\

&\text{use Shell when slicing parallel;}\\

&\text{use Cross‐Section when slices have known area }A(x).

\end{aligned}

Alternating Series Error Bound

\begin{aligned}

&\text{For an alternating series satisfying the AST conditions,}\\

&\quad\bigl|R_n\bigr|\;=\;\Bigl|\sum_{k=n+1}^\infty(-1)^k a_k\Bigr|

\;\le\;a_{n+1}.\\[6pt]

&\text{Hint: The magnitude of the remainder is at most the next term.}

\end{aligned}

Lagrange Error Bound

\begin{aligned}

&\text{If }f\in C^{\,n+1}[a,b],\text{ then the Taylor remainder }R_n\text{ at }b\text{ satisfies}\\

&\quad

\bigl|R_n\bigr|\;\le\;\frac{\max_{c\in[a,b]}\bigl|f^{(n+1)}(c)\bigr|}{(n+1)!}\,(b-a)^{n+1}.\\[6pt]

&\text{Hint: Bound the \((n+1)\)th derivative on \([a,b]\) and divide by \((n+1)!\).}

\end{aligned}

Logistical Growth

\begin{aligned}

&\frac{dP}{dt} = k\,P\,(M - P),\\

&P(t) = \frac{M}{1 + C e^{-k t}},\\[6pt]

&\text{where }P(t)\text{ is the population at time }t,\\

&\quad k\text{ is the intrinsic growth rate constant},\\

&\quad M\text{ is the carrying capacity},\\

&\quad C\text{ is set by the initial condition }P(0).

\end{aligned}

Parametric First Derivative

\frac{dy}{dx}

= \frac{\displaystyle \frac{dy}{dt}}

{\displaystyle \frac{dx}{dt}}

\quad(\text{provided }dx/dt\neq0).

Parametric Second Derivative

\frac{d^2y}{dx^2}

= \frac{d}{dx}\!\Bigl(\frac{dy}{dx}\Bigr)

= \frac{\displaystyle \frac{d}{dt}\!\bigl(\frac{dy}{dx}\bigr)}

{\displaystyle \frac{dx}{dt}}

Polar ↔ Rectangular Conversion

\begin{aligned}

&\text{Rectangular to Polar:}

&&r = \sqrt{x^2 + y^2},

&\quad\theta = \arctan\!\bigl(\tfrac{y}{x}\bigr).\\

&\text{Polar to Rectangular:}

&&x = r\cos\theta,

&\quad y = r\sin\theta.

\end{aligned}

Arc Length

\begin{aligned}

\text{Cartesian:} \quad

&L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^{2}}\,dx, \\[8pt]

\text{Parametric:} \quad

&L = \int_{t_{1}}^{t_{2}} \sqrt{\left(\frac{dx}{dt}\right)^{2} + \left(\frac{dy}{dt}\right)^{2}}\,dt. \\[8pt]

&\text{Hint: In the parametric form, } \frac{dx}{dt} \text{ and } \frac{dy}{dt} \text{ give the components of motion.}

\end{aligned}

Speed & Distance Traveled

\begin{aligned}

\text{Cartesian:} \quad

&L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^{2}}\,dx, \\[8pt]

\text{Parametric:} \quad

&L = \int_{t_{1}}^{t_{2}} \sqrt{\left(\frac{dx}{dt}\right)^{2} + \left(\frac{dy}{dt}\right)^{2}}\,dt. \\[8pt]

&\text{Hint: In the parametric form, } \frac{dx}{dt} \text{ and } \frac{dy}{dt} \text{ give the components of motion.}

\end{aligned}

Remove the integral operative for speed

Polar Area

\begin{aligned}

\text{Polar Area:} \quad

&A = \tfrac{1}{2} \int_{\theta_{1}}^{\theta_{2}} \bigl[r(\theta)\bigr]^2\, d\theta \\[6pt]

\text{Hint:} \quad

&\text{Divide the region into infinitesimal sectors of angle } d\theta\,.

\end{aligned}