Overview & units
Unit Title Exam % Key Topics BC-1 Advanced Integration 17–20% (shared) Integration by parts, partial fractions, improper integrals BC-2 Logistic & Euler's Method 6–12% (shared) Logistic growth model, Euler's method for DE approximation BC-3 Parametric & Vectors 11–12% BC-only Parametric derivatives, arc length, vector-valued functions BC-4 Polar Coordinates 5–6% BC-only Polar graphs, polar area, intersections BC-5 Sequences & Series 17–18% BC-only Convergence tests (integral, comparison, ratio, alternating) BC-6 Power, Taylor & Maclaurin 17–18% BC-only Taylor/Maclaurin series, radius & interval of convergence, error bound
BC contains everything in AP Calculus AB plus the additional topics below — advanced integration, logistic models and Euler's method, parametric, polar and vector functions, and the full theory of sequences and series. The six BC chapters, worked free-response questions, and a 50-question practice exam are all here.
AP Calculus BC is a superset of AB. It shares Units 1–8 and adds:
The exam format matches AB (multiple choice 50%, free response 50%), and a BC result also reports an AB subscore.
Beyond AB
For a parametric curve, \( \dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt} \). Polar area is \( \displaystyle A=\tfrac12\int_\alpha^\beta r^2\,d\theta. \)
The ratio test: \( \sum a_n \) converges absolutely if \( \displaystyle\lim_{n\to\infty}\Big|\dfrac{a_{n+1}}{a_n}\Big|<1 \). Taylor series and their intervals of convergence are the centrepiece of Unit 10.
AP Calculus BC extends AB with additional integration techniques, convergence tests, parametric/polar calculus, and infinite series — approximately 40% of the exam is BC-exclusive material.
Advanced Integration Techniques
📌 AP Calc BC BC-1 — Advanced Integration 17–20% of exam · Integration by parts · Partial fractions · Improper integrals
BC extends the integral toolkit with integration by parts, partial fractions, and improper integrals over infinite or unbounded intervals.
By parts: \( \displaystyle\int u\,dv=uv-\int v\,du \). Partial fractions split a rational integrand into simpler pieces. Improper integrals are limits: \( \displaystyle\int_a^{\infty} f=\lim_{b\to\infty}\int_a^{b} f \).
Evaluate \( \displaystyle\int x e^{x}\,dx \) with \( u=x,\ dv=e^{x}dx \). \[ x e^{x}-\int e^{x}\,dx=(x-1)e^{x}+C. \]
Evaluate \(\displaystyle\int x^2 e^x\,dx\). Apply IBP twice. Let \(u=x^2,\,dv=e^x dx\): \(x^2e^x-2\int xe^x dx\). For \(\int xe^x dx\): \(u=x,\,dv=e^x dx\Rightarrow xe^x-e^x\).
Logistic Models & Euler's Method
📌 AP Calc BC BC-2 — Logistic & Euler's Method 6–12% of exam · Logistic growth model · Euler's method · Long-run behaviour
BC adds a richer population model and a numerical method for approximating solutions to differential equations.
Logistic: \( \dfrac{dP}{dt}=kP\!\left(1-\dfrac{P}{M}\right) \), carrying capacity \( M \), fastest growth at \( P=\tfrac{M}{2} \). Euler's method steps forward: \( y_{n+1}=y_n+h\,f(x_n,y_n) \).
For \( \dfrac{dy}{dx}=y,\ y(0)=1 \), take one Euler step with \( h=0.5 \). \[ y(0.5)\approx 1+0.5\cdot 1=1.5. \]
Use Euler's method with step \(h=0.1\) to approximate \(y(0.2)\) for \(\dfrac{dy}{dx}=y-x\), \(y(0)=1\).
\(y_1=y_0+hf(x_0,y_0)=1+0.1(1-0)=1.1\).
\(y_2=y_1+hf(x_1,y_1)=1.1+0.1(1.1-0.1)=1.1+0.1=1.2\).
Parametric & Vector-Valued Functions
📌 AP Calc BC BC-3 — Parametric & Vectors 11–12% of exam · Parametric derivatives & arc length · Vector-valued functions & motion
When a curve is traced by a parameter \( t \), motion in the plane is captured by component functions and their derivatives.
\( \dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt} \). Speed is \( \sqrt{(dx/dt)^{2}+(dy/dt)^{2}} \), and arc length is \( \displaystyle\int\sqrt{(dx/dt)^{2}+(dy/dt)^{2}}\,dt \). A vector position \( \mathbf r(t)=\langle x(t),y(t)\rangle \) has velocity \( \mathbf r'(t) \).
For \( x=t^{2},\ y=t^{3} \), find \( \dfrac{dy}{dx} \). \[ \frac{dy}{dx}=\frac{3t^{2}}{2t}=\frac{3t}{2}. \]
Find the arc length of \(x=t^2,\;y=\tfrac{2}{3}t^3\) from \(t=0\) to \(t=1\).\[\frac{dx}{dt}=2t,\quad\frac{dy}{dt}=2t^2.\]\[L=\int_0^1\sqrt{4t^2+4t^4}\,dt=\int_0^1 2t\sqrt{1+t^2}\,dt=\left[\tfrac{2}{3}(1+t^2)^{3/2}\right]_0^1=\tfrac{2}{3}(2^{3/2}-1)=\tfrac{2(2\sqrt2-1)}{3}.\]
Polar Coordinates
📌 AP Calc BC BC-4 — Polar Coordinates 5–6% of exam · Converting polar/rectangular · Polar area formula · Polar curve derivatives
Polar coordinates locate points by radius and angle, and turn area into a sweep of the radius about the origin.
The polar curve \( r=\sin(k\theta) \) traces a rose. Change \( k \) and count the petals — odd \( k \) gives \( k \) petals, even \( k \) gives \( 2k \).
| θ (rad) | r = sin(kθ) | (x, y) |
|---|
Polar area is \( \tfrac12\displaystyle\int r^{2}\,d\theta \) — one of the headline BC integrals.
Convert with \( x=r\cos\theta,\ y=r\sin\theta \). The area swept is \( A=\tfrac12\displaystyle\int_\alpha^\beta r^{2}\,d\theta \). Slope follows from treating \( x \) and \( y \) as parametric in \( \theta \).
Find the area enclosed by \( r=2 \). \[ \tfrac12\int_0^{2\pi}2^{2}\,d\theta=\tfrac12\cdot4\cdot2\pi=4\pi. \]
Find the area enclosed by one petal of \(r=\sin(2\theta)\). One petal: \(0\leq\theta\leq\pi/2\).
Infinite Sequences & Series — Convergence
📌 AP Calc BC BC-5 — Sequences & Series 17–18% of exam · Convergence tests · Alternating series error bound · Absolute vs. conditional
The heart of BC: deciding whether an infinite sum settles to a finite value, using a battery of convergence tests.
Watch the geometric series \( \sum_{n\ge1}\left(\tfrac12\right)^{n} \) converge: each partial sum \( S_N \) climbs toward the limit \( 1 \) (the amber line).
| n | S(n) |
|---|
\( S_N=1-\left(\tfrac12\right)^{N}\to1 \); a geometric series converges because \( |r|<1 \).
nth-term test: if \( a_n\not\to0 \), the series diverges. A geometric series converges when \( |r|<1 \); a p-series when \( p>1 \). The comparison, ratio, and alternating series tests handle the rest.
Evaluate \( \displaystyle\sum_{n=1}^{\infty}\left(\tfrac12\right)^{n} \). \[ \frac{\tfrac12}{1-\tfrac12}=1. \]
Determine whether \(\displaystyle\sum_{n=1}^\infty\dfrac{n!}{n^n}\) converges.\[L=\lim_{n\to\infty}\frac{(n+1)!/(n+1)^{n+1}}{n!/n^n}=\lim_{n\to\infty}\frac{n^n}{(n+1)^n}=\lim_{n\to\infty}\left(\frac{n}{n+1}\right)^n=\frac{1}{e}<1.\] Ratio test: \(L<1\), so the series converges.
Power, Taylor & Maclaurin Series
📌 AP Calc BC BC-6 — Power, Taylor & Maclaurin 17–18% of exam · Taylor/Maclaurin series · Radius & interval of convergence · Lagrange error
Functions can be rebuilt as infinite polynomials, exact within an interval of convergence — the capstone of the course.
The amber Maclaurin polynomial for \( \sin x \) hugs the true curve (indigo) ever more widely as you add terms.
| x | sin(x) | T(x) |
|---|
\( \sin x = x-\dfrac{x^{3}}{3!}+\dfrac{x^{5}}{5!}-\cdots \) — convergent for all \( x \).
The Taylor series about \( a \) is \( \displaystyle\sum \frac{f^{(n)}(a)}{n!}(x-a)^{n} \); centred at \( 0 \) it is a Maclaurin series. Memorise the series for \( e^{x} \), \( \sin x \), \( \cos x \) and \( \tfrac{1}{1-x} \), each with its interval of convergence.
Write the Maclaurin series for \( e^{x} \). \[ e^{x}=1+x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+\cdots=\sum_{n=0}^{\infty}\frac{x^{n}}{n!}. \]
The Maclaurin series for \(\cos x\) is \(1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}-\cdots\). Use the Lagrange error bound to bound the error when approximating \(\cos(0.1)\) with the first two non-zero terms. Using \(P_2(x)=1-\dfrac{x^2}{2}\). Error \(\leq\dfrac{|x|^4}{4!}\cdot\max|f^{(4)}|\). Since \(|f^{(4)}|=|\cos c|\leq1\):
Worked FRQs
A curve is given by \( x(t)=t^2 \), \( y(t)=t^3-3t \). Find the slope \( \dfrac{dy}{dx} \) at \( t=2 \). Differentiate each: \( \dfrac{dx}{dt}=2t \), \( \dfrac{dy}{dt}=3t^2-3 \). Then
Find the interval of convergence of \( \displaystyle\sum_{n=1}^{\infty}\frac{x^n}{n} \). Ratio test: \( \left|\dfrac{x^{n+1}/(n+1)}{x^n/n}\right|=|x|\cdot\dfrac{n}{n+1}\to|x| \), so it converges for \( |x|<1 \). Check endpoints: \( x=1 \) gives the divergent harmonic series; \( x=-1 \) gives the convergent alternating harmonic series.
Exam practice
For \( x=\cos t,\ y=\sin t \), find \( \dfrac{dy}{dx} \) at \( t=\tfrac{\pi}{2} \).
FRQ\( \dfrac{dy}{dx}=\dfrac{\cos t}{-\sin t}=-\cot t \); at \( t=\tfrac{\pi}{2} \), \( \cot\tfrac{\pi}{2}=0 \).
\[ \frac{dy}{dx}=0. \]Does \( \displaystyle\sum_{n=1}^{\infty}\frac{1}{n^2} \) converge or diverge?
MCQA \( p \)-series with \( p=2>1 \), so it converges (in fact to \( \pi^2/6 \)).
Find the area enclosed by the polar curve \( r=2 \).
FRQ\( A=\tfrac12\displaystyle\int_0^{2\pi} r^2\,d\theta=\tfrac12\int_0^{2\pi}4\,d\theta=\tfrac12\cdot 4\cdot 2\pi=4\pi \) — the circle of radius 2.
Differentiate \( y=x^4-3x^2 \).
MCQEvaluate \( \displaystyle\int_0^{2} 6x^2\,dx \).
MCQEvaluate \( \displaystyle\lim_{x\to 0}\frac{\tan x}{x} \).
MCQDifferentiate \( y=e^{x^2} \).
MCQEvaluate \( \displaystyle\int x\,e^{x^2}\,dx \).
MCQLet \( u=x^2 \), \( x\,dx=\tfrac12\,du \):
\[ \tfrac12 e^{x^2}+C. \]Differentiate \( y=\arctan x \).
MCQEvaluate \( \displaystyle\int \frac{1}{1+x^2}\,dx \).
MCQEvaluate \( \displaystyle\int x\,e^{x}\,dx \).
FRQBy parts:
\[ e^{x}(x-1)+C. \]Evaluate \( \displaystyle\int \ln x\,dx \).
FRQEvaluate \( \displaystyle\int x\sin x\,dx \).
FRQBy parts, \( u=x,\ dv=\sin x\,dx \):
\[ -x\cos x+\sin x+C. \]Evaluate \( \displaystyle\int \frac{1}{x^2-1}\,dx \).
FRQPartial fractions \( \tfrac12\big(\tfrac{1}{x-1}-\tfrac{1}{x+1}\big) \):
\[ \tfrac12\ln\left|\frac{x-1}{x+1}\right|+C. \]Evaluate \( \displaystyle\int_1^{\infty}\frac{1}{x^2}\,dx \).
FRQImproper: \( \big[-\tfrac1x\big]_1^{\infty}=0-(-1) \):
\[ 1. \]Evaluate \( \displaystyle\int_0^{\infty} e^{-x}\,dx \).
FRQEvaluate \( \displaystyle\sum_{n=0}^{\infty}\Big(\tfrac12\Big)^{n} \).
MCQEvaluate \( \displaystyle\sum_{n=0}^{\infty}\Big(\tfrac13\Big)^{n} \).
MCQEvaluate \( \displaystyle\sum_{n=1}^{\infty}\Big(\tfrac25\Big)^{n} \).
MCQFirst term \( \tfrac25 \), ratio \( \tfrac25 \):
\[ \frac{2/5}{1-2/5}=\frac23. \]Does \( \displaystyle\sum_{n=1}^{\infty}\frac1n \) converge?
MCQThe harmonic series:
\[ \text{diverges}. \]Does \( \displaystyle\sum_{n=0}^{\infty}(0.9)^{n} \) converge, and to what?
MCQGeometric with \( |r|<1 \):
\[ \frac{1}{1-0.9}=10. \]Does \( \displaystyle\sum_{n=1}^{\infty}\frac{1}{\sqrt{n}} \) converge?
MCQ\( p \)-series with \( p=\tfrac12\le 1 \):
\[ \text{diverges}. \]Does \( \displaystyle\sum_{n=1}^{\infty}\frac{n}{n+1} \) converge?
MCQTerms \( \to 1\neq 0 \), so by the \( n \)th-term test:
\[ \text{diverges}. \]Evaluate \( \displaystyle\sum_{n=0}^{\infty} 3\Big(\tfrac14\Big)^{n} \).
MCQFor what \( x \) does \( \displaystyle\sum_{n=0}^{\infty}\frac{x^n}{n!} \) converge?
FRQRatio test: \( \big|\tfrac{x}{n+1}\big|\to 0<1 \) always:
\[ \text{all real } x. \]Write the Maclaurin series for \( e^{x} \).
MCQGive the first three nonzero terms of the Maclaurin series for \( \sin x \).
FRQGive the first three terms of the Maclaurin series for \( \cos x \).
FRQUse a two-term Maclaurin polynomial for \( e^{x} \) to approximate \( e^{0.1} \).
FRQ\( e^{x}\approx 1+x \) at \( x=0.1 \):
\[ 1+0.1=1.1. \]Find the interval of convergence of \( \displaystyle\sum_{n=0}^{\infty} x^{n} \).
FRQGeometric, converges for \( |x|<1 \):
\[ (-1,1). \]Find the interval of convergence of \( \displaystyle\sum_{n=1}^{\infty}\frac{x^n}{n} \).
FRQRatio test gives \( |x|<1 \). Endpoints: \( x=1 \) diverges (harmonic), \( x=-1 \) converges (alternating):
\[ [-1,1). \]Evaluate \( \displaystyle\sum_{n=1}^{\infty}\frac{1}{n(n+1)} \).
FRQTelescoping: \( \tfrac1n-\tfrac{1}{n+1} \):
\[ \sum_{n=1}^{\infty}\Big(\tfrac1n-\tfrac{1}{n+1}\Big)=1. \]Does \( \displaystyle\sum_{n=1}^{\infty}\frac{(-1)^n}{n} \) converge?
MCQThe alternating harmonic series satisfies the alternating series test:
\[ \text{converges}. \]Find the radius of convergence of \( \displaystyle\sum_{n=0}^{\infty}\frac{x^n}{2^n} \).
FRQGeometric in \( \tfrac{x}{2} \); converges for \( \big|\tfrac{x}{2}\big|<1 \):
\[ R=2. \]For \( x=t^2,\ y=t^3 \), find \( \dfrac{dy}{dx} \).
MCQFor \( x=e^{t},\ y=e^{2t} \), find \( \dfrac{dy}{dx} \).
MCQFor \( x=t^2,\ y=2t \), find \( \dfrac{dy}{dx} \).
MCQFor \( x=3t,\ y=t^2-1 \), find \( \dfrac{dy}{dx} \).
MCQFind the arc length of \( x=t,\ y=t \) for \( t\in[0,1] \).
FRQFor \( x=t^3,\ y=t^2 \), find \( \dfrac{dy}{dx} \) at \( t=1 \).
MCQFind the speed of a particle with \( x=t^2,\ y=t^2 \) at \( t=1 \).
FRQSpeed \( =\sqrt{(2t)^2+(2t)^2}=2\sqrt2\,t \) at \( t=1 \):
\[ 2\sqrt2. \]Convert the polar point \( \big(2,\tfrac{\pi}{2}\big) \) to rectangular coordinates.
MCQ\( x=2\cos\tfrac{\pi}{2}=0,\ y=2\sin\tfrac{\pi}{2}=2 \):
\[ (0,2). \]Find the area enclosed by \( r=4 \).
MCQConvert the rectangular point \( (0,-3) \) to polar coordinates.
MCQ\( r=3 \), and the point is on the negative \( y \)-axis:
\[ \big(3,\tfrac{3\pi}{2}\big). \]Find the area enclosed by the cardioid \( r=1+\cos\theta \).
FRQConvert the rectangular point \( (1,1) \) to polar coordinates.
MCQ\( r=\sqrt2 \), \( \theta=\arctan 1=\tfrac{\pi}{4} \):
\[ \big(\sqrt2,\tfrac{\pi}{4}\big). \]Find the area enclosed by \( r=3 \).
MCQThe polar curve \( r=2\cos\theta \) is a circle. Find its enclosed area.
FRQIt is a circle of radius 1; equivalently \( \tfrac12\int_{-\pi/2}^{\pi/2}4\cos^2\theta\,d\theta \):
\[ \pi. \]Solve \( \dfrac{dy}{dx}=3y \) with \( y(0)=1 \).
FRQSolve \( \dfrac{dy}{dx}=-2y \) with \( y(0)=10 \).
MCQExponential decay, \( k=-2 \):
\[ y=10e^{-2x}. \]Test generator
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Mirrors the College Board AP Calculus BC exam: covers all AB topics plus series, parametric, polar, and vector-valued functions.
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BC chapters, FRQs & practice
All six BC topic chapters, the worked free-response questions, and the full practice exam with solutions.
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