AP Calculus BC

Core Problems · Key Topics · 20 Must-Know Questions

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QUESTION 01Chain Rule & Implicit Differentiation

UNIT 3 · DIFFERENTIATION

Implicit + Chain Combined

🔑 INSTANT KEY

DRAG-THE-Y

Every time you differentiate a y-term, drag out a dy/dx by Chain Rule.

Find \(\dfrac{dy}{dx}\) if \(x^2 + 3xy + y^3 = 7\)

Differentiate both sides with respect to \(x\):

\(2x + 3y + 3x\dfrac{dy}{dx} + 3y^2\dfrac{dy}{dx} = 0\)

Factor: \(\dfrac{dy}{dx}(3x + 3y^2) = -2x - 3y\)

\(\dfrac{dy}{dx} = \dfrac{-2x-3y}{3x+3y^2}\)

Forgetting to apply the product rule to \(3xy\) term! It gives TWO terms.

Find \(\dfrac{dy}{dx}\): \(\sin(xy) + y^2 = x\)

QUESTION 02L'Hôpital's Rule

UNIT 1 · LIMITS

Indeterminate Forms

🔑 INSTANT KEY

0/0 or ∞/∞ → HOPITAL

Check the form FIRST, then differentiate top and bottom separately. Never use quotient rule here!

\(\displaystyle\lim_{x\to 0}\frac{\sin 3x}{5x}\)

Form: \(\frac{0}{0}\) ✓ → Apply L'Hôpital:

\(= \displaystyle\lim_{x\to 0}\frac{3\cos 3x}{5} = \frac{3(1)}{5} = \boxed{\frac{3}{5}}\)

Applying L'Hôpital when the form is NOT \(0/0\) or \(\infty/\infty\) — always verify first!

\(\displaystyle\lim_{x\to\infty}\frac{3x^2 - 1}{7x^2 + 5x}\) = ?

Bonus: What if the exponent in the numerator were larger?

QUESTION 03Fundamental Theorem of Calculus

UNIT 5 · INTEGRATION

FTC Part 1 — Derivative of an Integral

🔑 INSTANT KEY

PLUG-IN + CHAIN

\(\dfrac{d}{dx}\int_a^{g(x)}f(t)\,dt = f\big(g(x)\big)\cdot g'(x)\)

If \(F(x)=\displaystyle\int_1^{x^3}\cos(t^2)\,dt\), find \(F'(x)\).

Upper limit is \(g(x)=x^3\), so \(g'(x)=3x^2\)

\(F'(x) = \cos\!\left((x^3)^2\right)\cdot 3x^2 = 3x^2\cos(x^6)\)

Substituting \(x^3\) into the integrand but forgetting to multiply by \(3x^2\) (chain rule!).

If \(G(x)=\displaystyle\int_0^{\sin x}\sqrt{1+t^4}\,dt\), find \(G'(x)\).

QUESTION 04Integration by Parts

UNIT 6 · TECHNIQUES OF INTEGRATION

IBP — LIATE Priority

🔑 INSTANT KEY

LIATE

Logarithm · Inverse trig · Algebraic · Trig · Exponential
Pick \(u\) from the LEFT side of the list.

\(\displaystyle\int x\,e^x\,dx\)

\(u=x,\quad dv=e^x dx\)

\(du=dx,\quad v=e^x\)

\(\displaystyle = xe^x - \int e^x\,dx = xe^x - e^x + C\)

\(= e^x(x-1)+C\)

Choosing \(u=e^x\) (exponential) over \(x\) (algebraic) — LIATE says algebra wins!

\(\displaystyle\int x^2\ln x\,dx\)

QUESTION 05U-Substitution

UNIT 6 · INTEGRATION

Definite Integral with U-Sub

🔑 INSTANT KEY

CHANGE THE BOUNDS

For definite integrals: convert bounds using u = g(x). Never switch back to x!

\(\displaystyle\int_0^2 x\,e^{x^2}\,dx\)

Let \(u=x^2 \Rightarrow du=2x\,dx \Rightarrow x\,dx = \tfrac{1}{2}du\)

Bounds: \(x=0\Rightarrow u=0\); \(x=2\Rightarrow u=4\)

\(= \dfrac{1}{2}\displaystyle\int_0^4 e^u\,du = \dfrac{1}{2}\big[e^u\big]_0^4 = \dfrac{e^4-1}{2}\)

Keeping the original x-bounds after substituting u — always convert bounds!

\(\displaystyle\int_1^e \frac{(\ln x)^2}{x}\,dx\)

QUESTION 06Partial Fractions

UNIT 6 · INTEGRATION TECHNIQUES

Decompose, Then Integrate

🔑 INSTANT KEY

COVER-UP METHOD

To find \(A\): cover the \((x-a)\) factor in denominator, substitute \(x=a\) into the rest.

\(\displaystyle\int\frac{3x+1}{(x-1)(x+2)}\,dx\)

Write: \(\dfrac{3x+1}{(x-1)(x+2)} = \dfrac{A}{x-1}+\dfrac{B}{x+2}\)

Cover-up: \(A=\frac{3(1)+1}{1+2}=\frac{4}{3}\), \(B=\frac{3(-2)+1}{-2-1}=\frac{5}{3}\)

\(=\dfrac{4}{3}\ln|x-1|+\dfrac{5}{3}\ln|x+2|+C\)

Forgetting absolute value signs in \(\ln|\,\cdot\,|\) — the AP graders will notice!

\(\displaystyle\int\frac{5}{x^2-4}\,dx\)

QUESTION 07Differential Equations — Separable

UNIT 7 · DIFFERENTIAL EQUATIONS

Separate & Solve with Initial Condition

🔑 INSTANT KEY

SPLIT-INTEGRATE-EXPONENTIATE

1) Move all y's left, all x's right. 2) Integrate both sides. 3) Solve for y using \(e^{\ln}\) trick. 4) Use IC to find C.

\(\dfrac{dy}{dx}=2xy\), \(y(0)=3\)

\(\dfrac{dy}{y}=2x\,dx \Rightarrow \ln|y|=x^2+C\)

\(y=Ae^{x^2}\). At \(x=0\): \(3=A\)

\(y=3e^{x^2}\)

Writing \(y = e^{x^2+C}\) then forgetting that \(e^C\) is just another constant \(A\).

Solve: \(\dfrac{dy}{dx}=\dfrac{x}{y}\), \(y(0)=4\)

QUESTION 08Slope Fields

UNIT 7 · DIFFERENTIAL EQUATIONS

Reading and Matching Slope Fields

🔑 INSTANT KEY

PLUG-AND-TILT

Plug specific \((x,y)\) points into \(\frac{dy}{dx}\). The number you get = slope = tilt of the tiny line segment at that point.

For \(\dfrac{dy}{dx}=x-y\), what is the slope at \((2,1)\)?

\(\dfrac{dy}{dx}\bigg|_{(2,1)} = 2-1 = 1\) → line tilts at 45° upward

Points where slope = 0: \(x=y\) (diagonal line, all horizontal segments)

Confusing where \(dy/dx=0\) (horizontal segments) with where the solution curve is constant.

For \(\dfrac{dy}{dx}=\dfrac{x}{2}-1\), find all points where the slope field shows horizontal segments. Then sketch the solution through \((0,0)\).

QUESTION 09Euler's Method

UNIT 7 · NUMERICAL METHODS

Step-by-Step Approximation

🔑 INSTANT KEY

NEW = OLD + SLOPE × STEP

\(y_{n+1} = y_n + f(x_n,\,y_n)\cdot\Delta x\) Repeat. That's literally it.

\(\dfrac{dy}{dx}=x+y\), \(y(0)=1\), \(\Delta x = 0.1\). Approximate \(y(0.2)\).

Step 1: slope at \((0,1)\) = \(0+1=1\). \(y_1 = 1+1(0.1)=1.1\)

Step 2: slope at \((0.1,\,1.1)\) = \(0.1+1.1=1.2\). \(y_2=1.1+1.2(0.1)=\mathbf{1.22}\)

Using the new x-value but the old y-value for the slope — always use the same step's \((x_n,y_n)\) pair!

\(\dfrac{dy}{dx}=y-x\), \(y(0)=2\), \(\Delta x=0.5\). Find \(y(1)\).

QUESTION 10Taylor & Maclaurin Series

UNIT 10 · INFINITE SERIES

Building the Series from Known Ones

🔑 MEMORIZE THESE 4

BIG FOUR

\(e^x = \sum_{n=0}^{\infty}\dfrac{x^n}{n!}\) \(\sin x = \sum_{n=0}^{\infty}\dfrac{(-1)^n x^{2n+1}}{(2n+1)!}\)
\(\cos x = \sum_{n=0}^{\infty}\dfrac{(-1)^n x^{2n}}{(2n)!}\) \(\dfrac{1}{1-x}=\sum_{n=0}^{\infty}x^n,\;|x|<1\)

Write the first 3 nonzero terms of \(e^{-x^2}\)

Substitute \(-x^2\) for \(x\) in \(e^x\) series:

\(e^{-x^2} = 1 + (-x^2) + \dfrac{(-x^2)^2}{2!} + \cdots = 1 - x^2 + \dfrac{x^4}{2} - \cdots\)

Computing derivatives over and over instead of substituting into a known series — much slower and error-prone!

Write the first 3 nonzero terms of \(\sin(3x)\)

QUESTION 11Convergence Tests

UNIT 10 · SERIES CONVERGENCE

Ratio Test & Alternating Series Test

🔑 INSTANT KEY

RATIO → L < 1 CONVERGES

Ratio Test: \(L = \lim_{n\to\infty}\left|\dfrac{a_{n+1}}{a_n}\right|\). If \(L<1\): converges. If \(L>1\): diverges. If \(L=1\): inconclusive (use another test).

Does \(\displaystyle\sum_{n=1}^{\infty}\frac{n!}{3^n}\) converge?

\(L=\lim_{n\to\infty}\left|\dfrac{(n+1)!/3^{n+1}}{n!/3^n}\right|=\lim_{n\to\infty}\dfrac{n+1}{3}=\infty\)

\(L>1\) → Diverges

When L = 1, many students incorrectly conclude convergence. You MUST use a different test!

Test convergence: \(\displaystyle\sum_{n=1}^{\infty}\frac{2^n}{n!}\)

QUESTION 12Interval of Convergence

UNIT 10 · POWER SERIES

Finding Radius & Interval of Convergence

🔑 INSTANT KEY

RATIO → ENDPOINTS CHECK

1) Apply Ratio Test, set \(L<1\), solve for \(x\). 2) ALWAYS check each endpoint separately by plugging back in.

Find the IOC of \(\displaystyle\sum_{n=1}^{\infty}\frac{x^n}{n}\)

Ratio Test: \(L=|x|\). Converges when \(|x|<1\), i.e., \(-1

Check \(x=1\): \(\sum\frac{1}{n}\) — Harmonic, diverges

Check \(x=-1\): \(\sum\frac{(-1)^n}{n}\) — Alternating Series, converges

IOC: \([-1,\;1)\)

Auto-using open brackets \((\;)\) without testing endpoints — endpoints can be included or excluded!

Find the IOC of \(\displaystyle\sum_{n=0}^{\infty}\frac{(x-2)^n}{3^n}\)

QUESTION 13Parametric Derivatives

UNIT 9 · PARAMETRIC & VECTOR

First and Second Derivative from Parametric

🔑 INSTANT KEY

DY/DX = (dy/dt) ÷ (dx/dt)

For 2nd derivative: \(\dfrac{d^2y}{dx^2} = \dfrac{d}{dt}\!\left(\dfrac{dy}{dx}\right)\div\dfrac{dx}{dt}\) — divide by \(dx/dt\) again!

\(x=t^2,\; y=t^3-3t\). Find \(\dfrac{dy}{dx}\) and \(\dfrac{d^2y}{dx^2}\).

\(\dfrac{dy}{dx}=\dfrac{3t^2-3}{2t}\)

\(\dfrac{d}{dt}\!\left(\dfrac{3t^2-3}{2t}\right)=\dfrac{3t^2+3}{2t^2}\)

\(\dfrac{d^2y}{dx^2}=\dfrac{(3t^2+3)/(2t^2)}{2t}=\dfrac{3t^2+3}{4t^3}\)

Taking \(d^2y/dx^2\) as \(\frac{d^2y/dt^2}{d^2x/dt^2}\) — this is WRONG. You must use the chain rule formula!

\(x=e^t, y=e^{2t}\). Find \(\dfrac{dy}{dx}\). Is the curve concave up?

QUESTION 14Arc Length — Parametric

UNIT 9 · PARAMETRIC

Arc Length Formula

🔑 INSTANT KEY

PYTHAGORAS IN TIME

\(L=\displaystyle\int_a^b\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt\) — tiny Pythagorean steps added up!

Find arc length: \(x=3t,\;y=4t\), \(0\le t\le 1\)

\(\dfrac{dx}{dt}=3,\;\dfrac{dy}{dt}=4\)

\(L=\displaystyle\int_0^1\sqrt{9+16}\,dt = 5\displaystyle\int_0^1 dt = \mathbf{5}\)

(Makes sense — it's a straight line segment!)

Forgetting to square both derivatives under the radical. It's \((dx/dt)^2+(dy/dt)^2\), not \(dx/dt+dy/dt\).

Set up (do not evaluate) the arc length integral for \(x=\cos t,\;y=\sin t\) on \([0,\,\pi/2]\)

QUESTION 15Polar Area

UNIT 9 · POLAR COORDINATES

Area Enclosed by Polar Curve

🔑 INSTANT KEY

HALF-R-SQUARED

\(A = \dfrac{1}{2}\displaystyle\int_\alpha^\beta [r(\theta)]^2\,d\theta\) — the \(\tfrac{1}{2}\) is always there. Never forget it.

Area inside \(r=2\sin\theta\)

One full loop: \(\theta\) from \(0\) to \(\pi\)

\(A=\dfrac{1}{2}\displaystyle\int_0^{\pi}4\sin^2\theta\,d\theta=2\displaystyle\int_0^{\pi}\dfrac{1-\cos 2\theta}{2}\,d\theta = \pi\)

Using wrong bounds — you MUST identify where the curve starts and ends one full loop (\(r=0\) to \(r=0\)).

Set up the area inside \(r=1+\cos\theta\) (one full loop: \(0\) to \(2\pi\))

QUESTION 16Improper Integrals

UNIT 6 · INTEGRATION

Infinite Bounds — Limit Definition

🔑 INSTANT KEY

REPLACE ∞ WITH LIMIT

\(\displaystyle\int_a^{\infty}f\,dx = \lim_{b\to\infty}\int_a^b f\,dx\). If the limit exists → converges. If not → diverges.

\(\displaystyle\int_1^{\infty}\frac{1}{x^2}\,dx\)

\(=\lim_{b\to\infty}\left[-\dfrac{1}{x}\right]_1^b = \lim_{b\to\infty}\left(-\dfrac{1}{b}+1\right)=1\) ✓ Converges

Compare: \(\displaystyle\int_1^{\infty}\frac{1}{x}\,dx = \lim_{b\to\infty}\ln b = \infty\) ✗ Diverges

Simply writing \(\big[-1/x\big]_1^{\infty}\) without the limit notation — this is not rigorous and loses points!

Evaluate: \(\displaystyle\int_0^{\infty}e^{-3x}\,dx\)

QUESTION 17Logistic Growth

UNIT 7 · DIFFERENTIAL EQUATIONS

Logistic Model — Key Features

🔑 INSTANT KEY

FASTEST AT HALF-K

Logistic: \(\dfrac{dP}{dt}=kP\!\left(1-\dfrac{P}{M}\right)\). Population grows fastest when \(P=\dfrac{M}{2}\) (half the carrying capacity).

\(\dfrac{dP}{dt}=0.4P\!\left(1-\dfrac{P}{500}\right)\)

Carrying capacity \(M=500\)

Fastest growth: \(P=250\), at that point \(\dfrac{dP}{dt}=0.4(250)(1-\tfrac{1}{2})=50\)

As \(t\to\infty\): \(P\to 500\) (horizontal asymptote)

Confusing \(k\) (growth rate) with \(M\) (carrying capacity). In the formula \(k\) is always the coefficient of \(P\).

\(\dfrac{dP}{dt}=2P\!\left(1-\dfrac{P}{300}\right)\)

a) Carrying capacity? b) When is \(dP/dt\) maximum?

QUESTION 18Lagrange Error Bound

UNIT 10 · SERIES

Bounding the Taylor Remainder

🔑 INSTANT KEY

MAX-DERIV OVER FACTORIAL

\(|R_n(x)| \le \dfrac{M\,|x-a|^{n+1}}{(n+1)!}\) where \(M\) = max of \(|f^{(n+1)}|\) on the interval.

Approximate \(\cos(0.1)\) using the degree-2 Maclaurin polynomial. Bound the error.

\(P_2(x)=1-\dfrac{x^2}{2}\). At \(x=0.1\): \(P_2=1-0.005=0.995\)

\(f^{(3)}(x)=\sin x\), max on \([0,0.1]\) is \(\sin(0.1)<0.1\)

\(|R_2|\le\dfrac{0.1\cdot(0.1)^3}{3!}=\dfrac{0.0001}{6}\approx 0.0000167\)

Using \(n\) instead of \(n+1\) in the exponent and denominator. The error uses one degree HIGHER than the polynomial.

Using degree-3 Maclaurin for \(e^x\), bound the error for \(e^{0.5}\).

QUESTION 19Vector-Valued Functions

UNIT 9 · VECTORS

Position, Velocity, Speed, Acceleration

🔑 INSTANT KEY

PVA CHAIN

Position \(\vec{r}(t)\) → differentiate → Velocity \(\vec{v}(t)\) → differentiate → Acceleration \(\vec{a}(t)\).
Speed \(=|\vec{v}(t)|=\sqrt{(x')^2+(y')^2}\) (a scalar!)

\(\vec{r}(t)=\langle t^2,\,t^3\rangle\)

\(\vec{v}(t)=\langle 2t,\,3t^2\rangle\)

\(\vec{a}(t)=\langle 2,\,6t\rangle\)

Speed at \(t=1\): \(|\vec{v}(1)|=\sqrt{4+9}=\sqrt{13}\)

Calling \(|\vec{v}|\) the "velocity" — velocity is a vector; speed is its magnitude (always ≥ 0).

\(\vec{r}(t)=\langle \sin t,\,e^{2t}\rangle\). Find speed at \(t=0\).

QUESTION 20Area Between Curves & Accumulation

UNIT 8 · APPLICATIONS OF INTEGRATION

Top − Bottom, Find Intersections First

🔑 INSTANT KEY

TOP MINUS BOTTOM

\(A=\displaystyle\int_a^b\big[f(x)-g(x)\big]\,dx\) where \(f\ge g\).
Step 0: Set \(f=g\) to find the bounds \(a\) and \(b\)!

Area between \(f(x)=x+2\) and \(g(x)=x^2\)

Intersections: \(x^2=x+2 \Rightarrow x=-1,\,2\)

\(A=\displaystyle\int_{-1}^{2}\big[(x+2)-x^2\big]\,dx=\left[\dfrac{x^2}{2}+2x-\dfrac{x^3}{3}\right]_{-1}^{2}=\dfrac{9}{2}\)

Integrating without finding intersections first — you'll use wrong bounds and get a completely wrong answer!

Find the area between \(y=x^2-4\) and \(y=x+2\).

QUICK REFERENCE · KEY WORDS TO REMEMBER

Master Keyword List

#	Topic	Magic Word	Never Forget
1	Implicit Diff.	DRAG-THE-Y	chain rule on every y-term
2	L'Hôpital	0/0 or ∞/∞ → HOPITAL	check form first!
3	FTC Part 1	PLUG-IN + CHAIN	multiply by upper limit's derivative
4	IBP	LIATE	Log wins over Exponential
5	U-Sub (definite)	CHANGE THE BOUNDS	never switch back to x
6	Partial Fractions	COVER-UP	\|absolute value\| in ln
7	Sep. ODE	SPLIT-INTEGRATE-EXPONENTIATE	\(e^C = A\)
8	Slope Fields	PLUG-AND-TILT	slope = 0 → horizontal segment
9	Euler's Method	NEW = OLD + SLOPE × STEP	same step pair \((x_n,y_n)\)
10	Taylor Series	BIG FOUR	substitute, don't re-derive
11	Ratio Test	L < 1 CONVERGES	L=1 is INCONCLUSIVE
12	IOC	RATIO → ENDPOINTS CHECK	always test endpoints
13	Parametric dy/dx	DY/DX = (dy/dt)÷(dx/dt)	2nd deriv: divide by dx/dt again
14	Arc Length	PYTHAGORAS IN TIME	square BOTH derivatives
15	Polar Area	HALF-R-SQUARED	\(\frac{1}{2}\int r^2 d\theta\) — always ½!
16	Improper Integral	REPLACE ∞ WITH LIMIT	write the lim notation!
17	Logistic Growth	FASTEST AT HALF-K	max growth at \(P=M/2\)
18	Lagrange Error	MAX-DERIV OVER FACTORIAL	use degree \(n+1\)
19	Vector Functions	PVA CHAIN	speed = magnitude (scalar)
20	Area Between Curves	TOP MINUS BOTTOM	find intersections FIRST

📝 NOTES

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