AP CALC
——
BC
STUDY
NOTEBOOK
📐 Self-Study Workbook

AP Calculus BC

20 Core Problems — High-Frequency Mistakes
Fill in the blanks · Work the problems · Memorize the keys
📋 Topics Covered:
■ Limits & Continuity   ■ Derivatives & Applications   ■ Integration Techniques
■ Differential Equations   ■ Series & Convergence   ■ Parametric & Polar
Name
Date
Score Goal
UNIT 1 · Limits & Continuity
LIMITS L'Hôpital's Rule & Indeterminate Forms p. 1
# 01
Evaluate: \(\displaystyle\lim_{x \to 0} \frac{e^{3x} - 1 - 3x}{x^2}\) ⚠ trap!
IF you get \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\) → L'HÔP
DIFFERENTIATE top & bottom SEPARATELY
🚫 Do NOT use quotient rule on L'Hôpital!
✏ EXAMPLE (Worked)
\(\displaystyle\lim_{x\to 0}\frac{e^x - 1}{x}\) — plug in: \(\frac{0}{0}\) ✓ use L'Hôpital
\(\to \displaystyle\lim_{x\to 0}\frac{e^x}{1} = e^0 = \boxed{1}\)
For \(\frac{e^{3x}-1-3x}{x^2}\): plug in first — get \(\frac{0}{0}\). Apply L'Hôpital TWICE.
🔗 RELATED PROBLEM
Find \(\displaystyle\lim_{x \to \infty} x \cdot e^{-x}\).   (Rewrite as \(\frac{x}{e^x}\) first, then L'Hôpital)
Your work ↓
Q1
LIMITS Squeeze Theorem & Trig Limits p. 2
# 02
Evaluate: \(\displaystyle\lim_{x \to 0} \frac{\sin(5x)}{3x}\) ⚠ coefficient trap
GOLDEN TRIG LIMIT: \(\displaystyle\lim_{\theta \to 0}\frac{\sin\theta}{\theta} = 1\)
Key trick: MULTIPLY & DIVIDE to match form
\(\frac{\sin(5x)}{3x} = \frac{5}{3}\cdot\frac{\sin(5x)}{5x}\)
⚡ TRIG LIMIT TWINS
\(\lim_{x\to0}\frac{\sin x}{x}=1\)
\(\lim_{x\to0}\frac{1-\cos x}{x}=0\)
🔗 RELATED PROBLEM
Evaluate \(\displaystyle\lim_{x \to 0} \frac{\tan(4x)}{\sin(2x)}\). Hint: \(\tan = \frac{\sin}{\cos}\). Split and apply golden limit twice.
Your work ↓
UNIT 2 · Derivatives & Applications
DERIVATIVES Chain Rule — Multi-Level p. 3
# 03
Let \(f(x) = \sin^3(e^{2x})\). Find \(f'(x)\). ⚠ 3-layer chain
CHAIN RULE = OUTSIDE → INSIDE
Peel like an onion 🧅
Layer 1: \((\ )^3\)   Layer 2: \(\sin(\ )\)   Layer 3: \(e^{2x}\)
\(f'= 3\sin^2(e^{2x})\cdot\cos(e^{2x})\cdot e^{2x}\cdot 2\)
✏ EXAMPLE (Shorter chain)
\(g(x) = \cos(3x^2)\)
\(g'(x) = -\sin(3x^2) \cdot 6x\) ← don't forget the inner derivative!
🔗 RELATED PROBLEM
Differentiate \(h(x) = e^{\cos(x^2)}\).
Your work ↓
DERIVATIVES Implicit Differentiation p. 4
# 04
Given \(x^2y + y^3 = 5\), find \(\dfrac{dy}{dx}\). ⚠ product rule inside
Every time you see \(y\), attach \(\tfrac{dy}{dx}\)
PRODUCT RULE when you have \(xy\) together
Then ISOLATE \(\tfrac{dy}{dx}\) — factor it out!
Differentiate term by term: \(\frac{d}{dx}[x^2y] = 2xy + x^2\frac{dy}{dx}\) (product rule!)
🔗 RELATED PROBLEM
Find \(\dfrac{dy}{dx}\) if \(\sin(xy) = x + y\).
Your work ↓
DERIVATIVES Mean Value Theorem p. 5
# 05
Let \(f(x) = x^3 - 2x\) on \([0, 2]\). Find all values of \(c\) satisfying the MVT. ⚠ setup trap
MVT FORMULA: \(f'(c) = \dfrac{f(b)-f(a)}{b-a}\)
= "instantaneous = average"
✅ Check: f must be CONT on [a,b] & DIFF on (a,b)
(A) \(c = \frac{2\sqrt{3}}{3}\)
(B) \(c = \frac{\sqrt{6}}{3}\)
(C) \(c = 1\)
(D) No such \(c\) exists
🔗 RELATED PROBLEM
Explain why Rolle's Theorem is a special case of MVT. When does Rolle's apply?
Your work ↓
UNIT 3 · Integration Techniques
INTEGRATION U-Substitution p. 6
# 06
Evaluate: \(\displaystyle\int x\sqrt{x^2+4}\;dx\) ⚠ change limits for definite!
Look for: f(g(x)) · g'(x)
Let \(u = \) [the ugly inside thing]
Find \(du\), replace EVERYTHING including \(dx\)
DON'T FORGET +C for indefinite!
✏ EXAMPLE
\(\int 2x\cos(x^2)\,dx\): let \(u=x^2\), \(du=2x\,dx\)
\(= \int\cos(u)\,du = \sin(u)+C = \sin(x^2)+C\)
🔗 RELATED PROBLEM
Evaluate \(\displaystyle\int_0^1 \frac{x}{(x^2+1)^2}\,dx\). ⚠ Remember to change limits of integration when using u-sub on definite integrals!
Your work ↓
INTEGRATION Integration by Parts p. 7
# 07
Evaluate: \(\displaystyle\int x^2 e^x\,dx\) ⚠ must apply IBP TWICE
LIATE priority for \(u\):
Log · Inverse trig · Algebra · Trig · Ex
Formula: \(\int u\,dv = uv - \int v\,du\)
⚡ TABULAR METHOD saves time when you apply IBP repeatedly!
Signs alternate: + − + − ...
🔗 RELATED PROBLEM
Evaluate \(\displaystyle\int e^x\sin(x)\,dx\). The integral appears on BOTH sides — solve algebraically!
Your work ↓
INTEGRATION Partial Fraction Decomposition p. 8
# 08
Evaluate: \(\displaystyle\int \frac{3x+1}{x^2-x-2}\,dx\) ⚠ factor denominator first!
Step 1: FACTOR denominator
Step 2: Write \(\frac{A}{\text{factor}_1}+\frac{B}{\text{factor}_2}\)
Step 3: COVER UP or substitute roots to find A, B
Result: \(\int \frac{A}{x-r}\,dx = A\ln|x-r|+C\)
🔗 RELATED PROBLEM
Evaluate \(\displaystyle\int \frac{x^2}{x^2-4}\,dx\). ⚠ Degree of top ≥ bottom → do polynomial long division FIRST!
Your work ↓
INTEGRATION Improper Integrals p. 9
# 09
Determine if \(\displaystyle\int_1^{\infty} \frac{1}{x^2}\,dx\) converges. If so, find its value. ⚠ ∞ must be a LIMIT
REPLACE ∞ with variable t, take lim as t→∞
\(\int_1^\infty \frac{1}{x^p}\,dx\): CONV if p > 1, DIV if p ≤ 1
p-series test for quick check!
(A) Diverges
(B) Converges to \(\frac{1}{2}\)
(C) Converges to \(1\)
(D) Converges to \(2\)
🔗 RELATED PROBLEM
Does \(\displaystyle\int_1^\infty \frac{1}{x}\,dx\) converge? Compare with \(\frac{1}{x^2}\) and explain the difference.
Your work ↓
UNIT 4 · Differential Equations
DIFF EQ Separable Differential Equations p. 10
# 10
Solve: \(\dfrac{dy}{dx} = \dfrac{x}{y}\), given \(y(0) = 3\). ⚠ don't forget the ±
SEPARATE: y's left, x's right
\(y\,dy = x\,dx\) → integrate both sides
Use initial condition to find C
⚠ \(e^{x+C} = e^C \cdot e^x = Ae^x\) — rename \(e^C = A\)
🔗 RELATED PROBLEM
A population \(P\) grows at rate \(\dfrac{dP}{dt} = 0.04P\), \(P(0)=200\). Find \(P(t)\) and \(P(50)\).
Your work ↓
DIFF EQ Euler's Method p. 11
# 11
Use Euler's method with step size \(h = 0.5\) to approximate \(y(1)\) given \(\dfrac{dy}{dx} = x + y\), \(y(0) = 1\). ⚠ use updated values each step
EULER FORMULA: \(y_{n+1} = y_n + h \cdot f(x_n, y_n)\)
Build a table: \(x_n\) | \(y_n\) | \(f(x_n,y_n)\) | \(h\cdot f\)
Each new \(y\) uses the PREVIOUS \(y\) — update each step!
\(n\) \(x_n\) \(y_n\) \(f(x_n,y_n)\) \(h \cdot f\)
0 0 1
1 0.5
2 1.0 ? — —
Your work ↓
UNIT 5 · Series & Convergence
SERIES Convergence Tests — Ratio Test p. 12
# 12
Determine convergence: \(\displaystyle\sum_{n=1}^{\infty} \frac{n!}{n^n}\) ⚠ most students use wrong test
RATIO TEST: \(L = \lim_{n\to\infty}\left|\dfrac{a_{n+1}}{a_n}\right|\)
L < 1: CONVERGES   L > 1: DIVERGES   L = 1: inconclusive
✅ Use when you see: n!, r^n, n^n
TEST SELECTOR CHEAT SHEET
• \(n!\) or \(n^n\) → Ratio
• \(b_n^n\) → Root
• \(\frac{1}{n^p}\) → p-series
• Compare to known → Comparison
• Alt. signs → Alt. Series
🔗 RELATED PROBLEM
Does \(\displaystyle\sum_{n=1}^{\infty}\frac{2^n}{n!}\) converge? Apply the ratio test and simplify carefully.
Your work ↓
SERIES Alternating Series Test & Error Bound p. 13
# 13
For \(\displaystyle S = \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^2}\), approximate \(S\) using the first 4 terms and find an upper bound for the error. ⚠ error = next term's abs value
AST CONDITIONS:
1. Terms DECREASE in absolute value
2. \(\lim_{n\to\infty} b_n = 0\)
ERROR BOUND = \(|a_{N+1}|\) = first omitted term's abs value
🔗 RELATED PROBLEM
Distinguish: absolute convergence vs conditional convergence. Is \(\sum \frac{(-1)^n}{\sqrt{n}}\) absolutely or conditionally convergent?
Your work ↓
SERIES Power Series — Radius & Interval of Convergence p. 14
# 14
Find the interval of convergence: \(\displaystyle\sum_{n=0}^{\infty}\frac{(x-2)^n}{n\cdot 3^n}\) ⚠ CHECK ENDPOINTS!
Step 1: RATIO TEST → find R (radius)
Step 2: Check BOTH ENDPOINTS x = a±R separately
Step 3: Write interval with correct [ ] or ( )
⚠ Endpoints often converge one side, diverge the other!
✏ EXAMPLE (checking endpoints)
After finding \(|x-2| < 3\), i.e. \(x \in (-1, 5)\):
Check \(x = -1\): gives \(\sum\frac{(-3)^n}{n\cdot 3^n} = \sum\frac{(-1)^n}{n}\) → converges (Alt. Series)
Check \(x = 5\): gives \(\sum\frac{1}{n}\) → diverges (p-series, p=1)
Your work ↓
SERIES Taylor & Maclaurin Series p. 15
# 15
Write the Maclaurin series for \(f(x) = \cos(x^2)\) up to the \(x^8\) term. Find \(f^{(4)}(0)\). ⚠ DON'T differentiate 4 times!
KEY MACLAURIN SERIES — MEMORIZE:
\(e^x = \sum\frac{x^n}{n!}\)   \(\sin x = \sum\frac{(-1)^n x^{2n+1}}{(2n+1)!}\)
\(\cos x = \sum\frac{(-1)^n x^{2n}}{(2n)!}\)   \(\frac{1}{1-x}=\sum x^n\)
SUBSTITUTE into known series instead of computing derivatives!
Replace \(x\) with \(x^2\) in the \(\cos x\) series. The coefficient of \(x^4\) term = \(\frac{f^{(4)}(0)}{4!}\)
🔗 RELATED PROBLEM
Use the Maclaurin series to evaluate \(\displaystyle\lim_{x \to 0}\frac{\cos x - 1 + \frac{x^2}{2}}{x^4}\).
Your work ↓
UNIT 6 · Parametric, Vector & Polar
PARAMETRIC Derivatives & Arc Length of Parametric Curves p. 16
# 16
Given \(x = t^2 - 1\), \(y = t^3 - 3t\), find \(\dfrac{dy}{dx}\) and \(\dfrac{d^2y}{dx^2}\) at \(t = 2\). ⚠ 2nd deriv formula
\(\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}\) ← divide the derivatives

\(\dfrac{d^2y}{dx^2} = \dfrac{\dfrac{d}{dt}\!\left(\dfrac{dy}{dx}\right)}{dx/dt}\) ← NOT \(\frac{d^2y/dt^2}{d^2x/dt^2}\)!!
🔗 RELATED PROBLEM
For the same curve, find the arc length from \(t=0\) to \(t=1\).
\(L = \displaystyle\int_0^1\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt\)
Your work ↓
POLAR Area in Polar Coordinates p. 17
# 17
Find the area enclosed by \(r = 2\cos\theta\). ⚠ get the limits right!
POLAR AREA: \(A = \dfrac{1}{2}\displaystyle\int_{\alpha}^{\beta} r^2\,d\theta\)
Find limits: set \(r=0\) or trace one full loop
\(r=2\cos\theta\): circle of radius 1, center \((1,0)\)
Limits: 0 to π traces it once
Use the identity \(\cos^2\theta = \frac{1+\cos(2\theta)}{2}\) to integrate \(r^2 = 4\cos^2\theta\)
🔗 RELATED PROBLEM
Find the area of one petal of \(r = \sin(3\theta)\). How many petals does it have? What are the limits for ONE petal?
Your work ↓
UNIT 7 · Fundamental Theorem & Applications
FTC Fundamental Theorem of Calculus Part 2 p. 18
# 18
Let \(g(x) = \displaystyle\int_1^{x^3}\!\!\sqrt{1+t^4}\,dt\). Find \(g'(x)\). ⚠ chain rule with upper limit!
FTC Part 2: \(\dfrac{d}{dx}\displaystyle\int_a^x f(t)\,dt = f(x)\)
CHAIN RULE VERSION: upper limit is \(u(x)\)
\(\to f(u(x))\cdot u'(x)\)
Here: \(g'(x) = \sqrt{1+(x^3)^4}\cdot 3x^2\)
🔗 RELATED PROBLEM
Let \(F(x) = \displaystyle\int_x^{x^2}\cos(t^2)\,dt\). Find \(F'(x)\). ⚠ BOTH limits depend on x — split into two integrals using \(\int_x^{x^2} = \int_x^a + \int_a^{x^2}\)
Your work ↓
APPLICATIONS Volume — Washer vs Shell Method p. 19
# 19
The region bounded by \(y = x^2\) and \(y = x\) is revolved about the x-axis. Find the volume. ⚠ which is outer / inner?
WASHER METHOD (about x-axis):
\(V = \pi\displaystyle\int_a^b\!\left[R(x)^2 - r(x)^2\right]dx\)
\(R\) = OUTER radius, \(r\) = INNER radius
Check which function is on TOP in the region!
DISK vs WASHER vs SHELL
— No hole → DISK
— Hole in middle → WASHER
— Rotating about y / parallel axis → SHELL often easier
🔗 RELATED PROBLEM
Same region, but revolved about the y-axis. Use the shell method: \(V = 2\pi\displaystyle\int_a^b x\cdot[f(x)-g(x)]\,dx\)
Your work ↓
DIFF EQ Logistic Growth p. 20
# 20
A population follows \(\dfrac{dP}{dt} = 0.3P\!\left(1-\dfrac{P}{500}\right)\). ⚠ 5 things to know!
(a) What is the carrying capacity?   (b) When is growth fastest?   (c) Does the population always increase?
\(\dfrac{dP}{dt} = kP(1-\frac{P}{M})\)  ← LOGISTIC MODEL
\(M\) = CARRYING CAPACITY (max population)
Fastest growth at P = M/2
Growth SLOWS as P → M   Growth = 0 at P = M or P = 0
(A) Carrying capacity: 300
(B) Carrying capacity: 500
(C) Fastest at P = 250
(D) Both (B) and (C)
🔗 RELATED PROBLEM
Solve the logistic equation by partial fractions: \(\dfrac{dP}{P(1-P/M)} = k\,dt\). Show that \(P(t) = \dfrac{M}{1+Ae^{-kt}}\) where \(A = \dfrac{M-P_0}{P_0}\).
Your work ↓
⚡ QUICK REFERENCE — Memory Keys
REFERENCE 20 Golden Memory Keys ref
#1 L'Hôpital → only \(\frac{0}{0},\frac{\infty}{\infty}\)
INDETERMINATE CHECK FIRST
#2 \(\lim\frac{\sin\theta}{\theta}=1\)
GOLDEN TRIG LIMIT
#3 Chain Rule → OUTSIDE→IN
PEEL THE ONION
#4 Implicit → every \(y\) gets \(\frac{dy}{dx}\)
PRODUCT RULE on xy terms
#5 MVT: \(f'(c)=\frac{f(b)-f(a)}{b-a}\)
INST = AVERAGE
#6 U-sub: let u = ugly inside
CHANGE LIMITS for definite!
#7 IBP: LIATE for choosing u
\(\int u\,dv = uv - \int v\,du\)
#8 Partial fractions: FACTOR first
DEGREE TOP ≥ BOTTOM → divide!
#9 Improper: ∞ → LIMIT
p-series: conv if \(p>1\)
#10 Separable: SEPARATE, INTEGRATE
\(e^{C}\cdot e^x = Ae^x\)
#11 Euler: \(y_{n+1}=y_n+hf(x_n,y_n)\)
TABLE method — update each step
#12 Ratio test: \(L=\lim|\frac{a_{n+1}}{a_n}|\)
Use for \(n!, r^n, n^n\)
#13 Alt. series error = \(|a_{N+1}|\)
NEXT TERM is the bound
#14 Power series: Ratio Test → R
CHECK BOTH ENDPOINTS!
#15 Maclaurin: SUBSTITUTE not compute
Coeff of \(x^n\) = \(\frac{f^{(n)}(0)}{n!}\)
#16 Parametric 2nd deriv:
\(\frac{d(dy/dx)/dt}{dx/dt}\) — NOT \(\frac{d^2y}{d^2x}\)
#17 Polar area: \(\frac{1}{2}\int r^2\,d\theta\)
LIMITS = one full trace of curve
#18 FTC2 + chain: \(f(u(x))\cdot u'(x)\)
Both limits vary → SPLIT integral
#19 Washer: \(\pi\int(R^2-r^2)dx\)
OUTER² minus INNER²
#20 Logistic: carrying cap = M
Fastest growth at P = M/2
✨ You've got this. Work every problem. Write every step. ✨