AP Calculus AB  ·  Study Notebook
Calculus
Core Problems
20 Must-Know Questions 🎯
∫ lim d/dx
★ Worked Examples ⚡ Tricky Spots
Name:
Fill in answers, notes & work in the spaces provided ✏️
1 Limits — Direct Substitution vs. Factoring Unit 1
Memory Key
SIFT
Substitute first → If 0/0, Factor & cancel → Try again
Worked Example
Evaluate:  \(\displaystyle\lim_{x \to 3} \frac{x^2 - 9}{x - 3}\)
Step 1: Plug in \(x=3\) → get \(\frac{0}{0}\)  →  indeterminate! Use SIFT.
Step 2: Factor → \(\dfrac{(x+3)(x-3)}{x-3}\) → cancel \((x-3)\)
Step 3: \(\lim_{x\to 3}(x+3) = \boxed{6}\)
Your Turn ⭐⭐
Q1.
Evaluate:  \(\displaystyle\lim_{x \to -2} \frac{x^2 + 5x + 6}{x + 2}\)
Plugging in gives \(\frac{0}{0}\)! Don't stop here — factor first!
Answer: ___________
2 Limits at Infinity Unit 1
Memory Key
HATH
Highest degree on top vs. bottom → Ask: same, top bigger, or bottom bigger? → Top=Bottom: ratio of coefficients → Horizontal asymptote answer
Your Turn ⭐⭐
Q2.
Find:  \(\displaystyle\lim_{x \to \infty} \frac{4x^3 - 2x}{7x^3 + x^2}\)
Hint: divide every term by the highest power of \(x\) in the denominator.
  • \(0\)
  • \(\dfrac{4}{7}\)
  • \(\infty\)
  • \(-\dfrac{2}{7}\)
Answer: ( ) Reason: ___________________________
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3 Continuity — The 3-Step Check Unit 1
Memory Key
DEL
Defined at \(c\) → Exists (limit exists) → Limit equals \(f(c)\)
All three? ✅ Continuous. Any one fails? ❌ Discontinuous.
Worked Example
Is \(f(x)=\begin{cases} \dfrac{x^2-4}{x-2} & x\neq 2 \\ 5 & x=2 \end{cases}\) continuous at \(x=2\)?
D: \(f(2)=5\) ✅
E: \(\lim_{x\to 2}\frac{x^2-4}{x-2}=\lim_{x\to 2}(x+2)=4\) ✅
L: \(4 \neq 5\) ❌ → NOT continuous (removable discontinuity)
Your Turn ⭐⭐⭐
Q3.
Find the value of \(k\) that makes \(f\) continuous at \(x=1\):

\(f(x)=\begin{cases} 3x+k & x < 1 \\ x^2+4 & x \geq 1 \end{cases}\)
Both pieces must give the SAME value at \(x=1\). Set them equal!
\(k\) = ___________
4 Definition of the Derivative Unit 2
Memory Key
SLIDE
\(f'(x)=\displaystyle\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}\)
Slope · Limit · Increment \(h\) · Difference · Equals derivative
Your Turn ⭐⭐
Q4.
Use the limit definition to find \(f'(x)\) if \(f(x)=x^2+3x\).
Show every algebra step — AP exam graders look for this!
\(f'(x)\) = ___________
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5 Basic Differentiation Rules Unit 2
Memory Key
PECSS
Power rule: \(\frac{d}{dx}x^n = nx^{n-1}\)  ·  Exponential: \((e^x)'=e^x\)  ·  Constant: \((c)'=0\)  ·  Sin/Cos: \((\sin x)'=\cos x,\; (\cos x)'=-\sin x\)  ·  Sum: derivative distributes over \(+\) and \(-\)
Your Turn ⭐
Q5.
Differentiate each. Fill in the blanks.
a)  \(f(x)=5x^4-3x^2+7\)    \(f'(x)=\)
b)  \(g(x)=\sqrt{x}+\dfrac{1}{x^2}\)    \(g'(x)=\)
c)  \(h(x)=3e^x - 2\sin x\)    \(h'(x)=\)
Rewrite \(\sqrt{x}=x^{1/2}\) and \(\frac{1}{x^2}=x^{-2}\) before differentiating!
6 Product & Quotient Rule Unit 2
Memory Key
HI · DLO – LO · DHI / LO²
Quotient Rule:  \(\displaystyle\left(\frac{f}{g}\right)'=\frac{g\cdot f'-f\cdot g'}{g^2}\)
Product Rule:   \((fg)'=f'g+fg'\)  → mnemonic: "first D-second + second D-first"
Your Turn ⭐⭐
Q6.
Find \(y'\) for:  \(\displaystyle y=\frac{x^2\sin x}{\cos x + 1}\)
💡 Hint: Use quotient rule. The numerator itself needs the product rule!
Don't forget to differentiate the ENTIRE numerator \(x^2\sin x\) as a product first.
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7 Chain Rule Unit 2
Memory Key
OUTSIDE-IN
Derivative of the OUTSIDE (keep inside) × derivative of the INSIDE
\(\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\)
Think: "peel the layers like an onion 🧅"
Worked Example
Find \(\dfrac{d}{dx}[\sin(3x^2)]\)
Outside: \(\sin(\square)\) → derivative: \(\cos(\square)\)
Inside: \(3x^2\) → derivative: \(6x\)
Answer: \(\cos(3x^2) \cdot 6x = 6x\cos(3x^2)\) ✓
Your Turn ⭐⭐
Q7.
Differentiate each using the Chain Rule:
a)  \(y=(4x^3-1)^5\)    \(y'=\)
b)  \(y=e^{-2x^2}\)    \(y'=\)
c)  \(y=\ln(\cos x)\)    \(y'=\)
8 Implicit Differentiation Unit 3
Memory Key
TAG-IT
Every time you differentiate a "y" term, TAG it with \(\cdot\dfrac{dy}{dx}\)
Then collect all \(\frac{dy}{dx}\) terms on one side and solve!
Your Turn ⭐⭐⭐
Q8.
Find \(\dfrac{dy}{dx}\) given:  \(x^2+y^2=25\)
When you differentiate \(y^2\), you get \(2y\cdot\frac{dy}{dx}\), not just \(2y\)!
\(\dfrac{dy}{dx}=\) ___________
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9 Related Rates Unit 4
Memory Key
DRIVE
Draw a picture → Relate variables (write equation) → Implicitly differentiate with respect to \(t\) → Values: plug in known values → Evaluate the unknown rate
Worked Example
A spherical balloon is being inflated. The radius is increasing at \(2\) cm/s. How fast is the volume increasing when \(r=3\) cm?
Relate: \(V=\frac{4}{3}\pi r^3\)
Differentiate wrt \(t\): \(\frac{dV}{dt}=4\pi r^2\cdot\frac{dr}{dt}\)
Plug in: \(\frac{dV}{dt}=4\pi(3)^2(2)=72\pi\) cm³/s
Your Turn ⭐⭐⭐
Q9.
A ladder 10 ft long leans against a wall. The bottom slides away at 2 ft/s. How fast is the top sliding down when the bottom is 6 ft from the wall?
💡 Use Pythagorean theorem: \(x^2+y^2=100\)
Differentiate \(x^2+y^2=100\) implicitly with respect to \(t\), not \(x\)!
\(\dfrac{dy}{dt}=\) ___________ ft/s
10 Mean Value Theorem (MVT) Unit 5
Memory Key
SLOPE-MATCH
MVT says: somewhere in \((a,b)\) the instantaneous slope = average slope
\(f'(c) = \dfrac{f(b)-f(a)}{b-a}\)   for some \(c \in (a,b)\)
Your Turn ⭐⭐
Q10.
Find the value of \(c\) guaranteed by the MVT for \(f(x)=x^3-x\) on \([0,2]\).
  • \(c=\dfrac{2}{\sqrt{3}}\)
  • \(c=1\)
  • \(c=\dfrac{2}{3}\)
  • \(c=\sqrt{2}\)
Answer: ( ) Show your work above.
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11 Increasing / Decreasing & Critical Points Unit 5
Memory Key
SIGN-CHART
1. Set \(f'(x)=0\) or undefined → Critical numbers
2. Make a number line; test \(f'\) on each interval
\(f'>0\) → increasing ↗  |  \(f'<0\) → decreasing ↘
Your Turn ⭐⭐
Q11.
For \(f(x)=x^3-6x^2+9x+1\):
a) Find all critical numbers.
b) On which intervals is \(f\) increasing? Decreasing?
c) Classify each critical point as local max, local min, or neither.
Finding critical numbers requires \(f'(x)=0\) — then TEST each interval, don't just assume!
Critical #s: _______ Increasing: _______ Decreasing: _______ Local max/min: _______
12 Concavity & Inflection Points Unit 5
Memory Key
CUP / CAP
\(f''>0\) → concave UP ∪ (holds water like a CUP)
\(f''<0\) → concave DOWN ∩ (like a CAP / upside-down bowl)
Inflection Point: where concavity changes (not just \(f''=0\)!)
Your Turn ⭐⭐
Q12.
For \(f(x)=x^4-4x^3\), find:
a) \(f''(x)=\)
b) Intervals where \(f\) is concave up:
c) Inflection point(s):
An inflection point requires the concavity to actually CHANGE — verify with a sign chart for \(f''\)!
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13 Optimization (Applied Max/Min) Unit 5
Memory Key
RECODE
Read carefully → Express objective in 1 variable → Critical points → On closed interval? Check endpoints! → Decide max or min → Explain in context
Your Turn ⭐⭐⭐
Q13.
Classic AP Problem 🎯
A farmer has 200 meters of fencing and wants to enclose a rectangular field. What dimensions maximize the area?
Let width \(=x\). Then length \(=\frac{200-2x}{2}=\) _____ . Area \(A(x)=\) _____
Width = _____m, Length = _____m, Max Area = _____ m²
14 Basic Antiderivatives (Indefinite Integral) Unit 6
Memory Key
RAISE-N-DIVIDE
Power rule for integrals:  \(\displaystyle\int x^n\,dx = \frac{x^{n+1}}{n+1} + C\)
Raise the exponent by 1, then divide by the new exponent. Always add \(+C\)!
Your Turn ⭐
Q14.
Evaluate each integral:
a)  \(\displaystyle\int (6x^2 - 4x + 3)\,dx =\)
b)  \(\displaystyle\int \cos x\,dx =\)
c)  \(\displaystyle\int e^x\,dx =\)
Forgetting \(+C\) on an indefinite integral = automatic point deduction on FRQ!
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15 U-Substitution Unit 6
Memory Key
SWAP
See a composite? Pick the inside as \(u\) → Write \(du\) → All \(x\)'s gone? → Plug back in at end
Key: everything (including \(dx\)) must be replaced with \(u\) and \(du\)!
Worked Example
Evaluate: \(\displaystyle\int 2x(x^2+1)^4\,dx\)
Let \(u=x^2+1\) → \(du=2x\,dx\)
Substitute: \(\displaystyle\int u^4\,du = \frac{u^5}{5}+C=\frac{(x^2+1)^5}{5}+C\)
Your Turn ⭐⭐
Q15.
Evaluate:  \(\displaystyle\int \frac{3x^2}{x^3+5}\,dx\)
💡 What should \(u\) be? Check: does \(du\) appear in the integral?
Answer: ___________
16 Definite Integral & FTC Unit 6
Memory Key
ANTI-EVAL
FTC Part II:  \(\displaystyle\int_a^b f(x)\,dx = F(b)-F(a)\)
Find ANTIderivative \(F\), then EVALuate at top minus bottom.
No \(+C\) needed for definite integrals! ✅
Your Turn ⭐⭐
Q16.
Evaluate:
a)  \(\displaystyle\int_0^3 (2x+1)\,dx =\)
b)  \(\displaystyle\int_0^{\pi} \sin x\,dx =\)
For (b): \(\int\sin x\,dx = -\cos x\). Watch the negative sign when evaluating at bounds!
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17 FTC Part I — Derivative of an Integral Unit 6
Memory Key
PLUG-N-CHAIN
\(\dfrac{d}{dx}\displaystyle\int_a^{g(x)} f(t)\,dt = f(g(x))\cdot g'(x)\)
PLUG the upper limit into \(f\) → multiply by its CHAIN rule derivative
Worked Example
Find \(\dfrac{d}{dx}\displaystyle\int_1^{x^3}\cos(t)\,dt\)
Upper limit: \(g(x)=x^3\), so \(g'(x)=3x^2\)
Answer: \(\cos(x^3)\cdot 3x^2 = 3x^2\cos(x^3)\)
Your Turn ⭐⭐⭐
Q17.
Let \(\displaystyle G(x)=\int_2^{x^2} \sqrt{1+t^3}\,dt\).   Find \(G'(x)\).
  • \(\sqrt{1+x^3}\)
  • \(2x\sqrt{1+x^6}\)
  • \(\sqrt{1+x^6}\)
  • \(2x\sqrt{1+x^3}\)
Substitute \(t=x^2\) (not \(x\)) into the integrand, THEN multiply by \(\frac{d}{dx}(x^2)=2x\).
Answer: ( )
18 Area Between Curves Unit 8
Memory Key
TOP-MINUS-BOTTOM
\(\text{Area}=\displaystyle\int_a^b [\,f(x)-g(x)\,]\,dx\)
where \(f(x)\geq g(x)\) on \([a,b]\). Always TOP minus BOTTOM.
Step 0: Find intersection points — those are your limits \(a, b\)!
Your Turn ⭐⭐
Q18.
Find the area enclosed between \(y=x^2\) and \(y=x+2\).
Step 1: Set equal and find intersections → Step 2: Identify top function → Step 3: Integrate
If you're not sure which is on top, plug in a test point from the interval!
Area = ___________
— 9 —
19 Position, Velocity & Acceleration Unit 4 / 8
Memory Key
PVA LADDER
\(s(t)\) ←→ \(v(t)=s'(t)\) ←→ \(a(t)=v'(t)\)
Going down (→ right): differentiate    Going up (← left): integrate
Speed = \(|v(t)|\). Object at rest: \(v(t)=0\). Moving left: \(v(t)<0\).
Your Turn ⭐⭐⭐
Q19.
Full FRQ-Style Problem 📝
A particle moves along the \(x\)-axis with velocity \(v(t)=t^2-4t+3\) for \(t\geq 0\).
SHOW ALL WORK
a) When is the particle moving to the left?
b) Find the total distance traveled on \([0,4]\).
c) Find \(a(t)\) and determine when the particle is speeding up.
Total distance ≠ displacement! You must split the integral where \(v(t)=0\) and use absolute values.
20 Accumulation & Net Change Unit 6 / 8
Memory Key
NET = ANTI-EVAL
Net change in \(f\) from \(a\) to \(b\) = \(\displaystyle\int_a^b f'(x)\,dx = f(b)-f(a)\)
"Integral of the rate = total change in quantity" — works for anything: water, population, cost!
Your Turn ⭐⭐⭐
Q20.
Final Boss 🏆
Water flows into a tank at a rate of \(r(t)=3t^2+2t\) liters/min. The tank has 10 liters at \(t=0\).
a) How much water enters the tank from \(t=0\) to \(t=3\) minutes?
\(\displaystyle\int_0^3 r(t)\,dt =\) ___________
b) How much water is in the tank at \(t=3\)?
Answer: ___________ liters
For (b): Water at \(t=3\) = initial amount + amount added. Don't forget the initial 10 liters!
— 10 —

📚 Quick Memory Keys — All 20

Q1 · SIFT
Substitute → Indeterminate? → Factor → Try again
Q2 · HATH
Highest degree → Ask → Top=Bottom: ratio of coefficients → HA answer
Q3 · DEL
Defined → Exists → Limit=f(c)
Q4 · SLIDE
Slope · Limit · Increment · Difference · Equals
Q5 · PECSS
Power, Exponential, Constant, Sin/Cos, Sum rules
Q6 · HI·DLO–LO·DHI
Quotient rule song
Q7 · OUTSIDE-IN
Derivative of outside × derivative of inside
Q8 · TAG-IT
Every y term gets ·(dy/dx) tag
Q9 · DRIVE
Draw → Relate → Implicitly diff → Values → Evaluate
Q10 · SLOPE-MATCH
Instantaneous slope = average slope somewhere
Q11 · SIGN-CHART
Set f'=0 → test intervals → + increasing, − decreasing
Q12 · CUP/CAP
f''>0: concave up ∪ · f''<0: concave down ∩
Q13 · RECODE
Read → Express → Critical pts → On interval? → Decide → Explain
Q14 · RAISE-N-DIVIDE
∫xⁿdx = xⁿ⁺¹/(n+1) + C
Q15 · SWAP
See composite → let u = inside → write du → all x gone → sub back
Q16 · ANTI-EVAL
∫f = F(b)−F(a) · no +C needed
Q17 · PLUG-N-CHAIN
d/dx ∫ = f(g(x))·g'(x)
Q18 · TOP-MINUS-BOTTOM
Area = ∫[top − bottom] · find intersections first
Q19 · PVA LADDER
s ↔ v ↔ a · diff going down, integrate going up
Q20 · NET = ANTI-EVAL
Integral of rate = total net change + initial value
✏️ You've got this! Practice these daily and AP 5 is yours. 🚀
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