AP Calculus AB
Study Suite

Everything you need to score a 5. Flashcards, games, AI-graded mock tests, and more.

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📌 Score a 5: Key Strategy

  • Know core theorems (IVT, EVT, MVT, FTC) deeply — not just the formula
  • Connect graphical, numerical, analytical, and verbal representations
  • Always attach units in context problems
  • Justify with sign charts — College Board wants evidence
  • Practice FRQ justifications out loud (say WHY)
  • Units 5 and 6 carry the most weight — master them first
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Flashcards

50+ cards covering all 8 units, trig derivatives, and the unit circle

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Study Games

Term Match, Lightning Round T/F, and Derivative Drill minigames

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Mock Test

20 MC + 2 FRQ. AI-graded with personalized feedback on weak areas

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Reference

Interactive unit circle, all 6 trig derivatives, formula cheatsheet

📋 Exam Structure

  • MCQ Part A: 30 questions, 60 min, NO calculator
  • MCQ Part B: 15 questions, 45 min, calculator OK
  • FRQ Part A: 2 questions, 30 min, calculator OK
  • FRQ Part B: 4 questions, 60 min, NO calculator

Flashcards

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Unit 1

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Study Games

Learn by doing — games are the fastest way to lock in concepts.

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Term Match

Match formulas to their names. Beat the clock!

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Lightning Round

True or False — rapid fire calculus statements

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Derivative Drill

Type the derivative — practice with all rule types

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Time: 60s
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Question: 1/15
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Find the derivative of f(x) =

Type your answer using standard notation: x^2, sin(x), e^x, etc.

📝 Full Mock Test

  • 20 multiple-choice questions covering all 8 units
  • 2 free-response questions requiring written work
  • Your FRQ answers are graded by Gemini AI with detailed feedback
  • You'll receive a breakdown of exactly which topics to review
  • No calculator for the first 15 MC (Part A simulation)

Free Response Questions

Show all work. Write out your reasoning clearly.

Test Results

🤖 AI Feedback & Study Plan

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Reference Sheet

x y 1 -1 1 -1

Selected Point

Click any point on the circle

Degrees—
Radians—
cos θ (x)—
sin θ (y)—
tan θ—
Point—

Memory Tips

30-45-60 trick:
sin values: ½, √2/2, √3/2
(denominator always 2)

ASTC rule:
Q1: All positive
Q2: Sin positive
Q3: Tan positive
Q4: Cos positive

"All Students Take Calculus"

Derivatives of All 6 Trig Functions

d/dx[sin x]
= cos x
d/dx[cos x]
= −sin x
d/dx[tan x]
= sec²x
d/dx[csc x]
= −csc x · cot x
d/dx[sec x]
= sec x · tan x
d/dx[cot x]
= −csc²x

Inverse Trig Derivatives

d/dx[arcsin x]
= 1 / √(1 − x²)
d/dx[arccos x]
= −1 / √(1 − x²)
d/dx[arctan x]
= 1 / (1 + x²)

🧠 Memory Pattern

  • sin → cos (positive); cos → −sin (negative)
  • tan → sec²x; cot → −csc²x (co- versions are negative)
  • sec → sec·tan; csc → −csc·cot (product of the two related functions)
  • All "co-" derivatives (cos, csc, cot) are negative

Limits & Continuity

Continuity at x=a requires
f(a) exists AND lim f(x) exists AND lim f(x) = f(a)
IVT
f continuous on [a,b] → hits every value between f(a) and f(b)
EVT
f continuous on [a,b] → has absolute max AND min
lim sin(x)/x as x→0
= 1

Differentiation

Limit Definition
f'(a) = lim[h→0] (f(a+h)−f(a)) / h
Power Rule
d/dx[xⁿ] = nxⁿ⁻¹
Product Rule
d/dx[fg] = f'g + fg'
Quotient Rule
d/dx[f/g] = (f'g − fg') / g²
Chain Rule
d/dx[f(g(x))] = f'(g(x)) · g'(x)
d/dx[eˣ]
= eˣ
d/dx[ln x]
= 1/x
d/dx[aˣ]
= aˣ · ln a
Implicit: x² + y² = 9
2x + 2y(dy/dx) = 0 → dy/dx = −x/y
Inverse Function Derivative
(f⁻¹)'(a) = 1 / f'(f⁻¹(a))

Applications of Differentiation

Linearization
L(x) = f(a) + f'(a)(x − a)
Tangent Line
y − f(a) = f'(a)(x − a)
Average Rate of Change
[f(b) − f(a)] / (b − a)
Mean Value Theorem
f'(c) = [f(b)−f(a)]/(b−a) for some c in (a,b)
Critical Points
where f'(x) = 0 or f'(x) DNE

Integration

FTC Part 1
d/dx[∫ₐˣ f(t)dt] = f(x)
FTC Part 2
∫ₐᵇ f'(x)dx = f(b) − f(a)
∫xⁿ dx
= xⁿ⁺¹/(n+1) + C (n ≠ −1)
∫eˣ dx
= eˣ + C
∫(1/x) dx
= ln|x| + C
∫sin x dx
= −cos x + C
∫cos x dx
= sin x + C
∫sec²x dx
= tan x + C
Displacement
∫ₐᵇ v(t) dt
Total Distance
∫ₐᵇ |v(t)| dt

Differential Equations & Applications

Separation of Variables
dy/h(y) = g(x)dx → integrate both sides
Exponential Growth/Decay
dy/dt = ky → y = Ceᵏᵗ
Logistic Model
dP/dt = kP(1−P/M), carrying capacity = M
Area Between Curves
∫ₐᵇ (top − bottom) dx
Disk Method
V = π∫ₐᵇ [R(x)]² dx
Washer Method
V = π∫ₐᵇ [R(x)² − r(x)²] dx
Average Value
f_avg = (1/(b−a)) ∫ₐᵇ f(x) dx
Position from Velocity
s(b) = s(a) + ∫ₐᵇ v(t) dt

Standard Trig Values

Degrees Radians sin θ cos θ tan θ

🎯 Key Identities for AP

  • sin²x + cos²x = 1
  • 1 + tan²x = sec²x
  • 1 + cot²x = csc²x
  • sin(2x) = 2 sin x cos x
  • cos(2x) = cos²x − sin²x = 1 − 2sin²x