∫ AP Calculus AB Complete Cheatsheet 2026
Everything You Need to Know for AP Calculus AB Success
All 8 Units • Essential Formulas • Derivatives & Integrals • Exam Strategies
📈 Unit 1: Limits and Continuity 10-12%
📊 Basic Limits
Definition: limx→c f(x) approaches from BOTH sides
One-sided: limx→c+ f(x) from right, limx→c- f(x) from left
Properties: Linear, product, quotient rules
⚙️ Algebraic Techniques
Methods: Factoring, rationalization, completing square
L'Hôpital's Rule: For 0/0 or ∞/∞: lim f/g = lim f'/g'
📈 Growth Rates & Asymptotes
Order (fastest to slowest): x! > ax > xx > xln(x) > ln(x)
Horizontal Asymptotes: Compare degrees of numerator/denominator
🔗 Continuity & IVT
Continuity: f(c) = limx→c f(x)
Discontinuities: Removable (hole), jump, asymptote
IVT: Continuous f on [a,b], if k between f(a) and f(b), then ∃c where f(c) = k
📊 Unit 2: Fundamentals of Differentiation 10-12%
📐 Definition & Interpretation
Definition: f'(x) = limh→0 [f(x+h) - f(x)]/h
Meaning: Slope of tangent line, instantaneous rate of change
Key Fact: All differentiable functions are continuous
⚖️ Basic Derivative Rules
Power Rule: d/dx[xn] = nxn-1
Constant: d/dx[c] = 0
Sum/Difference: (f ± g)' = f' ± g'
Constant Multiple: (kf)' = kf'
📐 Trigonometric Derivatives
sin(x): cos(x)
cos(x): -sin(x)
tan(x): sec2(x)
sec(x): sec(x)tan(x)
📊 Exponential & Logarithmic
ex: ex
ln(x): 1/x
ax: ax ln(a)
loga(x): 1/(x ln(a))
🔗 Unit 3: Composite, Implicit, & Inverse Functions 9-13%
🔧 Advanced Rules
Product Rule: (uv)' = u'v + uv'
Quotient Rule: (u/v)' = (u'v - uv')/v2
Chain Rule: (f∘g)' = f'(g(x)) · g'(x)
Chain Rule Tip: Like unpeeling an onion - outside to inside
🔄 Implicit Differentiation
Method: Differentiate both sides with respect to x
Key: Multiply by dy/dx when differentiating y terms
Example: d/dx[xy] = y + x(dy/dx)
🔄 Inverse Trig Derivatives
arcsin(x): 1/√(1-x2)
arctan(x): 1/(1+x2)
arcsec(x): 1/(|x|√(x2-1))
Note: Co-functions are negatives of main functions
📊 Higher Order Derivatives
Method: Just repeat differentiation process
Notation: f''(x), f'''(x), f(n)(x)
Implicit 2nd: Functions of x, y, and dy/dx
🚀 Unit 4: Contextual Applications of Differentiation 10-15%
📈 Rate of Change Applications
Derivative: Rate of change of function
2nd Derivative: Rate of change of rate of change
Particle Motion: x(t) → v(t) = x'(t) → a(t) = v'(t)
🔗 Related Rates Steps
1. Draw picture
2. List knowns and unknowns
3. Write equation (don't plug in changing values)
4. Take d/dt of both sides
5. Substitute values and solve
📏 Linearization & L'Hôpital
Linearization: f(c + Δx) ≈ f(c) + f'(c)·Δx
L'Hôpital's Rule: Only for 0/0 or ∞/∞ forms
Remember: Check form before applying!
🎯 Unit 5: Analytical Applications of Differentiation 15-18%
📊 Mean Value Theorems
MVT: If f continuous on [a,b], differentiable on (a,b), then ∃c where f'(c) = [f(b)-f(a)]/(b-a)
Rolle's Theorem: Special MVT where f(a) = f(b)
EVT: Continuous function on closed interval has abs max/min
🔍 Finding Extrema
Critical Points: Where f'(x) = 0 or undefined
Local Extrema: Always at critical points or endpoints
Global Extrema: Absolute maximum or minimum
Remember: Check endpoints too!
📈 First & Second Derivative Tests
1st Derivative: + to - = max, - to + = min
2nd Derivative: f''(x) > 0 = min, f''(x) < 0 = max
Increasing/Decreasing: f'(x) > 0 = inc, f'(x) < 0 = dec
🏔️ Concavity & Optimization
Concavity: f''(x) > 0 = concave up, f''(x) < 0 = concave down
Inflection Points: Where f''(x) = 0 and concavity changes
Optimization: Draw picture → primary equation → constraints → solve
∫ Unit 6: Integration and Accumulation of Change 17-20%
📊 Basic Integration Concepts
Antiderivative: Reverse of differentiation
Definite Integral: Area under curve from a to b
Indefinite Integral: Family of antiderivatives (+ C)
📏 Riemann Sums
Left Sum: LRS = Σf(xi)Δx
Right Sum: RRS = Σf(xi+1)Δx
Midpoint: MRS = Σf((xi+xi+1)/2)Δx
Trapezoid: TRAP = Σ[(f(xi) + f(xi+1))/2]Δx
📜 Fundamental Theorems
FTOC Part 1: ∫ab f(x)dx = F(b) - F(a)
FTOC Part 2: d/dx[∫a(x)b(x) f(t)dt] = b'(x)f(b(x)) - a'(x)f(a(x))
🔧 Integration Techniques
U-Substitution: Choose inner functions or higher powers
Good u choices: Denominators, functions with derivatives present
Properties: Linearity, additivity, reversing limits
📐 Unit 7: Differential Equations and Mathematical Modeling 6-12%
📈 Slope Fields
Definition: Show tangent line segments for differential equations
Use: Visualize solutions without solving explicitly
🔄 Separation of Variables
When: DE can be written as g(y)dy = f(x)dx
Method: Integrate both sides separately
Result: Can leave answer implicit
📊 Solutions
General Solution: Contains arbitrary constants
Particular Solution: Uses initial conditions for specific constants
📈 Exponential Growth/Decay
Form: dy/dx = ky
Solution: y = cekx where c = y(0)
🏔️ Unit 8: Applications of Integration 10-15%
📊 Average Value & MVT for Integrals
Average Value: favg = (1/(b-a))∫ab f(x)dx
MVT for Integrals: ∃c ∈ (a,b) where f(c) = favg
🚀 Motion Applications
Displacement: ∫v(t)dt (can be negative)
Distance: ∫|v(t)|dt (always positive)
Key: Displacement ≠ Distance traveled
📐 Area Between Curves
Vertical strips: ∫[top - bottom]dx
Horizontal strips: ∫[right - left]dy
Note: Sometimes need to split region
🌊 Volume Applications
Disk Method: V = π∫[R(x)]2dx
Washer Method: V = π∫[(Router)2 - (Rinner)2]dx
Cross Sections: V = ∫[Area of cross section]dx
📐 Essential AP Calculus AB Formulas
🔧 Derivative Rules
Power Rule: d/dx[xn] = nxn-1
Product Rule: d/dx[uv] = u'v + uv'
Quotient Rule: d/dx[u/v] = (u'v - uv')/v2
Chain Rule: d/dx[f(g(x))] = f'(g(x)) · g'(x)
📐 Common Derivatives
d/dx[sin(x)] = cos(x)
d/dx[cos(x)] = -sin(x)
d/dx[tan(x)] = sec2(x)
d/dx[cot(x)] = -csc2(x)
d/dx[sec(x)] = sec(x)tan(x)
d/dx[csc(x)] = -csc(x)cot(x)
d/dx[ex] = ex
d/dx[ln(x)] = 1/x
d/dx[ax] = ax ln(a)
d/dx[arcsin(u)] = u' / √(1-u2)
d/dx[arctan(u)] = u' / (1+u2)
∫ Common Integrals
∫xn dx = xn+1/(n+1) + C, n≠-1
∫(1/x) dx = ln|x| + C
∫ex dx = ex + C
∫ax dx = ax/ln(a) + C
∫sin(x) dx = -cos(x) + C
∫cos(x) dx = sin(x) + C
∫sec2(x) dx = tan(x) + C
∫(1/√(a²-x²)) dx = arcsin(x/a) + C
∫(1/(a²+x²)) dx = (1/a)arctan(x/a) + C
📜 Fundamental Theorems
FTOC Part 1: ∫ab f(x)dx = F(b) - F(a)
FTOC Part 2: d/dx[∫ax f(t)dt] = f(x)
Average Value: favg = (1/(b-a))∫ab f(x)dx
📝 AP Calculus AB Exam Format
Total Time: 3 hours 15 minutes (Hybrid Digital Exam)
Section I: Multiple Choice
45 questions total
1 hour 45 minutes
50% of exam score
Part A: 30 questions, 60 min, NO calculator
Part B: 15 questions, 45 min, calculator REQUIRED
Section II: Free Response
6 questions total
1 hour 30 minutes
50% of exam score
Part A: 2 questions, 30 min, calculator REQUIRED
Part B: 4 questions, 60 min, NO calculator
Question Types
Algebraic, exponential, logarithmic functions
Trigonometric and general functions
Analytical, graphical, tabular representations
At least 2 real-world context FRQs
Mix of procedural and conceptual tasks
💡 AP Calculus AB Success Tips
Show all work, even when using calculator
Write equations/integrals before using calculator
Use standard notation, not calculator syntax (∫x²dx, not fnInt)
Label functions, graphs, tables clearly
Final answers accurate to 3 decimal places
Cross out incorrect work instead of erasing
Monitor time - don't spend too long on one question
Use shorthand: MVT, IVT, FTOC, EVT appropriately