Summary AP Calculus AB – High School Mathematics – 2025–20

AP Calculus AB Complete Study Guide
Created by Priece – High School Math Enthusiast​
Version: 1.0 (February 2026)​
Overview: This guide covers all 8 units of AP Calculus AB, aligned with the College Board
curriculum for the 2025–2026 exam. It includes key formulas, theorems, step-by-step examples,
common mistakes, AP exam tips, and practice problems with solutions. Use this for class
review, homework help, or AP exam prep.

How to Use This Guide:

●​ Read through units sequentially or jump to weak areas.
●​ Memorize the formula sheet at the end.
●​ Work practice problems without peeking at solutions.
●​ For visuals, graph functions on Desmos.com (free).

AP Exam Basics: 45% multiple-choice (45 questions, 105 min, some calculator), 55%
free-response (6 questions, 90 min, half calculator). Focus on justifying answers in FRQs.




Unit 1: Limits and Continuity (10–12% of Exam)
Key Concepts
●​ Limit: Value a function approaches as x approaches a point. Notation: lim_{x→a} f(x)
= L.
●​ One-sided limits: lim_{x→a⁻} (left), lim_{x→a⁺} (right). Limit exists if both match.
●​ Continuity: f is continuous at x=a if lim_{x→a} f(x) = f(a). Types: removable, jump,
infinite discontinuities.
●​ Theorems: Intermediate Value Theorem (IVT): If continuous on [a,b] and f(a) < k <
f(b), there’s c in (a,b) where f(c)=k.

Evaluation Methods
●​ Direct substitution (if defined).
●​ Factor/simplify for indeterminate forms (0/0).
●​ Special limits: lim_{x→0} sin(x)/x = 1, lim_{x→0} (1-cos(x))/x = 0.
●​ Asymptotes: Vertical (limit → ±∞), horizontal (limit as x→±∞).

Example 1: Algebraic Limit

, Find lim_{x→2} (x² - 4)/(x - 2).​
Step 1: Direct sub gives 0/0 (indeterminate).​
Step 2: Factor numerator: (x-2)(x+2)/(x-2).​
Step 3: Cancel (x-2): lim_{x→2} (x+2) = 4.

Example 2: One-Sided Limit
lim_{x→0⁺} √x = 0 (approaches from right).​
lim_{x→0⁻} √x = undefined (can’t approach from left in reals).

Common Mistakes & Tips
●​ Mistake: Forgetting to check one-sided for piecewise functions.
●​ Tip: On AP, sketch graphs mentally—limits from graphs are common MCQs.

Practice Problems
1.​ lim_{x→3} (x³ - 27)/(x - 3) = ?​
Solution: Factor: (x-3)(x²+3x+9)/(x-3) = lim_{x→3} (x²+3x+9) = 27.
2.​ Is f(x) = {x+1 if x<1, 2 if x=1, x² if x>1} continuous at x=1?​
Solution: lim_{x→1⁻} = 2, lim_{x→1⁺} = 1, f(1)=2. Not continuous (sides differ).




Unit 2: Differentiation Basics (10–12% of Exam)
Key Formulas
●​ Derivative: f’(x) = lim_{h→0} [f(x+h) - f(x)] / h.
●​ Power rule: d/dx [x^n] = n x^{n-1}.
●​ Constant: d/dx [c] = 0. Constant multiple: d/dx [c f(x)] = c f’(x).
●​ Sum/diff: d/dx [f±g] = f’ ± g’.
●​ Product: (fg)’ = f’g + fg’.
●​ Quotient: (f/g)’ = (f’g - fg’) / g².
●​ Trig: d/dx sin(x)=cos(x), cos(x)=-sin(x), tan(x)=sec²(x).
●​ Exp/log: d/dx e^x = e^x, d/dx ln(x) = 1/x.

Example 1: Basic Derivative
Find d/dx (3x⁴ - 2x² + 5).​
= 12x³ - 4x.

Example 2: Product Rule
d/dx [x² sin(x)] = 2x sin(x) + x² cos(x).