AP Calculus study guides
Every guide answers the question solvers skip: which method applies, and why. Worked examples in every one.
AB and shared topics
- Chain Rule: Spot the Composition, Then Differentiate
Recognize when a function is a composition, then differentiate it with the chain rule: outer derivative times inner derivative, with trig examples.
- U-Substitution: How to Choose u and Integrate It
Recognize when u-substitution applies, choose u as the inner function whose derivative appears, convert definite-integral bounds, and avoid failed choices.
- U-Sub vs Integration by Parts vs Partial Fractions
Decide between u-substitution, integration by parts, and partial fractions by reading the integrand. Includes a decision table and worked examples.
- Related Rates: The 5-Step Method for Any Word Problem
The 5-step method for related rates: diagram, equation, differentiate with respect to time, substitute, solve. Ladder and cone examples fully worked.
- Optimization Word Problems: Setup, Constraint, Solve
How to set up optimization word problems: objective vs constraint, reduce to one variable, then justify the max or min with a derivative test.
- How to Find Limits: Every Method, in Order
The order to evaluate AP Calculus limits: direct substitution, factoring, conjugates, trig limits, and the squeeze theorem, plus when to use each.
- L'Hopital's Rule: When You Can and Cannot Use It
L'Hopital's Rule works only on 0/0 or infinity/infinity forms. Learn to verify the indeterminate form, apply the rule, and avoid the classic misuse.
- Implicit Differentiation: Finding dy/dx Step by Step
Learn to recognize when to use implicit differentiation and find dy/dx step by step, with a circle example, a product-rule case, and the second derivative.
- The Fundamental Theorem of Calculus: Part 1 vs Part 2
How FTC Part 1 (accumulation functions) and Part 2 (evaluating definite integrals) differ, when to use each, and how to tell which the problem wants.
- Riemann Sums: Left, Right, Midpoint, and Trapezoidal
Approximate definite integrals with left, right, midpoint, and trapezoidal Riemann sums, and tell instantly whether your estimate is too high or too low.
- Disk vs Washer vs Shell: Choosing the Volume Method
Pick disk vs washer fast by matching a perpendicular slice to the axis; shell (a college topic) uses a parallel slice and is not on the AP exam.
- Area Between Two Curves: Setup, dx vs dy, Examples
Find the area between two curves: top minus bottom vs right minus left, choosing dx or dy, finding intersections, and handling curves that cross.
- Product Rule vs Quotient Rule: How and When to Use Each
Learn when to use the product rule versus the quotient rule in AP Calculus, plus when rewriting a fraction lets you skip the quotient rule entirely.
- Derivatives of Trig Functions: Table and Memory Tricks
All six trig derivatives, inverse trig (Topic 3.4), and the co-function negative trick, with complete worked examples for AP Calculus AB and BC.
- Derivative of ln x, e^x, and a^x: Answers and Proofs
Derivatives of ln x, e^x, and a^x with answer-first formulas and full proofs via inverse functions and rewriting a^x = e^(x ln a). AP Calc CED 2.7 and 3.3.
- Separable Differential Equations: Step-by-Step Method
Recognize, separate, integrate, and solve any separable differential equation, with initial conditions and the domain restrictions AP rubrics require.
- Slope Fields and Euler's Method Explained
How to read and sketch slope fields (AB and BC) and run Euler's method (BC only), with a full worked Euler table and the over/underestimate rule.
- Continuity and the Three Types of Discontinuity
Test continuity at a point with the 3-part definition, classify removable, jump, and infinite discontinuities, and apply the Intermediate Value Theorem.
- How to Study for AP Calculus AB: Unit-by-Unit Plan
A weight-by-unit AP Calculus AB study plan: where the points are, how to spot which method each problem needs, and where students lose easy marks.
- AP Calculus AB vs BC: Difficulty, Units, Which to Take
How AP Calculus AB and BC differ: the two BC-only units, the identical exam format, what the 2025 score gap really means, and how to choose.
- AP Calculus Exam Format: Sections, Timing, and Scoring
The AP Calculus AB and BC exam format: two sections, four parts, 3 hours 15 minutes, the calculator rules per part, and how the 1 to 5 score is built.
- AP Calculus Calculator Policy: What Is Allowed and When
Which AP Calculus exam parts allow a graphing calculator, the four calculator skills the exam expects, and the radian-mode and rounding mistakes to avoid.
BC only
- Integration by Parts: LIATE, When to Use It, Examples
How to recognize when integration by parts applies, use LIATE to choose u, and work the classic integrals of x e^x and ln x. BC-only, CED Topic 6.11.
- Which Series Convergence Test Should I Use?
A decision tree for AP Calculus BC: choose the right series convergence test (nth-term, geometric, p-series, alternating, ratio, comparison, integral) fast.
- The Ratio Test: How It Works and When to Use It
How the ratio test decides convergence: L < 1, L > 1, and the inconclusive L = 1 case, plus how to recognize when it is the right tool. AP Calc BC Topic 10.8.
- Taylor and Maclaurin Series: How to Build Them
Building a Taylor or Maclaurin series: the derivative formula, five series to memorize, and when to substitute instead. AP Calc BC Topics 10.11, 10.14, 10.15.
- Parametric and Polar Calculus: Derivatives and Area
How to find dy/dx and d^2y/dx^2 for parametric curves, compute arc length, and set up polar area integrals in AP Calculus BC Unit 9.