AP Calculus AB Foundations Course
A free 5-lesson AP Calculus AB course covering limits, continuity, derivatives, derivative applications, integrals and exam strategy with 80 linked practice questions.
What you will learn
- Interpret limits graphically, numerically and algebraically.
- Apply continuity conditions and the Intermediate Value Theorem.
- Compute derivatives with power, product, quotient and chain rules.
- Use derivatives to analyze rates, extrema and graph behavior.
- Evaluate integrals and apply the Fundamental Theorem of Calculus.
Before you start
- Strong algebra skills including factoring and equation solving.
- Comfort with functions, graphs and transformations.
- Basic trigonometry values and right-triangle ratios.
- Willingness to practise with paper and pencil.
Lesson 1 Limits and Continuity
Limits describe what a function approaches as the input gets close to a value. They are the foundation of derivatives and integrals, so AP Calculus AB starts by building precise limit intuition.
- Evaluate limits by substitution, factoring, conjugation and common limits.
- Recognize limits that do not exist and one-sided limits.
- Use limit notation to describe infinite behavior and asymptotes.
- Check continuity at a point and apply the Intermediate Value Theorem.
Limit tool order: try substitution first. If you get 0/0, factor and cancel, multiply by the conjugate for roots, or use a known limit. If substitution gives a number, that number is the limit; do not guess from a graph alone.
For continuity, check three things at the point: the function is defined, the limit exists, and the limit equals the function value. The Intermediate Value Theorem only applies when the function is continuous on the interval.
Example
Evaluate lim x->2 (x^2 - 4)/(x - 2). Factor the numerator: (x - 2)(x + 2)/(x - 2). Cancel x - 2, then substitute x = 2 to get 4.
Worked example: Evaluate lim x→2 (x² − 4)/(x − 2).
Substitution gives 0/0, so factor: (x − 2)(x + 2)/(x − 2) = x + 2. Then the limit is 2 + 2 = 4.
Check by plotting or by substituting values close to 2: the function approaches 4.
Lesson 2 Derivatives and Differentiation Rules
A derivative measures instantaneous rate of change. You need fast, reliable differentiation because every AB exam section contains multiple derivative questions.
- Use the power, product, quotient and chain rules.
- Differentiate exponential, logarithmic and trigonometric functions.
- Interpret the derivative as slope, velocity and marginal change.
- Write tangent-line equations and estimate derivatives from data.
Rule selection: decide which rule the expression needs before differentiating. Products need the product rule; quotients need the quotient rule; nested functions need the chain rule. For a sum of terms, differentiate term by term.
After finding the derivative, interpret it in context: slope of the tangent, velocity, or marginal cost. On multiple-choice questions, check that the answer uses the correct variable and has the right units.
Example
Find d/dx x^3 sin(x). Use the product rule: 3x^2 sin(x) + x^3 cos(x).
Worked example: Differentiate y = x³ sin x.
Product rule: u = x³, v = sin x, so y' = 3x² sin x + x³ cos x.
Check each term: derivative of u times v plus u times derivative of v.
Lesson 3 Applications of Derivatives
Derivatives become powerful when used to analyze motion, optimize quantities and describe graphs. AP Calculus AB frequently pairs these ideas with real-world word problems.
- Find critical points and classify local and absolute extrema.
- Use the first and second derivative tests.
- Analyze position, velocity and acceleration.
- Solve related rates and optimization problems.
Optimization routine: identify the quantity to maximize or minimize, write it as a function of one variable using any constraint, find critical points, and test endpoints. In related rates, write the equation first, then differentiate both sides with respect to time.
For graph analysis, connect signs: where the derivative is positive the function increases; where the second derivative is positive the graph is concave up. Sketch the graph to verify your conclusions.
Example
For f(x) = x^3 - 3x, f'(x) = 3x^2 - 3. Setting it to zero gives x = -1 and x = 1, the critical points where local extrema can occur.
Worked example: f(x) = x³ − 3x has f'(x) = 3x² − 3.
Critical points: x = −1 and x = 1. The second derivative is 6x, so f''(−1) = −6 (local max) and f''(1) = 6 (local min).
Sketch the cubic to confirm the shape.
Lesson 4 Integrals and the Fundamental Theorem
Integrals accumulate change. Indefinite integrals undo differentiation, while definite integrals compute areas and net change through the Fundamental Theorem of Calculus.
- Find antiderivatives of polynomial, exponential and trigonometric functions.
- Use u-substitution for composite functions.
- Evaluate definite integrals and signed areas.
- Use integrals to find areas between curves and average values.
Antiderivative checklist: add one to the exponent and divide by the new exponent for powers; keep constant factors outside the integral; for composite functions, consider u-substitution. For definite integrals, always apply the limits to the antiderivative.
Use the Fundamental Theorem to connect derivatives and integrals: the integral of f' over [a, b] equals f(b) − f(a). Check area questions for the sign: negative regions contribute negative signed area unless the question asks for total area.
Example
Evaluate integral 0 to 2 (3x^2) dx. An antiderivative is x^3, so the value is 2^3 - 0^3 = 8.
Worked example: Evaluate ∫₀² 3x² dx.
Antiderivative: x³. Apply limits: 2³ − 0³ = 8.
Check by differentiating x³ to get 3x².
Lesson 5 AP Calculus AB Exam Strategy
AP Calculus AB rewards clear reasoning and precise notation. Build a timed routine that lets you earn partial credit even when a final answer is uncertain.
- Write every limit, derivative and integral step in correct notation.
- Answer every multiple-choice question and eliminate unreasonable choices.
- For free-response, show setup, differentiation or integration, then conclusion.
- Review mistakes and repeat related practice sets.
Free-response skeleton: state the formula, substitute the given values, show the derivative or antiderivative, then box the final answer with units. Even if the final arithmetic is wrong, the setup earns partial credit.
For multiple choice, answer every question and use estimation when a calculation is long. Track errors by topic after each practice exam and spend the next session on the weakest topic.
Example
For a free-response question, state the formula, substitute values, show the derivative or antiderivative, then box the final answer with units.
Worked example: A free-response question asks for the area between a curve and the x-axis.
Write the integral, show the antiderivative, substitute the limits, and state the area with square units.
This complete structure earns method marks and makes the answer easy to check.