AP® Calculus AB & BC — Complete Course & Exam Prep

By Dr. M Categories: AP Calculus
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About Course

AP® Calculus AB and BC in one course. Calculus AB covers limits, derivatives, integrals and their applications, the equivalent of a first-semester university calculus course. Calculus BC adds parametric equations, polar coordinates, vector-valued functions, further integration techniques and infinite series, the equivalent of a full year. Every unit and lesson that is BC only is clearly marked (BC), so AB students can skip it. The course follows the College Board framework and prepares you for the May AP exam, aiming for a 5.

What you will learn

  • Units 1–2: Limits, continuity, and the definition of the derivative
  • Units 3–5: Differentiation rules, implicit and inverse functions, related rates, optimization, and curve analysis
  • Units 6–8: Integration, the Fundamental Theorem, integration by parts, partial fractions, improper integrals, differential equations, Euler’s method, logistic growth, areas and volumes
  • Unit 9: Parametric equations, polar coordinates, and vector-valued functions
  • Unit 10: Infinite sequences and series, convergence tests, Taylor and Maclaurin series, and error bounds

How the course works

  • Clear lessons with worked examples that build from basic to exam level
  • Interactive graphs so you can see what the calculus is doing
  • Practice at four levels, from foundational to challenge
  • A mastery check at the end of every lesson that tracks your progress
  • Full-length exam-style multiple-choice and free-response practice, calculator and non-calculator

Who this course is for

Students taking AP Calculus AB or BC in school, students self-studying for either exam, and strong MCV4U students who want university-level calculus before first year. You should be comfortable with functions, trigonometry, and logarithms (MHF4U level).

Need extra help?

If your mastery check results show you’re struggling with a topic, you can book a one-on-one session with Dr. Niyaifar directly from the course. Tutoring is targeted to exactly the skills your results show you need.

AP® is a registered trademark of the College Board, which is not affiliated with, and does not endorse, this course.

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Course Content

Unit 1: Limits and continuity
What a limit means, estimating limits from graphs and tables, limit laws and algebraic techniques, the squeeze theorem, continuity and its types of discontinuity, limits at infinity and asymptotes, and the Intermediate Value Theorem.

  • 1.1 Introducing Calculus: Change and Limits
  • 1.2 Estimating Limits from Graphs and Tables
  • 1.3 Limit Laws and Algebraic Manipulation
  • 1.4 The Squeeze Theorem
  • 1.5 Continuity and Types of Discontinuity
  • 1.6 Removing Discontinuities
  • 1.7 Infinite Limits and Vertical Asymptotes
  • 1.8 Limits at Infinity and Horizontal Asymptotes
  • 1.9 The Intermediate Value Theorem

Unit 2: Differentiation — definition and fundamental properties
Average and instantaneous rate of change, the derivative as a limit, differentiability and continuity, the power, constant, sum and difference rules, derivatives of sine, cosine, e^x and ln x, and the product and quotient rules.

Unit 3: Differentiation — composite, implicit and inverse functions
The chain rule, implicit differentiation, derivatives of inverse functions and inverse trigonometric functions, choosing the right differentiation technique, and higher-order derivatives.

Unit 4: Contextual applications of differentiation
Interpreting the derivative in context, straight-line motion (position, velocity, acceleration), rates of change in applied settings, related rates, local linearity and linear approximation, and L’Hôpital’s rule for indeterminate forms.

Unit 5: Analytical applications of differentiation
The Mean Value Theorem and Extreme Value Theorem, critical points, increasing and decreasing intervals, the first and second derivative tests, concavity and inflection points, sketching f, f′ and f″, optimization, and implicit relations.

Unit 6: Integration and accumulation of change
Accumulation as area, Riemann sums and the trapezoidal rule, the definite integral, both parts of the Fundamental Theorem of Calculus, antiderivatives, u-substitution, and long division and completing the square; BC: integration by parts, linear partial fractions and improper integrals.

Unit 7: Differential equations
Modelling with differential equations, verifying solutions, slope fields, separation of variables, particular solutions from initial conditions, and exponential growth and decay; BC: Euler’s method and logistic models.

Unit 8: Applications of integration
Average value of a function, position and total distance from velocity, accumulation in applied contexts, area between curves (in x and in y), volumes by cross-sections, and the disc and washer methods; BC: arc length of a smooth curve.

Unit 9: Parametric equations, polar coordinates and vector-valued functions (BC)
BC only. Derivatives and second derivatives of parametric curves, arc length of parametric curves, vector-valued functions and particle motion in the plane, polar coordinates and their derivatives, and area bounded by polar curves.

Unit 10: Infinite sequences and series (BC)
BC only. Convergence of sequences and series, geometric and p-series, the nth-term, integral, comparison, limit comparison, alternating series and ratio tests, absolute and conditional convergence, Taylor and Maclaurin polynomials and series, the Lagrange error bound, radius and interval of convergence, and representing functions as power series.

Exam review and full practice exams
Timed practice mirroring the AP Calculus AB and BC exams: calculator and non-calculator multiple choice, free-response questions with scoring guidelines, and a full practice exam for each; BC: series, parametric and polar free-response practice.

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