Subjects › AP Calculus BC
AP Calculus BC: work through limits, derivatives, integrals, and series by talking them ou
This page covers the full AP Calculus BC curriculum: limits and continuity, derivatives and their applications, integration techniques and their applications, differential equations, parametric and polar and vector functions, and infinite sequences and series. It includes everything in AP Calculus AB plus the BC-only material on convergence tests, Taylor series, and curves given parametrically or in polar form.
Each lesson starts from what you can already do and moves one question at a time. If you're asked to differentiate x²sin(x) and you reach for the chain rule instead of the product rule, the tutor doesn't just correct you — it asks what's being multiplied together, and lets you find the mismatch yourself. The same approach carries through to harder territory: setting up a related rates problem, choosing a substitution for an integral, or deciding whether the ratio test will actually resolve a series.
Who it is for
- Students taking or self-studying AP Calculus BC who want to talk through problems rather than just watch worked examples
- Students who finished AB-level material and need the BC topics: series, parametric/polar/vector calculus, advanced integration
- Parents looking for a way to check whether a student actually understands a topic, not just recognizes it
- Teachers wanting a supplementary resource that makes students explain their steps
- Students preparing for the AP exam who need practice justifying answers in words, not just computing them
What you will be able to do
- Compute limits, derivatives, and integrals fluently, by hand, using the appropriate rule or technique
- Justify conclusions using theorems like the Mean Value Theorem, Intermediate Value Theorem, and the Fundamental Theorem of Calculus
- Set up and solve applied problems: related rates, optimization, area and volume, motion along a line or in the plane
- Solve separable differential equations and interpret slope fields, logistic models, and Euler's method approximations
- Work with curves defined parametrically or in polar coordinates, including arc length and area
- Determine convergence or divergence of a series using the appropriate test, and construct and use Taylor series with error bounds
A taste of a lesson on Integration by Parts
The tutor asks, you answer, and it works from what you actually said. No videos to sit through.
The 47 topics in AP Calculus BC
Each topic builds on the ones before it. The tutor checks what you already know and starts there.
Introduction to Limits
What a limit means intuitively: the value a function approaches as input approaches a point. Estimating limits from graphs and tables, one-sided limits, and cases where limits fail to exist (jumps, oscillation, unbounded behavior).
Computing Limits Algebraically
Techniques for evaluating limits algebraically: direct substitution, factoring, rationalizing, and simplifying indeterminate forms 0/0. Includes limits involving piecewise functions.
Builds on Introduction to Limits
Limit Laws and Special Limits
Formal limit laws (sum, product, quotient, composition rules) and important special limits such as sin(x)/x as x approaches 0 and the squeeze theorem. Building justified computation of limits.
Builds on Computing Limits Algebraically
Continuity
Formal definition of continuity at a point and on an interval using limits. Types of discontinuities (removable, jump, infinite), and the Intermediate Value Theorem with applications to root-finding.
Builds on Limit Laws and Special Limits
Limits at Infinity and Asymptotes
Evaluating limits as x approaches infinity, determining horizontal and vertical asymptotes, and analyzing end behavior of rational, exponential, and logarithmic functions.
Builds on Limit Laws and Special Limits
The Derivative as a Limit
Defining the derivative as the limit of the difference quotient; interpreting it as instantaneous rate of change and slope of the tangent line. Computing derivatives directly from the limit definition for simple functions.
Builds on Continuity, Limits at Infinity and Asymptotes
Differentiability and Its Relation to Continuity
Understanding that differentiability implies continuity but not conversely; identifying points where a function is continuous but not differentiable (corners, cusps, vertical tangents). Interpreting derivative graphs.
Builds on The Derivative as a Limit
Basic Differentiation Rules
Power rule, constant multiple rule, sum/difference rule, and derivatives of exponential and basic trigonometric functions. Building fluency in fast symbolic differentiation.
Builds on Differentiability and Its Relation to Continuity
Product and Quotient Rules
Differentiating products and quotients of functions, with derivations and practice applying both rules together with basic rules in combination.
Builds on Basic Differentiation Rules
The Chain Rule
Differentiating composite functions; understanding the chain rule conceptually as a rate-of-change composition and applying it fluently, including nested compositions.
Builds on Product and Quotient Rules
Implicit Differentiation
Differentiating equations that are not solved explicitly for y, using the chain rule on implicit relations. Applying implicit differentiation to find tangent lines and derivatives of inverse functions.
Builds on The Chain Rule
Derivatives of Logarithmic, Inverse, and Inverse Trigonometric Functions
Derivatives of natural log and general logarithms, derivatives of inverse functions via implicit differentiation, and derivatives of inverse trigonometric functions.
Builds on Implicit Differentiation
Related Rates
Setting up and solving problems where multiple quantities change with respect to time, using implicit differentiation to relate their rates. Includes classic geometric and physical scenarios.
Builds on Derivatives of Logarithmic, Inverse, and Inverse Trigonometric Functions
Linear Approximation and Differentials
Using the tangent line to approximate function values near a point, and using differentials dy to estimate small changes. Understanding error in linear approximation.
Builds on Derivatives of Logarithmic, Inverse, and Inverse Trigonometric Functions
Mean Value Theorem and Rolle's Theorem
Statements and geometric meaning of the Mean Value Theorem and Rolle's Theorem; using them to justify existence of points with specific derivative values.
Builds on Linear Approximation and Differentials
Using Derivatives to Analyze Functions
Using the first derivative to determine increasing/decreasing intervals and local extrema, and the second derivative to determine concavity and inflection points. Includes the first and second derivative tests.
Builds on Mean Value Theorem and Rolle's Theorem
Optimization Problems
Applying derivative tests to solve real-world and geometric optimization problems: setting up an objective function, constraints, and finding absolute extrema on a domain.
Builds on Using Derivatives to Analyze Functions
L'Hôpital's Rule
Using derivatives to evaluate indeterminate limits of the forms 0/0 and infinity/infinity, and recognizing when to rewrite other indeterminate forms to apply the rule.
Builds on Using Derivatives to Analyze Functions
Position, Velocity, and Acceleration
Interpreting derivatives in the context of motion along a line: position, velocity, speed, and acceleration; analyzing motion using sign charts and graphs.
Builds on Using Derivatives to Analyze Functions
Riemann Sums and Approximating Area
Approximating the area under a curve using left, right, midpoint Riemann sums and trapezoidal sums; understanding the limit of these sums as the definition of a definite integral.
Builds on Position, Velocity, and Acceleration
The Definite Integral and Fundamental Theorem of Calculus
Defining the definite integral formally, and both parts of the Fundamental Theorem of Calculus: evaluating definite integrals via antiderivatives, and differentiating accumulation functions.
Builds on Riemann Sums and Approximating Area
Basic Antiderivatives and Indefinite Integrals
Finding antiderivatives using reverse power rule and known derivative formulas; notation of indefinite integrals plus constant of integration; initial value problems.
Builds on The Definite Integral and Fundamental Theorem of Calculus
Integration by u-Substitution
Using substitution to reverse the chain rule for integration; recognizing patterns for substitution in both indefinite and definite integrals, including adjusting limits of integration.
Builds on Basic Antiderivatives and Indefinite Integrals
Integration by Parts
Deriving and applying the integration by parts formula from the product rule; choosing u and dv effectively, including repeated application.
Builds on Integration by u-Substitution
Trigonometric Integrals and Trigonometric Substitution
Integrating products/powers of trigonometric functions using identities, and using trigonometric substitution to handle integrands with expressions like sqrt(a^2 - x^2).
Builds on Integration by Parts
Integration by Partial Fractions
Decomposing rational functions into partial fractions (distinct linear factors) to integrate them; recognizing when partial fractions is the appropriate technique.
Builds on Trigonometric Integrals and Trigonometric Substitution
Improper Integrals
Evaluating integrals with infinite limits of integration or with unbounded integrands, using limits; determining convergence or divergence.
Builds on Integration by Partial Fractions
Area Between Curves
Setting up and evaluating definite integrals to find area between two curves, including integrating with respect to y when appropriate.
Builds on Basic Antiderivatives and Indefinite Integrals
Volumes of Solids
Computing volumes using the disk, washer, and shell methods for solids of revolution, and volumes of solids with known cross-sections.
Builds on Area Between Curves
Arc Length
Deriving and applying the arc length formula for a curve y=f(x) using an integral involving the derivative of the function.
Builds on Volumes of Solids
Differential Equations and Slope Fields
Interpreting differential equations as describing rates of change; sketching and interpreting slope fields; matching slope fields to differential equations and solutions.
Builds on Basic Antiderivatives and Indefinite Integrals
Separable Differential Equations
Solving separable differential equations by separating variables and integrating both sides, including finding particular solutions with initial conditions. Application to exponential growth and decay models.
Builds on Integration by u-Substitution, Differential Equations and Slope Fields
Logistic Growth Models
Setting up and analyzing the logistic differential equation, understanding carrying capacity, and finding and interpreting its solution qualitatively and analytically.
Builds on Separable Differential Equations
Euler's Method
Using Euler's method to numerically approximate solutions to differential equations given an initial condition, including analyzing the accuracy of the approximation.
Builds on Differential Equations and Slope Fields
Parametric Equations and Calculus
Defining curves parametrically, finding dy/dx and d2y/dx2 for parametric curves, and computing arc length of parametric curves.
Builds on Arc Length
Vector-Valued Functions and Motion in the Plane
Representing motion in the plane using vector-valued functions; computing velocity and acceleration vectors, speed, and displacement for planar motion.
Builds on Parametric Equations and Calculus
Polar Coordinates Basics
Plotting points and curves in the polar coordinate system, converting between polar and rectangular coordinates, recognizing common polar curves (circles, roses, cardioids, limacons), and using trigonometric identities to analyze them. Foundational graphing skills needed before applying calculus to polar curves.
Builds on Vector-Valued Functions and Motion in the Plane
Polar Coordinates and Calculus
Finding dy/dx for polar curves, and computing area enclosed by polar curves (and between polar curves) using integration.
Builds on Polar Coordinates Basics
Sequences and Their Limits
Defining sequences, determining whether a sequence converges or diverges, and evaluating limits of sequences using known limit techniques including L'Hôpital's Rule where applicable.
Builds on Limits at Infinity and Asymptotes, L'Hôpital's Rule
Introduction to Series and Geometric Series
Defining an infinite series as the limit of partial sums; convergence and divergence of geometric series with the formula for the sum, and the nth-term test for divergence.
Builds on Sequences and Their Limits
Integral Test and Comparison Tests
Using the integral test to determine convergence of series (including p-series), and using direct and limit comparison tests with known series.
Builds on Improper Integrals, Introduction to Series and Geometric Series
Alternating Series, Ratio Test, and Root Test
Testing alternating series for convergence using the alternating series test and bounding the error; applying the ratio test and root test for convergence, especially for series involving factorials and powers.
Builds on Integral Test and Comparison Tests
Absolute and Conditional Convergence
Distinguishing absolute convergence from conditional convergence, and understanding implications for rearrangement and behavior of series.
Builds on Alternating Series, Ratio Test, and Root Test
Power Series and Interval of Convergence
Defining power series, using the ratio test to find radius and interval of convergence, and checking endpoint behavior.
Builds on Absolute and Conditional Convergence
Constructing Taylor and Maclaurin Series
Deriving the general Taylor series formula for a function centered at a point using successive derivatives, deriving the Maclaurin series as the special case centered at 0, and constructing series for given functions directly from the formula.
Builds on Power Series and Interval of Convergence
Manipulating Known Taylor Series
Memorizing common Maclaurin series (e^x, sin x, cos x, 1/(1-x), ln(1+x)) and generating new series from them via substitution, term-by-term differentiation, and term-by-term integration, including determining the resulting interval of convergence.
Builds on Constructing Taylor and Maclaurin Series
Taylor Polynomial Approximation and Error Bound
Using Taylor polynomials to approximate function values, and applying the Lagrange error bound (remainder) to estimate the accuracy of the approximation.
Builds on Manipulating Known Taylor Series
Practice questions in AP Calculus BC
A few of the kinds of questions you will work through, with answers.
Evaluate lim(x→0) sin(3x)/x.
See the answer
3
Write sin(3x)/x = 3·sin(3x)/(3x), and since sin(u)/u → 1 as u→0, the limit is 3·1 = 3.
Find d/dx of (3x² + 1)^5.
See the answer
30x(3x² + 1)^4
By the chain rule, bring down the exponent and multiply by the derivative of the inside: 5(3x²+1)^4·(6x) = 30x(3x²+1)^4.
Set up (do not evaluate) the integral for the volume when the region bounded by y = x² and y = 4 is revolved about the x-axis, using the washer method.
See the answer
V = π ∫ from -2 to 2 of (4² − (x²)²) dx
The outer radius is the distance to y=4 and the inner radius is the distance to y=x², and the curves meet at x = ±2.
Solve dy/dx = ky with y(0) = y0.
See the answer
y = y0·e^(kt)
Separating gives dy/y = k dx; integrating both sides gives ln|y| = kx + C, and applying the initial condition gives y = y0 e^(kx), the standard exponential growth/decay solution.
Does the series sum from n=1 to infinity of (-1)^n / n! converge, and if so, absolutely or conditionally?
See the answer
Converges absolutely
Applying the ratio test to the absolute values |1/n!| gives a limit of 0, which is less than 1, so the series converges absolutely.
Using the third-degree Maclaurin polynomial for e^x, estimate e^0.1, and explain how to bound the error using the Lagrange error bound.
See the answer
Approximately 1.10517; error bounded by max|f⁗(c)|·(0.1)^4/4! for c between 0 and 0.1
The third-degree Maclaurin polynomial is 1 + x + x²/2 + x³/6; plugging in x=0.1 gives the estimate, and the Lagrange error bound uses the fourth derivative of e^x, which is e^x itself, bounded by e^0.1 on the interval.
Questions about this subject
Does this cover both AP Calculus AB and BC topics?
Yes. The curriculum includes all AB topics (limits through basic integration and its applications) plus the BC-only topics: advanced integration techniques, parametric and polar and vector calculus, and infinite sequences and series.
Do I need to know any calculus before starting?
No. The course assumes a strong precalculus background — algebra, functions and their graphs, and trigonometric identities — but starts calculus itself from the definition of a limit.
How are series and convergence tests taught, since they're often the hardest BC topic?
The tutor builds up from sequences and geometric series to the integral, comparison, ratio, and root tests one at a time, asking you to decide which test applies to a given series before working through it, rather than presenting a flowchart to memorize.
Will this help with the free-response questions on the AP exam, which require written justification?
The lessons ask you to explain your reasoning in words at each step — stating which theorem applies, why a limit is indeterminate, or why a series test is appropriate — which is the same kind of justification the free-response questions expect.
Ready to learn AP Calculus BC?
Free to start. The tutor meets you where you are.
Start learning AP Calculus BC