AP-CALCBC

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Expected availability: Summer 2026

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AP-CALCBC College Board Available Summer 2026

AP^® Calculus BC

AP-CALCBC equips students with comprehensive AP Calculus BC mastery, covering AB foundations, advanced integration, differential equations, series convergence, Taylor polynomials, parametric, polar, and vector-valued functions, preparing them for exam success.

195

Minutes

Questions

3/5

Passing Score

$98

Exam Cost

Who Should Take This

High school juniors and seniors who have completed AP Calculus AB, or equivalent, and aim to earn college credit should enroll. It suits motivated learners with strong algebra and trigonometry skills seeking to deepen their calculus understanding, excel on the AP exam, and lay groundwork for STEM majors.

What's Covered

1 AB Foundation: Limits, Differentiation, and Integration

2 Advanced Integration Techniques

3 Advanced Differential Equations

4 Arc Length and Applications of Integration

5 Parametric Equations, Polar Coordinates, and Vector-Valued Functions

6 Infinite Sequences and Series

7 BC-Level Cross-Topic Synthesis

What's Included in AccelaStudy® AI

Adaptive Knowledge Graph

Practice Questions

Lesson Modules

Console Simulator Labs

Exam Tips & Strategy

20 Activity Formats

Course Outline

65 learning goals

1 AB Foundation: Limits, Differentiation, and Integration

5 topics

Limits and Continuity Foundations

Evaluate limits of functions using algebraic manipulation, limit laws, and the squeeze theorem, including limits involving indeterminate forms and limits at infinity.
Classify discontinuities as removable, jump, or infinite, and apply the Intermediate Value Theorem to establish existence of function values on continuous intervals.
Determine horizontal and vertical asymptotes of functions by evaluating limits at infinity and limits at points where the function is undefined.

Differentiation Foundations

Define the derivative as the limit of the difference quotient and apply basic differentiation rules (power, product, quotient, chain) to compute derivatives of polynomial, trigonometric, exponential, and logarithmic functions.
Apply implicit differentiation and the inverse function theorem to compute derivatives of implicitly defined functions and inverse trigonometric functions.
Interpret the derivative as an instantaneous rate of change in applied contexts and solve related rates problems using implicit differentiation with respect to time.
Apply L'Hopital's rule to evaluate limits of indeterminate forms 0/0 and infinity/infinity, verifying the conditions for its use.

Analytical Applications of Derivatives

Determine critical points, intervals of increase and decrease, and local extrema using the first derivative test for a given function.
Determine concavity and inflection points using the second derivative and apply the second derivative test to classify critical points as local maxima or minima.
Model and solve optimization problems by translating constraints into mathematical functions and applying calculus techniques to find global extrema with justification.
State the Mean Value Theorem and Extreme Value Theorem and apply them to justify the existence of specific function values and derivative values on closed intervals.

Integration Foundations

Approximate definite integrals using Riemann sums (left, right, midpoint, trapezoidal) and define the definite integral as the limit of Riemann sums.
State and apply the Fundamental Theorem of Calculus Parts I and II to evaluate definite integrals and to differentiate accumulation functions.
Apply u-substitution to evaluate indefinite and definite integrals, identifying the appropriate substitution and adjusting limits of integration.
Determine whether a Riemann sum overestimates or underestimates the definite integral based on the monotonicity and concavity of the integrand.

AB-Level Applications

Calculate the area between curves and volumes of solids of revolution using disc and washer methods, selecting the appropriate axis of revolution and limits of integration.
Solve separable differential equations using separation of variables and apply initial conditions to find particular solutions, including exponential growth and decay models.
Sketch and interpret slope fields for differential equations and use them to predict the qualitative behavior of solution curves.
Analyze particle motion along a line using position, velocity, and acceleration functions, distinguishing between displacement and total distance traveled.
Calculate the average value of a function on an interval and interpret definite integrals as net accumulation of a rate of change in applied contexts.

2 Advanced Integration Techniques

2 topics

Integration by Parts

State the integration by parts formula and identify appropriate choices for u and dv using the LIATE priority guideline to evaluate integrals of products of functions.
Apply integration by parts to evaluate definite and indefinite integrals involving products of polynomial, exponential, logarithmic, and trigonometric functions, including problems requiring repeated application.
Evaluate integrals using tabular integration by parts for problems involving polynomial factors multiplied by exponential or trigonometric functions where repeated application follows a predictable pattern.

Partial Fractions and Improper Integrals

Decompose rational expressions with distinct linear factors into partial fractions and integrate each resulting term using standard logarithmic antiderivatives.
Evaluate improper integrals with infinite limits of integration by expressing them as limits of definite integrals and determining whether the integral converges to a finite value or diverges.
Evaluate improper integrals with discontinuous integrands by splitting at the discontinuity, computing one-sided limits, and determining convergence or divergence.
Select the appropriate advanced integration technique (by parts, partial fractions, substitution) for a given integral by analyzing the structure of the integrand.

3 Advanced Differential Equations

2 topics

Euler's Method

Describe the Euler's method algorithm for numerically approximating solutions to first-order differential equations and execute the step-by-step procedure given a step size and initial condition.
Analyze the accuracy of Euler's method approximations by comparing with exact solutions and explaining how the concavity of the solution curve causes overestimates or underestimates.

Logistic Differential Equations

Identify the logistic differential equation dy/dt = ky(1 - y/L), describe its parameters (growth rate k, carrying capacity L), and state the general solution as a function of time.
Analyze the behavior of logistic growth models by identifying the inflection point at half the carrying capacity, describing the transition from exponential to bounded growth, and interpreting slope fields.
Construct and interpret logistic models for population dynamics scenarios, comparing the logistic model with exponential growth and explaining the biological significance of the carrying capacity.

4 Arc Length and Applications of Integration

1 topic

Arc Length and Cross-Sectional Volumes

Derive and apply the arc length formula for smooth curves defined as y = f(x) by integrating the square root of one plus the square of the derivative over the interval.
Calculate volumes of solids with known cross-sectional areas by integrating the cross-sectional area function along an axis, applying this to squares, semicircles, and equilateral triangles.

5 Parametric Equations, Polar Coordinates, and Vector-Valued Functions

3 topics

Parametric Equations and Calculus

Compute the first derivative dy/dx of a parametrically defined curve using the chain rule formula (dy/dt)/(dx/dt) and interpret it as the slope of the tangent line at a given parameter value.
Compute the second derivative d2y/dx2 of a parametrically defined curve and use it to determine the concavity of the curve at a given parameter value.
Calculate the speed of a particle moving along a parametric path as the magnitude of the velocity vector and set up integrals for total distance traveled over a parameter interval.
Set up arc length integrals for parametrically defined curves by integrating the square root of the sum of squared derivatives of the component functions with respect to the parameter.

Polar Coordinates and Area

Identify standard polar curves (circles, cardioids, rose curves, limacons) and convert between rectangular and polar representations using the relationships x = r cos(theta), y = r sin(theta).
Compute dy/dx for polar curves using the parametric derivative formula with x = r(theta)cos(theta) and y = r(theta)sin(theta) to find slopes of tangent lines.
Calculate the area enclosed by a polar curve or between two polar curves by setting up and evaluating the integral of (1/2)r^2 over the appropriate theta interval.
Determine the intersection points of two polar curves and set up integrals to compute the area of regions bounded by multiple polar curves, handling the geometry of overlapping regions.

Vector-Valued Functions

Differentiate and integrate vector-valued functions component-wise to find velocity and position vectors from acceleration and initial conditions.
Analyze planar motion by computing the velocity vector, acceleration vector, speed, and total distance traveled for a particle whose position is given by a vector-valued function.
Synthesize parametric, vector, and polar representations to solve multi-step motion and area problems, selecting the most efficient coordinate system and calculus technique for each component.

6 Infinite Sequences and Series

3 topics

Sequences and Series Fundamentals

Define convergence and divergence of infinite sequences using the limit definition and determine whether a given sequence converges by evaluating its limit as n approaches infinity.
Identify the nth partial sum of a series and determine convergence or divergence of the series based on the behavior of the sequence of partial sums.
Determine the sum of a convergent geometric series using the formula a/(1-r) and identify when a geometric series diverges based on the common ratio.
Apply the nth term test for divergence, recognizing that it can only confirm divergence and cannot establish convergence of a series.
Identify p-series and harmonic series, determine convergence or divergence based on the value of p, and use these as benchmark series for comparison tests.

Convergence Tests

Apply the integral test to determine convergence or divergence of a positive-term series by comparing it with the corresponding improper integral.
Apply the direct comparison test and the limit comparison test to determine convergence or divergence of a series by comparing with a known benchmark series.
Apply the alternating series test to determine convergence of alternating series and use the alternating series error bound to estimate the error of partial sum approximations.
Apply the ratio test to determine convergence or divergence of a series, especially those involving factorials, exponentials, or nth powers, identifying when the test is inconclusive.
Distinguish between absolute convergence and conditional convergence of a series and explain the implications for rearrangement and manipulation of series terms.
Select and justify the most appropriate convergence test for a given series by analyzing its structure, comparing multiple approaches, and providing a complete convergence argument.

Power Series and Taylor Series

Determine the radius of convergence and interval of convergence for a power series using the ratio test and checking endpoint behavior separately.
State the Maclaurin series for e^x, sin(x), cos(x), and 1/(1-x), and use substitution, multiplication, and algebraic manipulation to derive Maclaurin series for related functions.
Construct the Taylor polynomial of degree n centered at x = a for a given function by computing the function's value and its first n derivatives at x = a.
Apply the Lagrange error bound to estimate the maximum error when a Taylor polynomial of degree n is used to approximate a function value, determining the number of terms needed for a specified accuracy.
Differentiate and integrate power series term by term within the interval of convergence to create new series representations and evaluate definite integrals that lack elementary antiderivatives.
Synthesize series techniques to represent functions as power series, approximate function values and definite integrals using Taylor polynomials with error bounds, and explain the relationship between a function and its Taylor series representation.