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AP courses

AP Calculus

The full BC sequence, from limits and derivatives through integrals, differential equations, and series.

14 units · 125 lessons · 993 practice items · 275 videos

AP Calculus covers the full BC sequence. You begin with limits and continuity, build the derivative from its definition and the standard rules, and apply differentiation to motion and optimization. The second half turns to integration and accumulation, differential equations, and areas and volumes, then finishes with the BC topics: parametric and polar calculus, vector-valued functions, and infinite sequences and series. Each unit leans on the ideas the one before it established.

Practice stays visual where the subject is: you set up Riemann rectangles, shade the region an integral measures, or sketch a derivative from its function, and over two hundred short videos accompany the lessons. About half the problems carry hints. Each unit closes with a test, and the semester exams sit on the course map as units of their own.

This course follows the College Board's published AP Calculus BC framework.

Syllabus

Every published lesson in this course, by unit.

1Limits and continuity17 lessons
  1. 1Introducing calculus: can change occur at an instant?
  2. 2Defining limits and using limit notation
  3. 3Estimating limit values from graphs
  4. 4Estimating limit values from tables
  5. 5Determining limits using algebraic properties of limits
  6. 6Determining limits using algebraic manipulation
  7. 7Selecting procedures for determining limits
  8. 8Determining limits using the squeeze theorem
  9. 9Connecting multiple representations of limits
  10. 10Exploring types of discontinuities
  11. 11Defining continuity at a point
  12. 12Confirming continuity over an interval
  13. 13Removing discontinuities
  14. 14Connecting infinite limits and vertical asymptotes
  15. 15Connecting limits at infinity and horizontal asymptotes
  16. 16Working with the intermediate value theorem (IVT)
  17. 17Unit 1 test
2Differentiation: definition and basic derivative rules11 lessons
  1. 1Defining average and instantaneous rates of change at a point
  2. 2Defining the derivative of a function and using derivative notation
  3. 3Estimating derivatives of a function at a point
  4. 4Determining when derivatives do and do not exist
  5. 5Applying the power rule
  6. 6Derivative rules: constant, sum, difference, and multiple
  7. 7Derivatives of cos(x), sin(x), e^x, and ln(x)
  8. 8The product rule
  9. 9The quotient rule
  10. 10Finding the derivatives of tan(x), cot(x), sec(x), and csc(x)
  11. 11Unit 2 test
3Differentiation: composite, implicit, and inverse functions7 lessons
  1. 1The chain rule
  2. 2Implicit differentiation
  3. 3Differentiating inverse functions
  4. 4Differentiating inverse trigonometric functions
  5. 5Procedures for calculating derivatives
  6. 6Calculating higher-order derivatives
  7. 7Unit 3 test
ExamSemester 1 midterm1 lesson
  1. 1Semester 1 midterm exam
4Contextual applications of differentiation8 lessons
  1. 1Interpreting the meaning of the derivative in context
  2. 2Straight-line motion: connecting position, velocity, and acceleration
  3. 3Rates of change in applied contexts other than motion
  4. 4Introduction to related rates
  5. 5Solving related rates problems
  6. 6Approximating values of a function using local linearity and linearization
  7. 7Using L'Hôpital's rule for determining limits of indeterminate forms
  8. 8Unit 4 test
5Analytical applications of differentiation13 lessons
  1. 1Using the mean value theorem
  2. 2Extreme value theorem, global versus local extrema, and critical points
  3. 3Determining intervals on which a function is increasing or decreasing
  4. 4Using the first derivative test to determine relative (local) extrema
  5. 5Using the candidates test to determine absolute (global) extrema
  6. 6Determining concavity of functions over their domains
  7. 7Using the second derivative test to determine extrema
  8. 8Sketching graphs of functions and their derivatives
  9. 9Connecting a function, its first derivative, and its second derivative
  10. 10Introduction to optimization problems
  11. 11Solving optimization problems
  12. 12Exploring behaviors of implicit relations
  13. 13Unit 5 test
ExamSemester 1 final1 lesson
  1. 1Semester 1 final exam
6Integration and accumulation of change15 lessons
  1. 1Exploring accumulations of change
  2. 2Approximating areas with Riemann sums
  3. 3Riemann sums, summation notation, and definite integral notation
  4. 4The fundamental theorem of calculus and accumulation functions
  5. 5Interpreting the behavior of accumulation functions involving area
  6. 6Applying properties of definite integrals
  7. 7The fundamental theorem of calculus and definite integrals
  8. 8Finding antiderivatives and indefinite integrals: basic rules and notation
  9. 9Integrating using substitution
  10. 10Integrating functions using long division and completing the square
  11. 11Integrating using integration by parts (BC)
  12. 12Integrating using linear partial fractions (BC)
  13. 13Evaluating improper integrals (BC)
  14. 14Selecting techniques for antidifferentiation
  15. 15Unit 6 test
7Differential equations10 lessons
  1. 1Modeling situations with differential equations
  2. 2Verifying solutions for differential equations
  3. 3Sketching slope fields
  4. 4Reasoning with slope fields
  5. 5Approximating solutions using Euler's method (BC)
  6. 6Finding general solutions using separation of variables
  7. 7Finding particular solutions using separation of variables
  8. 8Exponential models with differential equations
  9. 9Logistic models with differential equations (BC)
  10. 10Unit 7 test
8Applications of integration14 lessons
  1. 1Finding the average value of a function on an interval
  2. 2Connecting position, velocity, and acceleration of functions using integrals
  3. 3Using accumulation functions and definite integrals in applied contexts
  4. 4Finding the area between curves expressed as functions of x
  5. 5Finding the area between curves expressed as functions of y
  6. 6Finding the area between curves intersecting at more than two points
  7. 7Volumes with cross sections: squares and rectangles
  8. 8Volumes with cross sections: triangles and semicircles
  9. 9Volume with disc method — revolving around the x- or y-axis
  10. 10Volume with disc method — revolving around other axes
  11. 11Volume with washer method — revolving around the x- or y-axis
  12. 12Volume with washer method — revolving around other axes
  13. 13The arc length of a smooth, planar curve and distance traveled (BC)
  14. 14Unit 8 test
ExamSemester 2 midterm1 lesson
  1. 1Semester 2 midterm exam
9Parametric equations, polar coordinates, and vector-valued functions (BC)10 lessons
  1. 1Defining and differentiating parametric equations (BC)
  2. 2Second derivatives of parametric equations (BC)
  3. 3Finding arc lengths of curves given by parametric equations (BC)
  4. 4Defining and differentiating vector-valued functions (BC)
  5. 5Integrating vector-valued functions (BC)
  6. 6Solving motion problems (BC)
  7. 7Defining polar coordinates and differentiating in polar form (BC)
  8. 8Finding the area of a polar region (BC)
  9. 9Finding the area of the region bounded by two polar curves (BC)
  10. 10Unit 9 test
10Infinite sequences and series (BC)16 lessons
  1. 1Defining convergent and divergent infinite series (BC)
  2. 2Working with geometric series (BC)
  3. 3The nth term test for divergence (BC)
  4. 4Integral test for convergence (BC)
  5. 5Harmonic series and p-series (BC)
  6. 6Comparison tests for convergence (BC)
  7. 7Alternating series test for convergence (BC)
  8. 8Ratio test for convergence (BC)
  9. 9Determining absolute or conditional convergence (BC)
  10. 10Alternating series error bound (BC)
  11. 11Finding Taylor polynomial approximations of functions (BC)
  12. 12Lagrange error bound (BC)
  13. 13Radius and interval of convergence of power series (BC)
  14. 14Finding Taylor or Maclaurin series for a function (BC)
  15. 15Representing functions as power series (BC)
  16. 16Unit 10 test
ExamSemester 2 final1 lesson
  1. 1Semester 2 final exam