AP® Calculus BC Self-Paced

Course Essential Questions

• How can we describe rate of change and net change using calculus?
• How is the concept of the limit the foundation of calculus?
• How does calculus allow us to analyze the behaviors of functions?

Course Learning Objectives:

• Work with functions represented in multiple ways: graphical, numerical, analytical, or verbal.
• Explain the meaning of the derivative in terms of a rate of change and local linear approximation and use derivatives to solve problems.
• Explain the meaning of the definite integral as a limit of Riemann sums and as the net accumulation of change and use integrals to solve problems.
• Describe the relationship between the derivative and the definite integral as expressed in both parts of the Fundamental Theorem of Calculus.
• Communicate mathematics and explain solutions to problems verbally and in writing.
• Model a written description of a physical situation with a function, a differential equation, or an integral.
• Use technology to solve problems, experiment, interpret results, and support conclusions.
• Determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement.

UNIT 1: Limits and Continuity (Topic 1)

Essential Questions

• How does knowing the value of a limit, or that a limit does not exist, help you to make sense of interesting features of functions and their graphs?
• How do we close loopholes so that a conclusion about a function is always true?

Topic 1: Limits and Continuity

• Estimate limits of functions
• Represent limits analytically using correct notation
• Interpret limits expressed in analytic notation
• Determine the limits of functions using limit theorems
• Determine the limits of functions using equivalent expressions for the function or the squeeze theorem
• Justify conclusions about continuity at a point using the definition
• Determine intervals over which a function is continuous
• Determine values of x or solve for parameters that make discontinuous functions continuous, if possible
• Explain the behavior of a function on an interval using the Intermediate Value Theorem
• Interpret the behavior of functions using limits involving infinity

UNIT 2: Differentiation - Definition and Fundamental Properties (Topics 2-3)

Essential Questions

• Can change occur at an instant?
• How can we determine the rate of change of a function?
• Why do mathematical properties and rules for simplifying and evaluating limits apply to differentiation?
• If you knew the rate of change of a function, what do you think it would tell you at a specific point?

Topic 2: Rate of Change, the Derivative and the Power Rule

• Determine average rates of change using difference quotients
• Represent the derivative of a function as the limit of a difference quotient
• Determine the equation of a line tangent to a curve at a given point
• Estimate derivatives
• Explain the relationship between differentiability and continuity
• Calculate the derivatives of familiar functions
• Interpret a limit as a definition of a derivative

Topic 3: Product Rule, Quotient Rule, and Derivatives of Trigonometric Functions

• Calculate the derivatives of familiar functions
• Interpret a limit as a definition of a derivative
• Calculate derivatives of products and quotients of differentiable functions

UNIT 3: Differentiation - Composite, Implicit, and Inverse Functions (Topics 4-6)

Essential Questions

• If pressure experienced by a diver is a function of depth and depth is a function of time, how might we find the rate of change in pressure with respect to time?

Topic 4: Chain Rule and Implicit Differentiation

• Calculate derivatives of compositions of differentiable functions.
• Calculate derivatives of implicitly defined functions.
• Determine higher-order derivatives of a function.

Topic 5: Derivatives of Inverse Functions and Logarithmic Functions

• Calculate derivatives of inverse functions
• Calculate derivatives of inverse trigonometric functions

Topic 6: Units 1-3 Review and Exam

• Review your knowledge of limits, continuity, and differentiation
• Demonstrate your knowledge of limits, continuity, and differentiation on the Units 1-3 Exam

UNIT 4: Contextual Applications of Differentiation (Topics 7-8)

Essential Questions

• How are problems about position, velocity, and acceleration of a particle in motion over time structurally similar to problems about volume over an interval of heights, population over a certain time period, or metabolism of a dose of medicine over time?
• Since certain indeterminate forms seem to actually approach a limit, how can we determine that limit, provided it exists?

Topic 7: Interpreting Derivatives and Related Rates

• Interpret the meaning of a derivative in context
• Calculate rates of change in applied contexts
• Interpret rates of change in applied contexts
• Calculate related rates in applied contexts
• Interpret related rates in applied contexts

Topic 8: Differentials, Linear Approximations, Newton’s Method, and L’Hopital’s Rule

• Interpret the meaning of a derivative in context.
• Calculate rates of change in applied contexts.
• Interpret rates of change in applied contexts.
• Approximate a value on a curve using the equation of a tangent line.

UNIT 5: Analytical Applications of Differentiation (Topics 9-11)

Essential Questions

• How might the Mean Value Theorem be used to justify a conclusion that you were speeding at some point on a certain stretch of highway, even without knowing the exact time you were speeding?
• What additional information is included in a sound mathematical argument about optimization that a simple description of an equivalent answer lacks?

Topic 9: Extrema, Mean Value Theorem, and Concavity

• Justify conclusions about functions by applying the Extreme Value Theorem
• Justify conclusions about the behavior of a function based on the behavior of its derivatives
• Justify conclusions about functions by applying the Mean Value Theorem over an interval

Topic 10: Explore Function Behavior Through Graphing, Optimization, and Differential Equations

• Justify conclusions about the behavior of a function based on the behavior of its derivatives
• Calculate minimum and maximum values in applied contexts or analysis of functions
• Interpret minimum and maximum values calculated in applied contexts
• Determine critical points of implicit relations
• Justify conclusions about the behavior of an implicitly defined functions based on evidence from its derivatives

Topic 11: Units 4-5 Review and Exam

• Review your knowledge of contextual and analytical applications of differentiation
• Demonstrate your knowledge of contextual and analytical applications of differentiation on the Units 4-5 Exam

UNIT 6: Integration and Accumulation of Change (Topics 12-16)

Essential Questions

• If compounding more often increases the amount in an account with a given rate of return and term, why doesn't compounding continuously result in an infinite account balance, all other things being equal?
• Given information about a rate of population growth over time, how can we determine how much the population changed over a given interval of time?
• How is integrating to find areas related to differentiating to find slopes?

Topic 12: Area and the Definite Integral

• Approximate a definite integral using geometric and numerical methods
• Interpret the limiting case of the Riemann sum as a definite integral
• Represent the limiting case of the Riemann sum as a definite integral

Topic 13: Fundamental Theorem of Calculus, Properties of Integrals, and Substitution Method

• Interpret the meaning of areas associated with the graph of a rate of change in context
• Determine the average value of a function using definite integrals
• Determine values for positions and rates of change using definite integrals in problems involving rectilinear motion
• Interpret the meaning of a definite integral in accumulation problems
• Determine net change using definite integrals in applied contexts
• Represent accumulation functions using definite integrals
• Calculate a definite integral using areas and properties of definite integrals
• Evaluate definite integrals analytically using the Fundamental Theorem of Calculus
• Determine antiderivatives of functions and indefinite integrals, using knowledge of derivatives
• For integrands requiring substitution or rearrangements into equivalent forms (a) determine indefinite integrals and (b) evaluate definite integrals

Topic 14: Integration Using Long Division, Integration by Parts, Integration Using Partial Fractions

• For integrands requiring integration by parts: (a) Determine indefinite integrals. (b) Evaluate definite integrals
• For integrands requiring integration by linear partial fractions: (a) Determine indefinite integrals. (b) Evaluate definite integrals

Topic 15: Improper Integrals and Integrals Containing Trigonometric Functions

• Evaluate an improper integral or determine that the integral diverges
• For integrands requiring substitution or rearrangements into equivalent forms (a) determine indefinite integrals and (b) evaluate definite integrals

Topic 16: Unit 6 Review and Exam

• Review your knowledge of integration and accumulation of change
• Demonstrate your knowledge of integration and accumulation of change on the Unit 6 Exam

UNIT 7: Differential Equations (Topics 17-18)

Essential Questions

• How can we derive a model for the number of computers, C, infected by a virus, given a model for how fast the computers are being infected, dC/dt, at a particular time?

Topic 17: Differential Equations and Slope Fields

• Verify solutions to differential equations.
• Determine general solutions to differential equations.
• Determine particular solutions to differential equations.
• Interpret verbal statements of problems as differential equations involving a derivative expression.
• Interpret the meaning of a differential equation and its variables in context.
• Determine general and particular solutions for problems involving differential equations in context.
• Estimate solutions to differential equations.

Topic 18: Euler’s Method and Logistic Model

• Estimate solutions to differential equations.
• Interpret the meaning of the logistic growth model in context.

UNIT 8: Applications of Integration (Topics 19-21)

Essential Questions

• How is finding the number of visitors to a museum over an interval of time based on information about the rate of entry similar to finding the area of a region between a curve and the x-axis?

Topic 19: Applications of Motion Along a Line and Areas Between Curves

• Determine the average value of a function using definite integrals
• Determine values for positions and rates of change using definite integrals in problems involving rectilinear motion
• Interpret the meaning of a definite integral in accumulation problems
• Determine net change using definite integrals in applied contexts
• Calculate areas in the plane using definite integral

Topic 20: Finding Volume with Disks, Washers, Shells, and Cross Sections

• Calculate volumes of solids of revolution using definite integrals
• Calculate volumes of solids with known cross sections using definite integrals

Topic 21: Units 7-8 Review and Exam

• Review your knowledge of differential equations and applications of integration
• Demonstrate your knowledge of differential equations and applications of integration on the Units 7-8 Exam

UNIT 9: Parametric Equations; Polar Coordinates; and Vector-Valued Functions (Topics 22-25)

Essential Questions

• How does the Chain Rule help us to analyze graphs defined using parametric equations or polar functions?
• How can we model motion not constrained to a linear path?

Topic 22: Parametric Equations

• Calculate derivatives of parametric functions

Topic 23: Tangent Lines and Parametric Equations

• Calculate derivatives of parametric functions
• Determine the length of a curve in the plane defined by parametric functions, using a definite integral
• Calculate derivatives of functions written in polar coordinates

Topic 24: Area in Polar Coordinates and Vector Functions

• Calculate areas of regions defined by polar curves using definite integrals
• Calculate derivatives of vector-valued functions

Topic 25: Motion Along a Curve

• Determine values for positions and rates of change in problems involving planar motion
• Determine a particular solution given a rate vector and initial conditions

UNIT 10: Infinite Sequences and Series (Topics 26-30)

Essential Questions

• How can the sum of infinitely many discrete terms be a finite value or represent a continuous function?

Topic 26: Sequences and Series

• Define sequences
• Determine whether a series converges or diverges

Topic 27: Exploring Convergence Tests for Series

• Determine whether a series converges or diverges
• Approximate the sum of a series

Topic 28: Summary of Tests and Power Series

• Determine whether a series converges or diverges
• Choose an appropriate test to determine whether a series converges
• Determine the radius of convergence and interval of convergence for a power series
• Represent a given function as a power series

Topic 29: Taylor Series, Maclaurin Series, and Polynomial Approximations

• Represent a function as a Taylor series or a Maclaurin series
• Interpret Taylor series and Maclaurin series
• Represent a function at a point as a Taylor polynomial
• Approximate function values using a Taylor polynomial
• Determine the error bound associated with a Taylor polynomial approximation

Topic 30: Units 9-10 Review and Exam

• Review your knowledge of parametric equations, polar coordinates, sequences, and series.
• Demonstrate your knowledge of parametric equations, polar coordinates, sequences, and series on the Units 9-10 Exam

UNIT 11: AP Practice Exam, Practice Exam Reflection, and Careers with Calculus (Topics 31-33)

Essential Questions

• How can we prepare for the AP Exam?

Topic 31: AP Review and AP Practice Exam

• Review unit topics
• Complete practice exam

Topic 32: AP Practice Exam Reflection

• Review graded practice exam
• Complete reflection activity on practice exam

Topic 33: Calculus Finish Line

• Identify the use of calculus in various careers
• Reflect on your course experience
• Complete end of course activities