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AP® Calculus AB Self-Paced

Syllabus

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 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 (Topics 1-3)

Essential Questions

How does knowing the value of a limit, or that a limit does not exist, help you 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: Finding Limits Numerically, Graphically, and Analytically

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

Topic 2: Continuity and Limits

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
Determine the limits of functions using equivalent expressions for the function or the squeeze theorem

Topic 3: Infinite Limits, Limits at Infinity, and Asymptotes

Interpret the behavior of functions using limits involving infinity

UNIT 2: Differentiation – Definition and Fundamental Properties (Topics 4-7)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 4: Rate of Change, the Derivative, and Differentiability

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

Topic 5: Find Derivatives Using the Power Rule, Product Rule, and Quotient Rule

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

Topic 6: 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

Topic 7: Units 1-2 Review and Exam

Review your knowledge of limits, continuity, and basic derivative rules
Demonstrate your knowledge of limits, continuity, and differentiation on the the Units 1-2 Exam

UNIT 3: Differentiation – Composite, Implicit, and Inverse Functions (Topics 8-10)

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 8: Chain Rule and Implicit Differentiation

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

Topic 9: The Chain Rule and Implicit Differentiation

Calculate derivatives of inverse functions
Calculate derivatives of inverse trigonometric functions

Topic 10: Unit 3 Review and Exam

Review your knowledge of differentiation of composite, implicit, and inverse functions
Demonstrate your knowledge of differentiation of composite, implicit, and inverse functions on the Unit 3 Exam

UNIT 4: Contextual Applications of Differentiation (Topics 11-13)

Essential Questions

How are problems about position, velocity, and accel;eration 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 11: 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 12: Differentials, Linear Approximations, and Newton's Method

Interpret the meaning of a derivative in context
Approximate a value on a curve using the equation of a tangent line

Topic 13: Indeterminate Forms and L’Hopital’s Rule

Determine limits of functions that result in indeterminate forms

UNIT 5: Analytical Applications of Differentiation (Topics 14-19)

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 14: Mean Value Theorem, Extrema, and Critical Numbers

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 15: Local Extrema and Concavity

Justify conclusions about the behavior of a function based on the behavior of its derivative

Topic 16: Using Calculus to Graph Functions

Justify conclusions about the behavior of a function based on the behavior of its derivative

Topic 17: Optimization

Calculate minimum and maximum values in applied contexts or analysis of functions
Interpret minimum and maximum values calculated in applied contexts

Topic 18: Antiderivatives

Calculate rates of change in applied contexts
Identify antiderivatives from a derivative

Topic 19: Units 4-5 Review and Exam

Review your knowledge of applications of derivatives
Demonstrate your knowledge of applications of differentiation on the Units 4-5 Exam

UNIT 6: Integration and Accumulation of Change (Topics 20-23)

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 20: 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 21: The Fundamental Theorem of Calculus and Properties of the Definite Integral

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

Topic 22: Indefinite Integrals, Substitution Method, and Integration Using Long Division

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 23: Integrals Containing Trigonometric Functions

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

UNIT 7: Differential Equations (Topics 24-26)

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 24: Introduction to Differential Equations

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

Topic 25: Slope Fields

Estimate solutions to differential equations.

Topic 26: Units 6-7 Review and Exam

Review your knowledge of integration, accumulation of change, and differential equations
Demonstrate your knowledge of integration, accumulation of change, and differential equations on the Units 6-7 Exam

UNIT 8: Applications of Integration (Topics 27-30)

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 27: 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 28: Volume of a Solid of Revolution

Calculate volumes of solids of revolution using definite integrals

Topic 29: Volume of a Solid with Known Cross Section

Calculate volumes of solids with known cross sections using definite integrals

Topic 30: Unit 8 Review and Exam

Review your knowledge of applications of integration
Demonstrate your knowledge of applications of integration on the Unit 8 Exam

UNIT 9: AP Practice Exam, Practice Exam Reflection, and Calculus Finish Line (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

Reflect on your course experience
Complete end of course activities