AP Calculus

College Board AP Calculus Standards

• LIM-5.A Approximate a definite integral using geometric and numerical methods.
• CHA-4.D Interpret the meaning of a definite integral in accumulation problems.
• CHA-5.A Calculate areas in the plane using the definite integral.
• CHA-5.B Calculate volumes of solids with known cross sections using definite integrals.
• CHA-5.C Calculate volumes of solids of revolution using definite integrals.
   

• Areas between curves in a two-dimensional plane.
• Volumes in a three-dimensional plane using cross sections.
• Volumes of solids of revolution using disks, washers, or shells.

• Definite integrals can be approximated using geometric and numerical methods.
• Areas of regions in the plane can be calculated with definite integrals.
• Definite integrals allow us to solve problems involving the accumulation of change in area or volume or an interval.

• Use integration to calculate areas of regions in a plane.
• Use integration to calculate volumes of solids with known cross sections.
• Use integration to calculate volumes of solids of revolution.
• Use integration to calculate volumes of solids using disks, washers, and shells.
• Use integration to calculate surface areas of solids of revolution. 

• How can calculus be used to calculate areas and volumes of regions and solids, and what methods are best suited for different geometric configurations?
• How can we use integration to determine the net or total area enclosed between two functions over a given interval?
• What are the bounds of integration and how do they relate to the geometry of the solid?
   

• How do we use integrals to find the areas between curves in a two-dimensional plane?
• How do we decide to integrate in terms of x or y?
• How do we set up and evaluate an integral in order to find the volumes of solids with known cross-sections?
• How do we set up and evaluate an integral in order to find the volumes of solids of revolution?
• How do we choose between the methods of discs, washers, or shells for finding the volume of a solid?

• Graphing utilities (graphing calculators, Desmos, etc.)
• Textbook - Calculus - Graphical, Numerical, and Algebraic 6th edition
• Teacher created notes packets
   

• Area between curves
• Volume by cylindrical shells
• Solid of revolution
• Volume by slicing
• Volume by disks
• Volume of a solid
• Volume by washers
• Cross section

• Unit Quizzes
• Unit Tests
• Informal and formal writing assignments
• Informal assessments through warm-ups and exit tickets
• Informal check of daily assignments

College Board AP Calculus Standards

• FUN-6.E For integrands requiring integration by parts: a) determine indefinite integrals and b) evaluate definite integrals.
• FUN-6.F For integrands requiting integration by linear partial fractions: a) determine indefinite integrals and b) evaluate definite integrals
• FUN-7.C Estimate solutions to differential equations.
• FUN-7.D Determine general solutions to differential equations.
• FUN-7.E Determine particular solutions to differential equations.
• FUN-7.F Interpret the meaning of a differential equation and its variables in context.
• FUN-7.G Determine general and particular solutions for problems involving differential equations in context.
• FUN-7.H Interpret the meaning of the logistic growth model in context.
• CHA-6.A Determine the length of a curve in the plane defined by a function, using a definite integral.

• Integration by parts.
• Partial fractions.
• Differential equations.
• Logistic growth.
• Length of a curve.

• Integration by parts is a technique for finding antiderivatives.
• Some rational functions can be decomposed into sums of ratios of linear, non repeating factors to which basic integration techniques can be applied.
• Euler’s method provides a procedure for approximating a solution to a differential equation or a point on a solution curve.
• Antidifferentiation can be used to find general solutions to differential equations and to describe infinitely many solutions to a differential equation. 
• Specific applications of finding general and particular solutions to differential equations include motion along a line and exponential growth and decay.
• The limiting value (carrying capacity) of a logistic differential equation as the independent variable approaches infinity can be determined using the logistic growth model and initial conditions.
• The length of a planar curve defined by a function can be calculated using a definite integral.

• Use integration by parts for antidifferentiation.
• Employ the method of partial fractions to complete antidifferentiation.
• Approximate a solution to a differential equation, using Euler’s method.
• Solve separable differential equations, including exponential growth and decay.
• Solve and apply logistic differential equations for real world models.
• Find the length of a curve using integration.

• How do you recognize which technique is most efficient when integrating?
• How does solving differential equations allow us to determine functions and develop models?
• How do definite integrals allow us to solve problems involving the accumulation of change in length over an interval?
   

• How do you know when integration by parts is the correct method for a specific problem?
• How do you know when the method of partial fractions is needed to complete integration?
• How is solving separable differential equations used for exponential growth and decay applications?
• How is a logistic differential equation used to model population growth?
• How do you apply a definite integral to find the length of a curve?

• Graphing utilities (graphing calculators, Desmos, etc.)
• Textbook - Calculus - Graphical, Numerical, and Algebraic 6th edition
• Teacher created notes packets
   

• Integration by parts
• Tabular integration
• Separable differential equation
• Exponential growth
• Exponential decay
• Law of exponential change
• half-life
• Newton’s law of cooling
• Partial fractions
• Logistic differential equation
• Logistic curve
• Carrying capacity
• Limit to growth
• Heaviside method
• Euler’s method
• Length of curve
• Smooth curve
• Arc length

• Unit Quizzes
• Unit Tests
• Informal and formal writing assignments
• Informal assessments through warm-ups and exit tickets
• Informal check of daily assignments
   

College Board AP Calculus Standards

• LIM-2.D Interpret the behavior of functions using limits involving infinity.
• LIM-4.A Determine limits of functions that result in indeterminate forms.
• LIM-6.A Evaluate an improper integral or determine that the integral diverges.

• Relative rates of growth.
• Indeterminate form.
• Improper integral.

• The concept of a limit can be extended to include limits at infinity.
• Limits at infinity describe end behavior.
• Relative magnitudes of functions and their rates of change can be compared using limits.
• When the ratio of two functions tends to zero over zero or infinity over infinity in the limit, such forms are said to be indeterminate and may be evaluated using L’Hospital’s Rule.
• An improper integral is an integral that has one or both limits infinite or has an integrand that is unbounded in the interval of integration.
• Improper integrals can be determined using limits of definite integrals.

• Compare growth rates of various function types as the independent variable increases without bounds.
• Use L’Hospital’s Rule to determine limits of indeterminate forms.
• Identify various indeterminate forms and the methods for finding their limits.
• Evaluate improper integrals of various kinds.
• Use comparison tests for convergence or divergence of improper integrals.

• How can we use derivatives to evaluate limits that result in indeterminate forms?.
• How can we determine whether an improper integral converges or diverges?
• How do we evaluate an improper integral when the interval or integrand is unbounded?
• How can we compare the long-term behavior of different functions to determine which grows faster as the input becomes very large?
   

• How do you identify the various indeterminate forms that a limit could approach?
• When can you use L’Hospital’s rule for an indeterminate form of a limit?
• When would you need to use logarithms when working with an indeterminate form?
• How are limits used to compare the relative rates of growth of various functions (faster, slower, same)?
• How can the transitive property help us determine the relative growth rates of multiple functions by comparing them pairwise?
• How do we evaluate an improper integral as a limit of a definite integral?
• How do we use a comparison test for convergence and divergence of improper integrals? 
   

• Graphing utilities (graphing calculators, Desmos, etc.)
• Textbook - Calculus - Graphical, Numerical, and Algebraic 6th edition
• Teacher created notes packets

• Comparison test
• Convergence of improper integral
• Divergence of improper integral
• Grows faster
• Grows slower
• Grows at the same rate
• Indeterminate form
• L’Hospital’s 
• Rule
• Transitivity of growing rates
• Improper integral
• Value of improper integral
• Improper integral with infinite discontinuity

• Unit Quizzes
• Unit Tests
• Informal and formal writing assignments
• Informal assessments through warm-ups and exit tickets
• Informal check of daily assignments
   

College Board AP Calculus Standards

• LIM-7.A Determine whether a series converges or diverges.
• LIM-7.B Approximate the sum of a series.
• LIM-8.A Represent a function at a point as a Taylor polynomial.
• LIM-8.B Approximate function values using a Taylor polynomial.
• LIM-8.C Determine the error bound associated with a Taylor polynomial approximation.
• LIM-8.D Determine the radius of convergence and interval of convergence for a power series.
• LIM-8.E Represent a function as a Taylor series or a Maclaurin series.
• LIM-8.F Interpret Taylor series and Maclaurin series.
• LIM-8.G Represent a given function as a power series.

• Arithmetic and geometric sequences and series.
• Convergence and divergence of a series.
• Power series.
• Taylor polynomial.
• Taylor series and Maclaurin series.
• Truncation error.
• Radius of convergence.
• Interval of convergence.
• Absolute and conditional convergence.
• Testing convergence at endpoints.

• The nth partial sum is defined as the sum of the first n terms of a series.
• An infinite series of numbers converges to a real number S (or has sum S) if and only if the limit of its sequence of partial sums exists and equals S.
• A geometric series is a series with a constant ratio between successive terms.
• If the absolute value of the common ratio r is less than one, the sum of the series exists.
• The nth term test, integral test, direct comparison test, limit comparison test, ratio test, p-series test, alternating series test, etc. are tests to determine whether a series converges or diverges.
• In addition to geometric series, common series of numbers include the harmonic series, the alternating harmonic series, and p-series.
• A series may be absolutely convergent, conditionally convergent, or divergent.
• If a series converges absolutely, then any series obtained from it by regrouping or rearranging the terms has the same value.
• If an alternating series converges by the alternating series test, then the alternating series error bound can be used to bound how far a partial sum is from the value of the infinite series.
• The coefficient of the nth degree term in a Taylor polynomial for a function f centered at x=a is f(n)(a)n!.
• In many cases, as the degree of a Taylor polynomial increases, the nth degree polynomial will approach the original function over some interval.
• Taylor polynomials for a function f centered at x=a can be used to approximate function values of f near x=a.
• The Lagrange error bound can be used to determine a maximum interval for the error of a Taylor polynomial approximation to a function.
• In some situations, the alternating series error bound can be used to bound the error of a Taylor polynomial approximation to the value of a function.
• A power series is a series of the form n=0anx-r, where n is a non-negative integer, an is a sequence of real numbers, and r is a real number.
• If a power series converges, it either converges at a single point or has an interval of convergence.
• The ratio test can be used to determine the radius of convergence of a power series.
• The radius of convergence of a power series can be used to identify an open interval on which the series converges, but it is necessary to test both endpoints of the interval to determine the interval of convergence.
• If a power series has a positive radius of convergence, then the power series is the Taylor series of the function to which it converges over the open interval.
• The radius of convergence of a power series obtained by term-by-term differentiation or term-by-term integration is the same as the radius of convergence of the original power series.
• A Taylor polynomial for f(x) is a partial sum of the Taylor series for f(x).
• The Maclaurin series for 11 - x is a geometric series.
• The Maclaurin series for sinx, cosx, and ex provides the foundation for constructing the Maclaurin series for other functions.
• Using a known series, a power series for a given function can be derived using operations such as term-by-term differentiation or term-by-term integration, and by various methods, such as algebraic processes, substitutions, or using properties of geometric series.

• Find the terms and write the rules for sequences and/or series.
• Tell whether a series converges or diverges.
• Find the interval of convergence of a geometric series.
• Find a power series that represents a function.
• Find a power series by differentiation and integration.
• Construct a Taylor series and/or Maclaurin series for a given function.
• Analyze truncation error for Taylor series using alternating series error bound and the Lagrange error bound found in Taylor’s Theorem.
• Derive Euler’s Formula from the Maclaurin series.
• Find a radius of convergence of a power series.
• Use various tests (nth term test, direct comparison test, ratio test, integral test, p-series test, limit comparison test, alternating series test, etc.) to determine convergence.

• How can we determine whether an infinite series converges or diverges, and what does that tell us about the behavior of functions or mathematical models?
• How does a power series allow us to represent associated functions on an appropriate interval?
• How can we use Taylor series to approximate complex functions, and what are the limits of that approximation?
   

• How do we analyze a geometric series to determine whether the series converges or diverges?
• How do we construct a power series to represent a function?
• What is the difference between a Taylor series and Maclaurin series?
• How do we find the truncation error for Taylor polynomials?
• How do we use the Lagrange error bound to determine a maximum interval for the error of a Taylor polynomial approximation to a function?
• How can we use the Remainder Bounding Theorem to prove that the Maclaurin series for a given function converges to that function for all real values of x?
• How is the Ratio Test used to find the radius of convergence for a power series?
• How are the various tests (nth term test, direct comparison test, ratio test, integral test, p-series test, limit comparison test, alternating series test, etc.) applied to determine convergence?
• How do you test for conditional and absolute convergence at the endpoints of intervals of convergence?

• Graphing utilities (graphing calculators, Desmos, etc.)
• Textbook - Calculus - Graphical, Numerical, and Algebraic 6th edition
• Teacher created notes packets

• Arithmetic sequence
• Common difference
• Geometric sequence 
• Common ratio
• Nth term of a sequence
• Terms of a sequence
• Power series 
• Center of power series
• Convergent series
• Divergent series
• Finite sum
• Infinite series
• Interval of convergence
• Nth term of a series
• Nth term test for divergence
• Partial sum
• Sum of a series
• Terms of a series
• Geometric series
• Differentiation of series
• Integration of series
• Maclaurin series
• Taylor series
• Taylor polynomial
• Term-by-term differentiation
• Term-by-term integration
• Terms of a series
• Error term
• Alternating series bound theorem
• Lagrange error bound
• Lagrange form of the remainder
• Remainder of order n
• Remainder bounding theorem
• Taylor’s formula
• Taylor’s theorem with remainder
• Truncation error
• Absolute convergence
• Convergence theorem for power series
• Converges absolutely
• Direct comparison test
• Euler’s formula
• Radius of convergence
• Ratio test
• Telescoping series
• Alternating series
• Alternating harmonic series
• Alternating series test
• Conditional convergence
• Harmonic series
• Integral test
• Limit comparison test
• Nth root test
• P-series
• P-series test

• Unit Quizzes
• Unit Tests
• Informal and formal writing assignments
• Informal assessments through warm-ups and exit tickets
• Informal check of daily assignments

College Board AP Calculus Standards

• CHA-3.G Calculate derivatives of parametric functions.
• CHA-6.B Determine the length of a curve in the plane defined by parametric functions using a definite integral.
• CHA-3.H Calculate derivatives of vector-valued functions.
• FUN-8.A Determine a particular solution given a rate vector and initial conditions.
• FUN-8.B Determine values for positions and rates of change in problems involving planar motion.
• FUN-3.G Calculate derivatives of functions written in polar coordinates.
• CHA-5.D Calculate areas of regions defined by polar curves using definite integrals.

• Slope and concavity of a curve given by parametric functions.
• Length of a curve given by parametric functions.
• Two-dimensional vectors.
• Position, velocity, and acceleration for vector-valued functions.
• Displacement and distance traveled for vector-valued functions.
• Slope of a curve given by a polar function.
• Areas bounded by polar curves.

• Methods for calculating derivatives or integrals of rectangular functions can be extended to parametric functions, vector-valued functions, or polar functions.
• For a curve defined parametrically, the value of dydx at a point on the curve is the slope of line tangent to the curve at that point.
• The length of a parametrically defined curve can be calculated using a definite integral.
• Derivatives can be used to determine velocity, speed, and acceleration for a particle moving along a curve in the plane defined using parametric or vector-valued functions.
• For a particle in planar motion over an interval of time, the definite integral of the velocity vector represents the particle’s displacement over the interval of time, from which we might determine its position.
• The definite integral of speed represents the particle’s total distance traveled over the interval of time.
• For a curve given by a polar equation r=f, derivatives of r, x, and y with respect to , and first and second derivatives of y with respect to x can provide information about the curve.
• The concept of calculating areas in rectangular coordinates can be extended to polar coordinates.
• Areas or regions bounded by polar curves can be calculated with definite integrals.

• Calculate the first and second derivatives of parametric functions.
• Eliminate the parameter for a set of parametric equations.
• Find the horizontal and vertical tangent lines to the curve defined by a set of parametric equations.
• Find the length of a curve given by parametric functions.
• Perform vector operations.
• Find the magnitude and direction angle of a vector.
• Find any of the following: position vector, velocity vector, and acceleration vector given various pieces of information, using differentiation or integration. 
• Use integration to find displacement and distance traveled for a particle, given a velocity vector.
• Convert between polar and rectangular coordinates/equations.
• Graph polar coordinates/equations.
• Calculate the slope of a curve given by a polar function.
• Find the areas bounded by polar curves.

• How can we use derivatives to solve real-world problems involving rates of change?
• How are definite integrals used to solve problems involving the accumulation of change in length over an interval, as well as area or volume over an interval?
• How can we solve an initial value problem in order to determine an expression for the position of a particle moving in the plane?
   

• How do we take the derivative of a parametric function, since there is an extra variable now?
• How do we find the arc length of a parametrized curve using derivatives and integrals?
• How do we use trigonometry skills to find the magnitude and direction of a vector?
• How do we decide whether to take a derivative or integral when working between position, velocity, and acceleration vectors?
• What is the difference between speed and velocity?
• How do we find displacement and distance traveled for a particle using derivatives and integrals?
• How do we use previous skills of finding derivatives of parametric functions in order to find the slope of a polar curve?
• How are integrals used to find the area between polar curves?
   

• Graphing utilities (graphing calculators, Desmos, etc.)
• Textbook - Calculus - Graphical, Numerical, and Algebraic 6th edition
• Teacher created notes packets

• Arc length of parametric curve
• Parametric function
• Parametric formula for derivative (1st and 2nd)
• Absolute value of a vector
• Acceleration vector
• Component form of a vector
• Direction angle of a vector
• Direction of motion
• Direction vector
• Displacement
• Distance traveled
• Head minus tail rule
• Initial point of an arrow
• Initial ray of angle direction
• Magnitude of a vector
• Opposite of a vector
• Path of a particle
• Position vector
• Properties of vectors
• Resultant vector
• Scalar multiple of a vector
• Speed
• Unit vector
• Velocity vector
• Zero vector
• Arc length of polar curve
• Area between polar curves
• Cardioid
• Convex limacon
• Dimpled limacon
• Lemniscate 
• Limacon 
• Inner loop limacon
• Parametric equations of a polar curve
• Polar coordinate
• Polar function
• Polar-rectangular conversion
• Pole
• Rose curve

• Unit Quizzes
• Unit Tests
• Informal and formal writing assignments
• Informal assessments through warm-ups and exit tickets
• Informal check of daily assignments