Calculus

Completion: 0%

Limits & Continuity

LIM-1.CLIM-1.DLIM-2.BLIM-2.C
0%
Limits by Direct Substitution--
Limits by Factoring--
Limits of Rational Functions--
Limits at Infinity--
One-Sided Limits--
Continuity at a Point--

Derivative Rules

FUN-3.AFUN-3.BFUN-3.CFUN-3.D
0%
Power Rule--
Derivatives of Polynomials--
Derivatives of Trig Functions--
Product Rule--
Quotient Rule--
Chain Rule (basic)--
Chain Rule (nested)--
Derivatives of e^x and ln(x)--
Implicit Differentiation--

Applications of Derivatives

CHA-3.CFUN-4.BFUN-4.CFUN-4.D
0%
Equation of Tangent Line--
Find Critical Points--
Increasing/Decreasing Intervals--
Relative Extrema--
Points of Inflection--
Optimization Problems--
Related Rates--

Integration

FUN-5.AFUN-6.ACHA-4.B
0%
Power Rule for Integrals--
Integrate Polynomials--
Integrate Trig Functions--
Definite Integrals (polynomial)--
Fundamental Theorem of Calculus--
Initial Value Problems--

Integration Techniques

FUN-6.BFUN-6.DFUN-6.E
0%
u-Substitution (basic)--
u-Substitution (definite integrals)--
Integration by Parts--
Partial Fractions--

Applications of Integration

CHA-5.ACHA-5.BCHA-5.D
0%
Area Between Curves--
Volume (Disc Method)--
Volume (Washer Method)--
Average Value of a Function--

Differential Equations

FUN-7.BFUN-7.CFUN-7.E
0%
Verify Solutions--
Separable Equations (basic)--
Separable Equations (with initial condition)--
Exponential Growth/Decay--

Sequences & Series

LIM-7.ALIM-7.BLIM-8.ALIM-8.B
0%
Convergence Tests--
Geometric Series Sum--
Ratio Test--
Taylor Polynomials (at 0)--
Taylor Polynomials (at a)--
Maclaurin Series--

AP Calculus Standards

AP Calculus AB

Unit 1 — Limits and Continuity

Determining Limits Using Algebraic Properties and Methods
  • LIM-1.ARepresent limits analytically using correct notation.
  • LIM-1.BInterpret limits expressed in analytic notation.
  • LIM-1.CDetermine the limits of functions using limit theorems and algebraic manipulation.
  • LIM-1.DDetermine the limits of functions using equivalent expressions for the function or the squeeze theorem.
  • LIM-1.EDetermine the limits of functions using L'Hôpital's rule.
Estimating Limit Values and Determining Continuity
  • LIM-2.AEstimate limits of functions from tables and graphs of functions.
  • LIM-2.BDetermine the applicability of important calculus theorems using continuity.
  • LIM-2.CDetermine intervals over which a function is continuous.
  • LIM-2.DDetermine values of x or conditions for which a function is discontinuous and classify the type of discontinuity.

Unit 2 — Differentiation: Definition and Fundamental Properties

Defining and Applying the Derivative
  • CHA-2.ADetermine average and instantaneous rates of change using the definition of the derivative.
  • CHA-2.BDetermine the equation of a line tangent to a curve at a given point.
  • CHA-2.CEstimate derivatives at a point using the slope of a secant line or a tangent line approximation.
  • FUN-3.ACalculate derivatives of familiar functions, including power, exponential, logarithmic, and trigonometric functions.

Unit 3 — Differentiation: Composite, Implicit, and Inverse Functions

Using the Chain Rule and Implicit Differentiation
  • FUN-3.CCalculate derivatives of compositions of differentiable functions using the chain rule.
  • FUN-3.DCalculate derivatives of implicitly defined functions.
  • FUN-3.ECalculate derivatives of inverse and inverse trigonometric functions.
Differentiation Rules
  • FUN-3.BCalculate derivatives of products and quotients of differentiable functions.

Unit 4 — Contextual Applications of Differentiation

Applying Derivatives to Real-World Contexts
  • CHA-3.AInterpret the meaning of the derivative in context, including units of measure.
  • CHA-3.BCalculate rates of change in applied contexts, including motion along a line (position, velocity, acceleration).
  • CHA-3.CSolve related rates problems by applying the chain rule and other differentiation techniques.
  • CHA-3.DCalculate related rates in applied contexts.
Linear Approximation and L'Hôpital's Rule
  • CHA-3.EApproximate function values using local linear approximation and differentials.
  • LIM-4.ADetermine limits of functions that result in indeterminate forms using L'Hôpital's rule.

Unit 5 — Analytical Applications of Differentiation

Using the Mean Value Theorem and Analyzing Functions
  • FUN-1.AJustify conclusions about functions using the Mean Value Theorem over an interval.
  • FUN-4.AJustify conclusions about the behavior of a function based on the behavior of its derivatives.
  • FUN-4.BDetermine critical points and absolute/relative extrema of a function on an interval.
  • FUN-4.CDetermine intervals over which a function is increasing or decreasing, and determine concavity of a function over an interval.
Optimization
  • FUN-4.DDetermine the optimal value (maximum or minimum) of a function in an applied context using calculus techniques.
  • FUN-4.EDetermine the location of points of inflection of a function.

Unit 6 — Integration and Accumulation of Change

Interpreting and Evaluating Definite Integrals
  • CHA-4.AInterpret the meaning of the definite integral as the limit of a Riemann sum in context.
  • CHA-4.BDetermine the value of a definite integral using the Fundamental Theorem of Calculus.
  • CHA-4.CDetermine the value of a definite integral using areas and properties of definite integrals.
  • CHA-4.DRepresent accumulation as a definite integral.
Antiderivatives and Indefinite Integrals
  • FUN-5.ADetermine antiderivatives of functions and indefinite integrals using knowledge of derivatives.
  • FUN-6.AEvaluate definite integrals analytically using the Fundamental Theorem of Calculus.
  • FUN-6.BEvaluate definite integrals using substitution.
  • FUN-6.CEvaluate improper integrals (BC only).

Unit 7 — Differential Equations

Solving Differential Equations
  • FUN-7.AInterpret verbal statements of problems as differential equations involving a derivative expression.
  • FUN-7.BVerify solutions to differential equations.
  • FUN-7.CDetermine general and particular solutions of separable differential equations.
  • FUN-7.DSketch slope fields and solution curves for differential equations.
Modeling with Differential Equations
  • FUN-7.EInterpret, create, and solve differential equations that model real-world phenomena, including exponential growth and decay.
  • FUN-7.FDetermine the behavior of a solution to a differential equation using slope fields.

Unit 8 — Applications of Integration

Finding Area and Volume
  • CHA-5.ACalculate areas of regions in the plane bounded by curves defined by functions.
  • CHA-5.BCalculate volumes of solids of revolution using disc and washer methods.
  • CHA-5.CCalculate volumes of solids with known cross sections perpendicular to an axis.
Average Value and Accumulation Functions
  • CHA-5.DDetermine the average value of a function over an interval using the definite integral.

AP Calculus BC (Additional Topics)

Unit 6 BC — Integration Techniques (BC)

Advanced Integration Methods
  • FUN-6.DEvaluate integrals using integration by parts.
  • FUN-6.EEvaluate integrals using partial fraction decomposition (linear, non-repeating factors).

Unit 9 — Parametric Equations, Polar Coordinates, and Vector-Valued Functions

Parametric and Vector-Valued Functions
  • CHA-6.ACalculate derivatives of parametric functions.
  • CHA-6.BCalculate the length of a curve defined by parametric equations.
  • FUN-8.ACalculate derivatives and integrals involving vector-valued functions.
Polar Functions
  • CHA-6.CCalculate derivatives of polar functions.
  • CHA-6.DCalculate areas of regions bounded by polar curves.

Unit 10 — Infinite Sequences and Series

Convergence and Divergence of Series
  • LIM-7.ADetermine whether a series converges or diverges using the nth-term test, integral test, comparison tests, alternating series test, and ratio test.
  • LIM-7.BDetermine or estimate the sum of a series (geometric series, telescoping series).
  • LIM-7.CDetermine the radius and interval of convergence of a power series.
Taylor and Maclaurin Series
  • LIM-8.ARepresent a function as a Taylor series or a Maclaurin series.
  • LIM-8.BDetermine the coefficients of a Taylor polynomial approximation of a function.
  • LIM-8.CDetermine the radius and interval of convergence of a Taylor series.
  • LIM-8.DUse Taylor polynomial approximations to estimate function values and error bounds (Lagrange error bound).