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High School AP Calculus BC

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Suggested Prerequisites

Algebra I, Geometry, Algebra II, Pre-Calculus or Trigonometry/Analytical Geometry.

Description

As you dive into this interactive online calculus course, you will follow in the footsteps of great mathematicians like Newton and Leibniz. This adventure covers many topics, including limits, continuity, differentiation, integration, differential equations, the applications of derivatives and integrals, parametric and polar equations, and infinite sequences and series, including Taylor, Maclaurin and power series. This Advanced Placement (AP) calculus course covers a full year of material equivalent to college-level calculus. Students who complete this course often seek to earn college credit or advanced placement. Colleges and universities generally assign students to appropriate calculus courses based on their preparation, which is often evaluated through AP exam results or other criteria.

Module One: Limits and Continuity

-Using limits to analyze instantaneous change

-Estimating limit values using graphs and tables

-Determining limits using algebraic properties and manipulation

-Evaluating limits of indeterminant form

-Evaluating limits using substitution

-Squeeze Theorem

-Intermediate Value Theorem

-Determining continuity and exploring discontinuity

-Connecting limits, infinity and asymptotes


Module Two: Differentiation: Definition and Fundamental Properties

-Definition of a derivative

-Average and instantaneous rates of change

-Determining differentiability

-Estimating derivatives

-Rules of differentiation

-Product rule

-Quotient rule

-Derivatives of trigonometric, exponential, and logarithmic functions


Module Three: Differentiation: Composite, Implicit, and Inverse Functions

-Chain rule

-Implicit differentiation

-Differentiating inverse functions

-Differentiating composite functions

-Differentiating inverse trigonometric functions

-Selecting procedures for calculating derivatives

-Calculating higher-order derivatives


Module Four: Contextual Applications of Differentiation

-Interpreting and applying the derivative in motion

-Rates of change in other applied contexts

-Related rates

-Approximating values using local linearity and linearization

-L'Hospital's Rule


Module Five: Contextual Applications of Differentiation

-Mean value and extreme value theorem

-Determining function behavior

-First derivative test

-Determining absolute extrema using the candidates test

-Determining concavity of functions

-Second derivative test

-Connecting graphs of functions and their derivatives

-Optimization problems

-Exploring behaviors of implicit relations

Module Six: Integration and Accumulation of Change

-Riemann Sums and the definite integral

-Accumulation functions involving area

-The Fundamental Theorem of Calculus

-Applying properties of the definite integrals

-Finding indefinite integrals and antiderivatives

-Integrating using substitution and integration by parts

-Integrating using linear partial fractions

-Evaluating improper integrals

-Integrating functions using various methods

-Selecting techniques for antidifferentiation


Module Seven: Differential Equations

-Solutions of differential equations

-Sketching and reasoning using slope fields

-Approximating solutions using Euler's Method

-Finding solutions using separation of variables

-Exponential models with differential equations

-Logistic models with differential equations


Module Eight: Applications of Integration

-Average value and connecting position using integrals

-Velocity and acceleration using integrals

-Using accumulation functions and definite integrals contextually

-Finding the area between curves

-Area between curves with multiple intersections

-Volumes with discs

-Volumes with washers

-Volumes with cross sections

-Arc length of a smooth planar curve

-Arc length and distance traveled


Module Nine: Parametric, Polar and Vector-Valued Equations

-Differentiating parametric equations and finding arc length

-Differentiating and integrating vector-valued functions

-Solving motion problems using parametric functions

-Solving motion problems using vector-valued functions

-Defining polar coordinates

-Differentiating in polar form to

find area bounded by polar curves


Module Ten: Infinite Sequences and Series

- Convergent and divergent infinite series 

-Convergent and divergent geometric series

-Integral test for convergence, harmonic and p-series

-Comparison test for convergence

-Additional tests to determine convergence

-Alternating series and their error bound

-Taylor polynomial approximations and evaluating error

-Radius and convergence interval of power series

-Taylor or Maclaurin series for a function

-Representing functions as power series