Calculus BC

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Calculus BC

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Calculus AB is the first calculus course typically taken after pre calculus / analysis. Historically, this class has been a high school level course that is often offered in junior or senior year

Students learn: Limits and Continuity, Differentiation, Integration, Differential equations and applications

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Description

Calculus BC

Calculus BC is calculus course typically taken after pre-calculus/ analysis or Algebra2/Trigonometry. Historically, this class has been a high school level course that is often offered in junior or senior year. Pre-requisite: Algebra2/Trig

Contents of the Algebra 1 Course:

  • Limits and Continuity
    • How limits help us to handle change at an instant
    • Definition and properties of limits in various representations
    • Definitions of continuity of a function at a point and over a domain
    • Asymptotes and limits at infinity
    • Reasoning using the Squeeze theorem and the Intermediate Value Theorem
  • Differentiation: Definition and fundamental properties
    • Defining the derivative of a function at a point and as a function
    • Connecting differentiability and continuity
    • Determining derivatives for elementary functions
    • Applying differentiation rules
  • Differentiation: Composite, implicit and inverse functions
    • The chain rule for differentiating composite functions
    • Implicit differentiation
    • Differentiation of general and particular inverse functions
    • Determining higher-order derivatives of functions
  • Contextual applications of differentiation
    • Identifying relevant mathematical information in verbal representations of real-world problems involving rates of change
    • Applying understandings of differentiation to problems involving motion
    • Generalizing understandings of motion problems to other situations involving rates of change
    • Solving related rates problems
    • Local linearity and approximation
    • L’Hospital’s rule
  • Analytical applications of differentiation
    • Mean Value Theorem and Extreme Value Theorem
    • Derivatives and properties of functions
    • How to use the first derivative test, second derivative test, and candidates test
    • Sketching graphs of functions and their derivatives
    • How to solve optimization problems
    • Behaviors of Implicit relations
  • Integration and accumulation of change
    • Using definite integrals to determine accumulated change over an interval
    • Approximating integrals using Riemann Sums
    • Accumulation functions, the Fundamental Theorem of Calculus, and definite integrals
    • Antiderivatives and indefinite integrals
    • Properties of integrals and integration techniques, extended
    • Determining improper integrals
  • Differential equations
    • Interpreting verbal descriptions of change as separable differential equations
    • Sketching slope fields and families of solution curves
    • Using Euler’s method to approximate values on a particular solution curve
    • Solving separable differential equations to find general and particular solutions
    • Deriving and applying exponential and logistic models
  • Applications of integrations
    • Determining the average value of a function using definite integrals
    • Modeling particle motion
    • Solving accumulation problems
    • Finding the area between curves
    • Determining volume with cross-sections, the disc method, and the washer method
    • Determining the length of a planar curve using a definite integral
  • Parametric Equations, Polar Coordinates and Vector-Valued Functions
    • Finding derivatives of parametric functions and vector-valued functions
    • Calculating the accumulation of change in length over an interval using a definite integral
    • Determining the position of a particle moving in a plane
    • Calculating velocity, speed, and acceleration of a particle moving along a curve
    • Finding derivatives of functions written in polar coordinates
    • Finding the area of regions bounded by polar curves
  • Infinite Sequences and Series
    • Applying limits to understand convergence of infinite series
    • Types of series: Geometric, harmonic, and p-series
    • A test for divergence and several tests for convergence
    • Approximating sums of convergent infinite series and associated error bounds
    • Determining the radius and interval of convergence for a series
    • Representing a function as a Taylor series or a Maclaurin series on an appropriate interval

Pre-requisite: Algebra2/Trig

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