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AP College Calculus BC

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AP College Calculus BC

Course Overview

Introducing Calculus: Can Change Occur at an Instant?

Defining limits and using limit notation
Estimating limit values from graphs
Estimating limit values from tables
Determining limits using algebraic properties
Determining limits using algebraic manipulation
Selecting procedures for determining limits
Determining limits using the Squeeze Theorem
Connecting multiple representations of limits
Exploring types of discontinuities
Defining continuity at a point
Confirming continuity over an interval
Removing discontinuities
Connecting infinite limits and vertical asymptotes
Connecting limits at infinity and horizontal asymptotes
Working with the Intermediate Value Theorem (IVT)

Differentiation: Definition and Fundamental Properties

Defining average and instantaneous rates of change at a point
Defining the derivative of a function and using derivative notation
Estimating derivatives of a function at a point
Connecting differentiability and continuity: determining when derivatives do and do not exist
Applying the Power Rule
Derivative rules: constant, sum, difference, and constant multiple derivatives of cos⁡x\cos xcosx, sin⁡x\sin xsinx, exe^xex, and ln⁡x\ln xlnx
The Product Rule
The Quotient Rule
Finding the derivatives of tangent, cotangent, secant, and cosecant functions

Differentiation: Composite, Implicit, and Inverse Functions

The Chain Rule
Implicit differentiation
Differentiating inverse functions
Differentiating inverse trigonometric functions
Selecting procedures for calculating derivatives
Calculating higher-order derivatives

Contextual Applications of Differentiation

Interpreting the meaning of the derivative in context
Straight-line motion: connecting position, velocity, and acceleration
Rates of change in applied contexts other than motion
Introduction to related rates
Solving related rates problems
Approximating values of a function using local linearity and linearization
Using L'Hospital's Rule for determining limits of indeterminate forms

Analytical Applications of Differentiation

Using the Mean Value Theorem
Extreme Value Theorem: global versus local extrema, and critical points
Determining intervals on which a function is increasing or decreasing
Using the First Derivative Test to determine relative (local) extrema
Using the Candidates Test to determine absolute (global) extrema
Determining concavity of functions over their domains
Using the Second Derivative Test to determine extrema
Sketching graphs of functions and their derivatives
Connecting a function, its first derivative, and its second derivative
Introduction to optimization problems
Solving optimization problems
Exploring behaviors of implicit relations

Integration and Accumulation of Change

Exploring accumulations of change
Approximating areas with Riemann sums
Riemann sums, summation notation, and definite integral notation
The Fundamental Theorem of Calculus and accumulation functions
Interpreting the behavior of accumulation functions involving area
Applying properties of definite integrals
The Fundamental Theorem of Calculus and definite integrals
Finding antiderivatives and indefinite integrals: basic rules and notation
Integrating using substitution
Integrating functions using long division and completing the square
Integrating using integration by parts
Using linear partial fractions
Evaluating improper integrals
Selecting techniques for antidifferentiation

Differential Equations

Modelling situations with differential equations
Verifying solutions for differential equations
Sketching slope fields
Reasoning using slope fields
Approximating solutions using Euler’s Method
Finding general solutions using separation of variables
Finding particular solutions using initial conditions and separation of variables
Exponential models with differential equations
Logistic models with differential equations

Applications of Integration

Finding the average value of a function on an interval
Connecting position, velocity, and acceleration of functions using integrals
Using accumulation functions and definite integrals in applied contexts
Finding the area between curves expressed as functions of xxx
Finding the area between curves expressed as functions of yyy
Finding the area between curves that intersect at more than two points
Volumes with cross sections: squares and rectangles
Volumes with cross sections: triangles and semicircles
Volume with the disc method: revolving around the xxx- or yyy-axis
Volume with the disc method: revolving around other axes
Volume with the washer method: revolving around the xxx- or yyy-axis
Volume with the washer method: revolving around other axes
The arc length of a smooth, planar curve and distance traveled

Parametric Equations, Polar Coordinates, and Vector-Valued Functions

Defining and differentiating parametric equations
Second derivatives of parametric equations
Finding arc lengths of curves given by parametric equations
Defining and differentiating vector-valued functions
Integrating vector-valued functions
Solving motion problems using parametric and vector-valued functions
Defining polar coordinates and differentiating in polar form
Finding the area of a polar region or the area bounded by a single polar curve
Finding the area of the region bounded by two polar curves

Infinite Sequences and Series

Defining convergent and divergent infinite series
Working with geometric series
The nnnth term test for divergence
Integral test for convergence
Harmonic series and ppp-series
Comparison tests for convergence
Alternating series test for convergence
Ratio test for convergence
Determining absolute or conditional convergence
Alternating series error bound
Finding Taylor polynomial approximations of functions
Lagrange error bound
Radius and interval of convergence of power series
Finding Taylor or Maclaurin series for a function
Representing functions as power series

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