AP Calculus BC
In short
AP Calculus BC has 10 units: Limits and Continuity, Differentiation: Definition and Fundamental Properties, Differentiation: Composite, Implicit, and Inverse Functions and 7 more. Each unit lists its topics in course framework order, with a self-check list and how each is tested on the exam. Start with Limits and Continuity or jump to the unit you are revising.
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What units are in AP Calculus BC?
1.1 Introducing Calculus: Can Change Occur at an Instant?
1.2 Defining Limits and Using Limit Notation
1.3 Estimating Limit Values from Graphs
- I can identify when a limit might not exist for some functions at particular values of x, such as when the function is unbounded, oscillating, or when the left and right limits are not equal.
1.4 Estimating Limit Values from Tables
- I can use numerical information (such as tables of values) to estimate limits.
1.5 Determining Limits Using Algebraic Properties of Limits
- I can determine one-sided limits analytically or graphically.
1.6 Determining Limits Using Algebraic Manipulation
1.7 Selecting Procedures for Determining Limits
1.8 Determining Limits Using the Squeeze Theorem
- I can use definitions, theorems, and properties to justify claims about limits.
- I can determine the limits of functions using the squeeze theorem.
- I can find the limit of a function by applying the squeeze theorem and verifying its conditions.
1.10 Exploring Types of Discontinuities
1.11 Defining Continuity at a Point
- I can justify whether a function is continuous at a specific point using the formal definition.
- I can verify the three conditions for continuity at a point: the function value exists, the limit exists, and the limit equals the function value.
1.12 Confirming Continuity over an Interval
1.13 Removing Discontinuities
- I can remove a discontinuity by defining or redefining the value of the function at a point to equal the limit of the function as x approaches that point.
- I can make a piecewise-defined function continuous at a boundary by ensuring the expressions on both sides of the boundary and the function value at the boundary are all equal.
1.14 Connecting Infinite Limits and Vertical Asymptotes
- I can describe and explain asymptotic and unbounded behavior of functions using limits.
1.15 Connecting Limits at Infinity and Horizontal Asymptotes
- I can describe the end behavior of a function using limits at infinity.
1.16 Working with the Intermediate Value Theorem (IVT)
- I can state the conditions and conclusion of the Intermediate Value Theorem for a continuous function on a closed interval.
2.1 Defining Average and Instantaneous Rates of Change at a Point
- I can express the instantaneous rate of change of a function at x = a using the limit definitions of the derivative.
2.2 Defining the Derivative of a Function and Using Derivative Notation
- I can recognize and use various notations for the derivative, including dy/dx, f'(x), and y'.
- I can represent the derivative graphically, numerically, analytically, and verbally.
- I can relate the derivative of a function at a point to the slope of the tangent line at that point.
2.3 Estimating Derivatives of a Function at a Point
2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
- I can state that if a function is differentiable at a point, it must be continuous at that point, and that a point outside the domain of a function is also outside the domain of its derivative.
- I can recognize and provide examples of continuous functions that fail to be differentiable at a point in their domain (e.g., corners, cusps, vertical tangents).
2.5 Applying the Power Rule
- I can use the power rule to calculate the derivative for functions of the form f(x) = x^r.
2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple
2.7 Derivatives of cos x, sin x, e^x, and ln x
- I can calculate the derivatives of sine, cosine, exponential, and logarithmic functions.
2.8 The Product Rule
- I can apply the product rule to find the derivative of a product of functions.
2.9 The Quotient Rule
- I can calculate the derivative of a quotient of differentiable functions using the quotient rule.
2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions
3.1 The Chain Rule
3.3 Differentiating Inverse Functions
- I can calculate derivatives of inverse functions.
3.4 Differentiating Inverse Trigonometric Functions
3.5 Selecting Procedures for Calculating Derivatives
- I can identify and select the appropriate mathematical rule or procedure to calculate a derivative based on the structure of a given expression.
3.6 Calculating Higher-Order Derivatives
- I can find the second derivative by differentiating the first derivative, and repeat the process to find higher-order derivatives.
- I can recognize and use various notations for higher-order derivatives, such as f''(x), y'', d^2y/dx^2, and f^(n)(x).
4.1 Interpreting the Meaning of the Derivative in Context
4.2 Straight-Line Motion: Connecting Position, Velocity, and Acceleration
4.3 Rates of Change in Applied Contexts Other Than Motion
4.4 Introduction to Related Rates
4.5 Solving Related Rates Problems
- I can solve related rates problems by finding the rate at which one quantity changes by relating it to other quantities whose rates of change are known, and I can explain the meaning of my solution in context.
4.6 Approximating Values of a Function Using Local Linearity and Linearization
4.7 Using L'Hospital's Rule for Determining Limits of Indeterminate Forms
5.1 Using the Mean Value Theorem
- I can state and verify the conditions of the Mean Value Theorem (continuity on [a, b] and differentiability on (a, b)) to guarantee a point where the instantaneous rate of change equals the average rate of change.
5.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points
- I can apply the Extreme Value Theorem to guarantee a minimum and maximum value for a continuous function on a closed interval.
- I can find critical points of a function by determining where its first derivative is zero or undefined.
- I can explain that all local extrema occur at critical points, but not all critical points result in local extrema.
5.3 Determining Intervals on Which a Function Is Increasing or Decreasing
- I can use the first derivative of a function to determine intervals where the function is increasing or decreasing.
5.4 Using the First Derivative Test to Determine Relative (Local) Extrema
- I can apply the first derivative test to determine and justify relative (local) extrema.
5.5 Using the Candidates Test to Determine Absolute (Global) Extrema
- I can use the candidates test to find absolute (global) extrema on a closed interval by evaluating the function at critical points and endpoints.
5.6 Determining Concavity of Functions over Their Domains
- I can determine if the graph of a function is concave up or down on an open interval by checking if its derivative is increasing or decreasing.
- I can use the second derivative of a function to find intervals of upward or downward concavity.
5.7 Using the Second Derivative Test to Determine Extrema
5.8 Sketching Graphs of Functions and Their Derivatives
5.9 Connecting a Function, Its First Derivative, and Its Second Derivative
5.10 Introduction to Optimization Problems
5.11 Solving Optimization Problems
5.12 Exploring Behaviors of Implicit Relations
6.1 Exploring Accumulations of Change
- I can evaluate accumulation of change by using geometry in some cases.
6.2 Approximating Areas with Riemann Sums
- I can approximate definite integrals for functions represented graphically, numerically, analytically, and verbally.
- I can approximate definite integrals using left, right, and midpoint Riemann sums, as well as trapezoidal sums, with uniform or nonuniform partitions.
6.3 Riemann Sums, Summation Notation, and Definite Integral Notation
6.4 The Fundamental Theorem of Calculus and Accumulation Functions
- I can apply the Fundamental Theorem of Calculus to differentiate an accumulation function.
6.5 Interpreting the Behavior of Accumulation Functions Involving Area
- I can represent accumulation functions using definite integrals and interpret their behavior from various representations of the integrand.
6.6 Applying Properties of Definite Integrals
- I can apply properties of definite integrals, including scalar multiplication, sum of functions, reversal of limits, and adjacent intervals.
- I can evaluate definite integrals for functions with removable or jump discontinuities.
6.7 The Fundamental Theorem of Calculus and Definite Integrals
- I can recognize that the function defined by the integral of f(t) from a to x is an antiderivative of f.
- I can apply the Fundamental Theorem of Calculus to evaluate the definite integral of a continuous function over an interval [a, b].
6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation
- I can use indefinite integral notation and remember to include the constant of integration.
- I can apply basic differentiation rules in reverse to find antiderivatives.
- I understand that not all functions have closed-form antiderivatives.
6.9 Integrating Using Substitution
6.10 Integrating Functions Using Long Division and Completing the Square
- I can determine indefinite integrals and evaluate definite integrals for integrands requiring rearrangements into equivalent forms.
- I can use techniques such as long division and completing the square to rearrange integrands into equivalent forms to find antiderivatives.
6.11 Integrating Using Integration by Parts
- I can determine indefinite integrals and evaluate definite integrals using the integration by parts technique.
6.12 Integrating Using Linear Partial Fractions
- I can determine indefinite and evaluate definite integrals for integrands requiring integration by linear partial fractions.
- I can decompose rational functions into sums of ratios of linear, nonrepeating factors to apply basic integration techniques.
6.13 Evaluating Improper Integrals
- I can identify an improper integral by checking if it has infinite limits or an unbounded integrand.
- I can evaluate improper integrals by setting up and calculating the limits of definite integrals.
7.1 Modeling Situations with Differential Equations
7.2 Verifying Solutions for Differential Equations
7.3 Sketching Slope Fields
7.4 Reasoning Using Slope Fields
7.5 Approximating Solutions Using Euler's Method
7.6 Finding General Solutions Using Separation of Variables
- I can use antidifferentiation to find the general solution to a differential equation.
7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables
- I can recognize and use an accumulation function as a particular solution to a differential equation.
7.8 Exponential Models with Differential Equations
7.9 Logistic Models with Differential Equations
- I can recognize the logistic differential equation that arises from a proportional rate of change.
- I can interpret the logistic differential equation and its initial conditions without needing to solve it.
- I can determine the carrying capacity (limiting value) of a logistic differential equation as the independent variable approaches infinity.
- I can determine the value of the dependent variable at the point when it is changing fastest.
8.1 Finding the Average Value of a Function on an Interval
- I can apply the formula 1/(b-a) ∫ f(x)dx to find the average value of a continuous function.
8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals
- I can use the definite integral of velocity to find a particle's displacement and the definite integral of speed to find its total distance traveled over an interval of time.
8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts
- I can interpret the meaning of a definite integral in the context of an accumulation problem.
- I can determine the net change of a quantity using definite integrals in applied contexts.
8.4 Finding the Area Between Curves Expressed as Functions of x
8.5 Finding the Area Between Curves Expressed as Functions of y
- I can calculate the area of a region in the plane by setting up and evaluating a definite integral with respect to y.
8.6 Finding the Area Between Curves That Intersect at More Than Two Points
- I can calculate the area between curves that intersect at more than two points by using a sum of definite integrals or the absolute value of their difference.
8.7 Volumes with Cross Sections: Squares and Rectangles
8.8 Volumes with Cross Sections: Triangles and Semicircles
- I can find the volume of a solid with triangular cross sections using definite integrals and the area formula for a triangle.
- I can find the volume of a solid with semicircular and other geometrically defined cross sections using definite integrals and the appropriate area formulas.
8.9 Volume with Disc Method: Revolving Around the x- or y-Axis
8.10 Volume with Disc Method: Revolving Around Other Axes
8.11 Volume with Washer Method: Revolving Around the x- or y-Axis
8.12 Volume with Washer Method: Revolving Around Other Axes
- I can find the volume of a solid of revolution around any horizontal or vertical line using the washer method.
8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled
9.1 Defining and Differentiating Parametric Equations
- I can calculate the derivative dy/dx of a parametrically defined curve by dividing dy/dt by dx/dt, and use it to find the slope of the tangent line.
9.2 Second Derivatives of Parametric Equations
- I can calculate the second derivative of a parametric function by dividing the derivative of dy/dx with respect to t by dx/dt.
9.3 Finding Arc Lengths of Curves Given by Parametric Equations
9.4 Defining and Differentiating Vector-Valued Functions
9.5 Integrating Vector-Valued Functions
9.6 Solving Motion Problems Using Parametric and Vector-Valued Functions
- I can use the definite integral of the velocity vector to find a particle's displacement and position, and the definite integral of speed to find the total distance traveled over an interval of time.
9.7 Defining Polar Coordinates and Differentiating in Polar Form
9.8 Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve
9.9 Finding the Area of the Region Bounded by Two Polar Curves
10.1 Defining Convergent and Divergent Infinite Series
10.2 Working with Geometric Series
- I can identify a geometric series by verifying that there is a constant ratio between successive terms.
- I can determine whether a geometric series converges by checking if |r| < 1, and I can calculate its sum using the formula a/(1-r).
10.3 The nth Term Test for Divergence
- I can use the nth term test to determine if a series diverges.
10.4 Integral Test for Convergence
10.5 Harmonic Series and p-Series
- I can identify the harmonic series, the alternating harmonic series, and p-series, and use them to determine whether a series converges or diverges.
10.6 Comparison Tests for Convergence
10.7 Alternating Series Test for Convergence
10.8 Ratio Test for Convergence
10.9 Determining Absolute or Conditional Convergence
- I know that rearranging or regrouping the terms of an absolutely convergent series does not change its value.
10.10 Alternating Series Error Bound
10.11 Finding Taylor Polynomial Approximations of Functions
- I can explain that as the degree of a Taylor polynomial increases, it approaches the original function over some interval.
- I can use a Taylor polynomial centered at x = a to approximate function values near x = a.
10.12 Lagrange Error Bound
10.13 Radius and Interval of Convergence of Power Series
- I can recognize the standard form of a power series.
- I know that a power series converges either at a single point or on an interval of convergence.
- I can test both endpoints of an open interval to determine the full interval of convergence for a power series.
- I know that term-by-term differentiation or integration of a power series does not change its radius of convergence.
10.14 Finding Taylor or Maclaurin Series for a Function
10.15 Representing Functions as Power Series
- I can derive a power series for a given function by applying operations like term-by-term differentiation, integration, algebraic processes, or substitutions to a known series.
How should you use these resources?
Revise
Work through your course unit by unit, in the order it is taught, so nothing gets missed.
Test yourself
Check each topic against what you should be able to do, and practise on exam-style questions as they are added.
Improve
Go back to what you could not do, or work with a tutor to close the gaps before your exams.
AP Calculus BC: common questions
What units are in AP Calculus BC?
AP Calculus BC has 10 units: Unit 1: Limits and Continuity; Unit 2: Differentiation: Definition and Fundamental Properties; Unit 3: Differentiation: Composite, Implicit, and Inverse Functions; Unit 4: Contextual Applications of Differentiation; Unit 5: Analytical Applications of Differentiation; Unit 6: Integration and Accumulation of Change; Unit 7: Differential Equations; Unit 8: Applications of Integration; Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions; Unit 10: Infinite Sequences and Series.
How should I revise AP Calculus BC?
Work through one unit at a time in order, since later units build on earlier ones. For each topic, check you can do everything on its list before moving on, and go back to the ones you could not.
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