AP Calculus BC Syllabus Contents

Curricular Requirements 2
AP Calculus BC Syllabus .. 3
Primary Textbook .. 3
Supplemental Materials . 3
Graphing Calculators . 3
Unit 1: Limits and Continuity ... 3
Unit 2: Derivatives .... 4
Unit 3: Applications of Derivatives, with Mean Value Theorem & Modeling 6
Unit 4: Further Applications of Derivatives ..7
Unit 5: Integrals and Applications 8
Unit 6: Derivatives and Integrals of Exponential and Logarithmic Functions 10
Unit 7: Area, Volume, and Arc Length ...11
Unit 8: Sequences and Series ..11
Unit 9: Parametric and Polar Functions ..14
AP Calculus BC Exam Preparation .15

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 Codes for Curricular Requirements
CR1a The course is structured around the enduring understandings within Big Idea 1: Limits.
See page 3

CR1b The course is structured around the enduring understandings within Big Idea 2: Derivatives.
See pages 4-9, 14

CR1c The course is structured around the enduring understandings within Big Idea 3: Integrals and the
Fundamental Theorem of Calculus.
See pages 7, 9, 11, 14

CR1d The course is structured around the enduring understandings within Big Idea 4: Series.
See page 11

CR2a The course provides opportunities for students to reason with definitions and theorems.
See pages 4, 5, 9, 10, 12, 13

CR2b The course provides opportunities for students to connect concepts and processes.
See pages 5, 7, 9-13

CR2c The course provides opportunities for students to implement algebraic/computational processes.
See pages 7, 9, 10, 12, 13

CR2d The course provides opportunities for students to engage with graphical, numerical, analytical, and verbal
representations and demonstrate connections among them.
See pages 4, 6-10

CR2e The course provides opportunities for students to build notational fluency.
See pages 5, 9-11

CR2f The course provides opportunities for students to communicate mathematical ideas in words, both spoken
and written.
See pages 4, 5, 7, 10

CR3a Students have access to graphing calculators.

See page 3

CR3b Students have opportunities to use calculators to solve problems.

See pages 7, 9

CR3c Students have opportunities to use a graphing calculator to explore and interpret calculus concepts.
See pages 3, 4, 5, 11

CR4 Students and teachers have access to a college-level textbook.

See pages 3, 4, 6, 7,9

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 AP Calculus BC Syllabus

PRIMARY TEXTBOOK [CR4]

Larson, R., and Bruce H. Edwards. Calculus of a Single Variable. 10th ed. Boston: Brooks/Cole, 2006.

SUPPLEMENTAL MATERIAL
Davidson Next. AP Calculus: Challenging Concepts from AB & BC.
https://edge.edx.org/courses/DavidsonNext/Cal_SMin/3T2014. (MOOC)

Stu Schwartz. Master Math Mentor. http://www.mastermathmentor.com/ABStepbyStep.ashx.

AP Central Resources

GRAPHING CALCULATORS [CR3a]

Each student is required to have a graphing calculator that is approved for the AP Calculus exam (TI-84,
TI-89). If students are unable to purchases one, our school has some that can be signed out for the year.
We also have class sets of TI-84 Plus CEs for use at school.

CONTENT
Highlights of each unit follows, including an overview of the content covered, sections from the
textbook where problems sets may be assigned, and examples of problems that might be found in class
examples, homework, and assessments. Activities are listed as calculator active (CA) or inactive (IA).

Unit 1: Limits and Continuity [CR1a] (8 days)

Understand the limit definition and symbolic representation of a limit.
From a given graph, evaluate a limit at a point. Include one-sided limits.
Use numeric and graphic information, with and without a graphing calculator, to estimate
a limit. Include one-sided limits, limits at , and functions that are unbounded,
oscillating, and contain different one-sided limits.
Use limit theorems to evaluate limits analytically, including limits of piecewise functions.
Evaluate limits analytically. Include finding limits of sums, differences, products,
quotients, and composite functions that use limit theorems and algebraic rules.
Evaluate the limit of a function involving algebraic manipulation, special trig functions,
and the squeeze theorem.
Use one-sided limits to recognize and find vertical asymptotes.
Use limits as x to recognize and find horizontal asymptotes.
Sketch a function given a set of limit conditions.
Use limits to describe asymptotic and unbounded behavior.

3
 Explore continuity visually, and understand and apply the definition of continuity.
Explore different types of discontinuity (removable, jump, infinite) using a graphing
calculator. [CR3c]
Determine types of discontinuity analytically. Include polynomial, rational, power,
exponential, logarithmic, and exponential functions.

Textbook Section: [CR4]

1.2 Finding Limits Graphically & Numerically
1.3 Evaluating Limits Analytically
1.4 Continuity and One-Sided Limits
1.5 Infinite Limits
3.5 Limits at Infinity

Sample Activity: Students work on these problems in pairs and will present their solutions
orally to the class and justify their reasoning. (IA) [CR2f]

1) Let a be a non-zero value and

a) Is g (t ) defined at t a ? If so, what is its value? Use the definition of a function to

show the analysis that leads to your conclusion.
b) Does the limit exists at t a ? If so, what is its value? Use the definition of a limit to
show the analysis that leads to your conclusion. [CR2a]
c) For what values of t will g (t ) be continuous for all reals? Use the definition of
continuity to show the analysis that leads to your conclusion. [CR2a]
1 1
2) Graph the functions y sin and y x sin on your calculator. Use both graphs and
x x
the table feature to explore whether these functions are continuous at x = 0.
[CR2d] [CR3c] [CR2f]

Unit 2: Derivatives [CR1b] (10 days)

Visually understand the difference between average rate of change and instantaneous rate
of change.
Recognize and use difference quotients when evaluating average rate of change, average
velocity, average acceleration, and approximation of slope or derivative.
Make connections between the limit of a difference quotient and rate of change at a point.

4
 Recognize the use of the limit of difference quotients. Use the limit of a difference
quotient when evaluating rate of change, slope of a tangent line, velocity, acceleration, or
derivative value at a point.
Define a derivative using limits. Understand the multiple notations for representing
derivatives.
Find equations of tangent and normal lines.
Use linear approximation to estimate a functions value at a point.
Find derivatives by using the power, product, quotient, and chain rules.
Use implicit differentiation to find dy/dx and the location of vertical/horizontal tangents.
Use implicit differentiation to find d2y/dx2 in terms of x and y.
Use the graphing calculator to find the derivative of a function at a given point. [CR3c]

Textbook Sections [CR4]

2.1 The Derivative and the Tangent Line Problem
2.2 Basic Differentiation Rules and Rates of Change
2.3 Product and Quotient Rules and Higher-Order Derivatives
2.4 The Chain Rule
2.5 Implicit Differentiation

Sample Activity: Students collaborate with a partner to assess their understanding of derivative
rules.

[CR1b], [CR2a], [CR2c], [CR2e], [CR2f]

5
Unit 3: Derivative Applications and the Mean Value Theorem [CR1b] (8 days)
Define and apply the Intermediate Value Theorem, Extreme Value Theorem, and Mean
Value Theorem. [CR2a]
Explore the connection between continuity and differentiability. [CR2b]
Determine differentiability of a variety of functions using limits.
Using a variety of notations, find higher-order derivatives.
Use the first derivative to determine critical points, horizontal and vertical tangent lines,
intervals of increase or decrease, and relative extrema.
Find absolute extrema.
Use the second derivative to determine intervals of upward/downward concavity and
points of inflection.
Given various representations of f ' or f ' ' (graph, table, formula), state the relationship
these functions have to the function f and sketch a possible graph of f .
Use the graphing calculator to graph f ' or f ' ' to find critical points, horizontal and
vertical tangent lines, intervals of increase/decrease, relative extrema, concavity, and
points of inflection.
Use the graphing calculator to confirm characteristics (e.g. concavity) of functions.
Explore and apply the second derivative test for relative extrema.

Textbook Sections: [CR4]

3.1 Relative Extrema and Critical Numbers
3.2 Rolles Theorem and The Mean Value Theorem
3.3 Increasing and Decreasing Functions and the First Derivative Test
3.4 Concavity and the Second Derivative Test
3.6 A Summary of Curve Sketching

Sample Activities:
1. Let f be a function that is continuous and differentiable for all real numbers. The table
below gives the value of f and its derivative f ' for selected point x in the closed interval
1.5 x 1.5 .
x -1.5 -1.0 -0.5 0 0.5 1.0 1.5
f (x) -1 -4 -6 -7 -6 -4 -1
f ' ( x) -7 -5 -3 0 3 5 7

a) Write an equation of the line tangent to the graph of f at the point where x = 1.
Use this line to approximate f (1.2) .
b) Find a number r having the property that there must exist a value c with
0 c 0.5 and f (c) r . Give a reason for your answer.

6
 2. Consider the curve defined by 2 y 3 6x 2 y 12x 2 6 y 1. Find dy/dx, then write and
equation of each horizontal tangent line to the curve.

3.

[CR1b], [CR2a], [CR2b], [CR2c], [CR2d], [CR2f], [CR3b]

Unit 4: Further Applications of Derivatives [CR1b] (9 days)

Solve rectilinear motion problems involving position, speed, velocity, and acceleration.
Identify variables, constants, and rates of change, and represent these symbolically in
related rates word problems. Find an unknown rate of change relating the given
information.
Use derivatives to solve optimization problems.
Use linear approximation and differentials to estimate the value of a function at a point.
Use differentials to discuss propagated error.
Use LHopitals Rule to evaluate limits in indeterminate form.

Textbook Sections: [CR4]

2.6 Related Rates
3.7 Optimization
3.9 Differentials
8.7 Indeterminate Forms and LHopitals Rule
Supplement: Motion Module

Sample Activities:
1. Groups of 3-4 students create a graphic organizer to teach their classmates how to solve a
related rates or optimization problem like the following:

Sand is falling off a conveyor onto a conical pile at the rate of 20 ft3/min. The diameter
of the base of the cone is four times its height. At what rate is the height of the pile
changing when it is 8 feet tall?

7

2. For time t 0 , let r (t ) 120 1 e 10t represent the speed, in kilometers per hour, at
2

which a car travels along a straight road. The number of liters of gas used by the car to

travel x kilometers is modeled by g ( x) 0.05x 1 e x / 2 . Use your calculator to find the
rate of change with respect to time of the number of liters of gas used by the car when
t = 2 hours if the car traveled 206.370 km in the first 2 hours. Indicate units of measure.

[CR3b], [CR2b], [CR2d]

Unit 5: Integrals and Applications [CR1c] (12 days)

Translate the symbolic representation of the definite integral to a graphical
representation.
Make a connection between definite integrals and area. Calculate the definite integrals of
linear functions over a given interval using geometry.
b b
Formulate and use properties of definite integrals. Include k f ( x) dx k f ( x) dx ,
a a
b b b b a

f ( x) g( x)dx f ( x) dx g( x) dx , f ( x) dx f ( x) dx , and
a a a a b
b c b

f ( x) dx f ( x) dx f ( x) dx for a c b .
a a c

Use the graphing calculator to explore Riemann sums with different partition sizes on
non-linear functions. Define the definite integral and the limit of a Riemann sum.
For functions represented graphically, numerically, algebraically, and verbally, use left,
right, and midpoint Riemann sums with different partition sizes to approximate definite
integrals.
For functions represented graphically, numerically, algebraically, and verbally, use a
trapezoidal sum with different partition sizes to approximate definite integrals.
Recognize a limit of a Riemann sum as the definite integral and translate to its symbolic
form.
Define and use the process of antidifferentiation to find the expression of an indefinite
integral. Develop techniques for finding antiderivatives by algebraic manipulation and
substitution of variables.
Understand and use the Fundamental Theorem of Calculus to evaluate definite integrals.
Use the graphing calculator to perform numerical integration.
Apply definite integrals to find the average value of a function.
Use initial conditions to find the value of the constant of integration in indefinite
integrals.
Given a graphical, numerical, analytical, or verbal representation of a function defined as
x
g ( x) f (t ) dt , form a conclusion about the function g (x) .
a

In rectilinear motion over an interval of time, determine displacement, distance, or speed.

8
 In rectilinear motion with initial condition given, use antiderivatives to find position or
velocity.
Find the derivative of a functions inverse, if the derivative exists.
[CR2a], [CR2b], [CR2c], [CR2d], [CR2e], [CR2f]

Textbook Sections [CR4]

4.1 Antiderivatives and Indefinite Ingegration
4.2 Area
4.3 Riemann Sums and Definite Integrals
4.4 The Fundamental Theorem of Calculus
4.5 Integration by Substitution
4.6 Numerical Integration
5.3 Inverse Functions

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Unit 6: Logarithmic, Exponential, and Other Transcendental Functions (28 days)
[CR1b], [CR1c]
Use the graphing calculator to draw slope fields and a solution curve.
Use slope fields to predict the behavior of the solution to first order differential equations.
Use Eulers Method to approximate the particular solution of a differential equation.
Solve differential equations by separation of variables. Support with slope fields and
verify with derivatives.
Solve separable differential equations with an initial condition to show a particular
solution; consider domain restrictions.
Understand continuity and differentiability when working with logarithmic and
exponential functions.
Apply derivative rules to find derivatives and solve applications involving logarithmic
and exponential functions.
Apply antiderivative rules involving logarithmic and exponential functions.
Solve application problems logarithmic and exponential functions involving
antiderivatives, integrations and definite integrals
Evaluate definite integrals by changing the limits of integration.
Find the derivative and antiderivative of inverse trigonometric functions.
Use antidifferentiation to solve application problems involving exponential growth.
Analyze the logistics curve to obtain information about growth, rate of growth, and
carrying capacity.
Use integrations by parts to find an antiderivative and to evaluate a definite integral.
Use partial fraction decomposition to find an antiderivative and to evaluate a definite
integral.
Define an improper integral and determine its value, if it exists, using limits of definite
integrals
[CR2a], [CR2b], [CR2c], [CR2d], [CR2e], [CR2f]

Textbook Sections [CR4]

5.1 The Natural Logarithmic Function: Differentiation
5.2 The Natural Logarithmic Function: Integration
5.4 Exponential Functions: Differentiation and Integration
5.5 Bases Other than e and Applications
5.6 Inverse Trigonometric Functions: Differentiation
5.7 Inverse Trigonometric Functions: Integration
6.1 Slope Fields and Eulers Method
6.2 Differential Equations: Growth and Decay
6.3 Separation of Variables and the Logistic Equation
6.4 First-Order Linear Differential Equations
8.2 Integration by Parts
8.5 Partial Fractions

10
 Sample Activity:

Unit 7: Area and Volume (8 days) [CR1c]

Determine the area of regions using definite integrals resulting from vertical or horizontal
slicing.
Use definite integrals to compute volume of solids of revolution around horizontal or
vertical lines using disks and washers.
Use definite integrals to compute volumes of solids having known cross sections.
Use definite integrals to find arc length.
[CR2b]

Textbook Sections [CR4]

7.1 Area of a Region Between Two Curves
7.2 Volume: The Disk Method
7.3 Volume: The Shell Method
7.4 Arc Length and Surfaces of Revolution

Unit 8: Sequences and Series (25 days) [CR1d]

Define a sequence using proper symbolic notation.
Explore a variety of sequences, including alternating, geometric, and arithmetic.

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 Use limit properties when determining convergence or divergence of a sequence.
Explore partial sums using a calculator. [CR3c]
Define infinite series. Understand that an infinite series converges to a number if and
only if the sequence of its partial sums exists and equals that number.
Use a graphing calculator to show partial sums for a telescoping series and its
convergence. Determine the value of a telescoping series. Use partial fractions to rewrite
certain rational series as telescoping series. [CR3c]
Determine whether a geometric series converges or diverges.

a r
a
Find the sum of an infinite geometric series: If r 1 , then n
.
n0 1 r
Apply the nth term test for divergence.
Identify p-series and use its test for convergence or divergence.
Apply the integral test to determine convergence or divergence.
Use the direct comparison test and the limit comparison test to determine convergence or
divergence.
Use the alternating series test to determine convergence or divergence.
Recognize the harmonic series and the alternating harmonic series. Determine whether
they converge or diverge.
Use the error bound for convergent alternating series to estimate how close a partial sum
is to the value of the infinite series.
Understand the difference between conditional and absolute convergence of a series.
Apply the ratio test and the nth root test to determine convergence or divergence.
Determine which test for convergence is most appropriate for a series, and then apply it
to determine convergence or divergence.
Taylor polynomials. [CR1d: Taylor polynomials]
o Construct Taylor polynomials for a variety of functions centered at x = a.
o Define a Maclaurin polynomial to construct a Taylor polynomial centered at
x = 0.
o Use the Taylor polynomial centered at x = a to approximate a functions value
near x = a.
o Use the Langrange error bound to bound the error of a Taylor polynomial
approximation to a function on a given interval.
1
Recognize that the Maclaurin series for is a geometric series.
1 x
Power Series [CR1d: power series]
o Define a power series and create Maclaurin series for sin x , cos x , and e x . Use
these series to construct Maclaurin series for other functions. Include algebraic
processes, substitutions, properties of geometric series, and differentiation or
antidifferentiation to construct.
o Determine the interval of convergence of a power series.
o Understand if a power series has a positive raius of convergence, then the power
series is a Taylor series of the function to which it converges over the open
interval.

12
 o Recognize that the radius of convergence does not change when a power series is
differentiated or integrated term by term.
[CR2a], [CR2b], [CR2c]

Textbook Sections [CR4]

9.1 Sequences
9.2 Series and Convergence
9.3 The Integral Test and p-Series
9.4 Comparison of Series
9.5 Alternating Series
9.6 The Ratio and Root Tests
9.7 Taylor Polynomials and Approximations
9.8 Power Series
9.9 Representation of Functions by Power Series
9.10 Taylor and Maclaurin Series

13
[CR2a], [CR2b], [CR2c]

Unit 9: Parametric and Polar Functions (12 days) [CR1b], [CR1c]

Find the derivative of vector-valued functions and parametric functions and graph those
functions.
Use derivatives to determine velocity, speed, and acceleration for a particle moving along
curves given by parametric or vector-valued functions.
Solve problems involving polar coordinates, their graphs, and calculus work with
derivatives, area, arc length, and surface area.
[CR2b], [CR2c]

Textbook Sections [CR4]

10.1 Conics and Calculus
10.2 Plane Curves and Parametric Equations
10.3 Parametric Equations and Calculus
10.4 Polar Coordinates and Polar Graphs
10.5 Area and Arc Length in Polar Coordinates

14
 Sample Activity

AP Calculus BC Exam Preparations (approximately 15 days)

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