Damien High School
Mathematics & Computer Science Department Curriculum Map
Course Title
Prerequisites
CSU/UC Approval
Length of Course
AP Calculus BC
“B” or better in AP Calculus AB and a score of 3 or higher on the AP Calculus AB exam
Yes – Category D
Year
Brief Course Description
This course will include a review of functions, an introduction to limits and continuity, derivatives and their applications, integrals and their applications,
and an introduction to differential equations. There is an emphasis on conceptual understanding and working with functions represented graphically,
numerically, analytically, and verbally.
Assigned Textbook(s)
Supplemental Material(s)
Single Variable Calculus, Early Transcendentals, 6th ed., Stewart
Change and Motion : Calculus Made Easy (DVD Lecture Series)
TI-89 Graphing Calculator
Common Assessments Utilized
Common Final each semester
Homework
Quizzes
Group Work
Tests
ISOs Addressed
Be academically prepared for a higher education …
Exhibit community and global awareness …
Overview of Course / Skill Outcomes
This section serves as a precursor to the Curriculum Map and, as such, should briefly describe the various units (major content chunks) that comprise the course as well as the skills / techniques
necessary to be successful in the course.
Major Content Outcomes
I. Functions, Graphs, and Limits
A. Analysis of graphs: With the aid of technology, graphs of functions are often easy to produce.
The emphasis is on the interplay between the geometric and analytic information and on the use of
calculus both to predict and to explain the observed local and global behavior of a function.
B. Limits of functions
1.
An intuitive understanding of the limiting process
2.
Calculating limits using algebra
3.
Estimating limits from graphs or tables of data
C. Asymptotic and unbounded behavior
1.
Understanding asymptotes in terms of graphical behavior
2.
Describing asymptotic behavior in terms of limits involving infinity
3.
Comparing relative magnitude of functions and their rates of change
D. Continuity as a property of functions
1.
An intuitive understanding of continuity.
2.
Understanding continuity in terms of limits
3.
Geometric understanding of graphs of continuous functions
II. Derivatives
Major Skill Outcomes
• Students should be able to work with functions represented in a variety of ways:
graphical, numerical, analytical, or verbal. They should understand the connections among these
representations.
• Students should understand the meaning of the derivative in terms of a rate of change and local linear
approximation, and should be able to use derivatives to solve a variety of problems.
• Students should understand the meaning of the definite integral both as a limit of Riemann sums and as
the net accumulation of change, and should be able to use integrals to solve a variety of problems.
• Students should understand the relationship between the derivative and the definite integral as expressed
in both parts of the Fundamental Theorem of Calculus.
• Students should be able to communicate mathematics and explain solutions to problems both verbally and
in written sentences.
• Students should be able to model a written description of a physical situation with a function, a differential
equation, or an integral.
• Students should be able to use technology to help solve problems, experiment, interpret results, and
support conclusions.
• Students should be able to determine the reasonableness of solutions, including
sign, size, relative accuracy, and units of measurement.
• Students should develop an appreciation of calculus as a coherent body of knowledge and as a human
accomplishment.
Unit 1
Students will understand and be able to determine the limit of a function both numerically and graphically
Students will understand and be able to calculate limits using the limit laws
Students will understand and be able to determine the continuity of a function
Students will understand and be able to determine limits at infinity.
A. Concept of the derivative
1. Derivative presented graphically, numerically, and analytically
2. Derivative interpreted as an instantaneous rate of change
3. Derivative defined as the limit of the difference quotient
4. Relationship between differentiability and continuity
B. Derivative at a point
1. Slope of a curve at a point.
2. Tangent line to a curve at a point and local linear approximation.
3. Instantaneous rate of change as the limit of average rate of change
4. Approximate rate of change from graphs and tables of values
C. Derivative as a function
Unit 2
Students will understand and be able to determine average rates of change on an interval
Students will understand the tangent line problem and be able to calculate rates of change using the
difference quotient
Students will understand and be able to determine the derivative of a function and view the result as a slope.
Unit 3
Students will be able to determine derivatives of polynomials
Students will be able to use the product rule and quotient rule to determine the derivative of a function
Students will be able to determine derivatives of trigonometric functions
Students will be able to us the chain rule to determine the derivative of a function
Students will understand and be able to apply implicit differentiation to find the derivative of functions
defined implicitly for a given variable.
Students will be able to determine derivatives of inverse functions, including exponential, logarithmic, and
inverse trigonometric functions
1. Corresponding characteristics of graphs of f and its derivative
2. Relationship between the increasing and decreasing behavior of f and the sign of the derivative
Unit 4
3. The Mean Value Theorem and its geometric interpretation
4. Equations involving derivatives. Verbal descriptions are translated into equations involving
derivatives and vice versa.
D. Second derivatives
1. Corresponding characteristics of the graphs of f, the 1 st derivative, and the 2nd derivative
2. Relationship between the concavity of f and the sign of the 2nd derivative
3. Points of inflection as places where concavity changes
E. Applications of derivatives
1. Analysis of curves, including the notions of monotonicity and concavity
2. Analysis of planar curves given in parametric form, polar form, and vector form, including
velocity and acceleration
3. Optimization, both absolute and relative extrema
4. Modeling rates of change, including related rate problems
5. Use of implicit differentiation to find the derivative of an inverse function
6. Interpretation of the derivative as a rate of change in varied applied contexts, including velocity,
speed, and acceleration
7. Geometric interpretation of differential equations via slope fields and the relationship between
slope fields and solution curves for differential equations
8. L’Hopital’s Rule, including its use in determining limits and convergence of improper integrals
and series
F. Computation of derivatives
1. Knowledge of derivatives of basic functions, including power, exponential, logarithmic,
trigonometric, and inverse trigonometric functions
2. Derivative rules for sums, products, and quotients of functions
3. Chain rule and implicit differentiation
III. Integrals
A. Interpretations and properties of definite integrals
Students will be able to demonstrate an understanding of differentiation to solve application problems
involving rates of change in the sciences
Students will be able to demonstrate an understanding of differentiation to solve application problems
involving related rates
Students will be able to find the linear approximation of a function and use differentials to approximate
function values
Students will be able to determine maximum and minimum values of a function
Students will be able to explain how the Mean Value Theorem applies to various situations
Students will be able to determine how derivatives affect the shape of a graph
Students will be able to demonstrate an understanding of differentiation to solve application problems
involving optimization
Unit 5
Students will be able to find an antiderivative of a function and use it to find the position of an object using
its velocity.
Students will be able to find an approximation of the area under a curve by using the left endpoint, right
endpoint, midpoint and trapezoidal rules.
Students will be able to describe the integral of a function as the exact area under a curve between two xvalues. Students will then be able to find both indefinite and definite integrals using the properties and
describe the difference between them.
Students will be able to use the Fundamental Theorem of Calculus to find the area under a curve and apply
it to application problems involving net change. Students will then be able to describe the relationship
between a derivative and an integral.
Students will be able to apply various integration techniques including pattern recognition, u-substitution,
integration by parts, and partial fraction decomposition
Students will be able to evaluate the two types of an improper integral: Infinite interval and discontinuous
integrand.
Students will be able to use L’Hospital’s Rule to calculate limits of functions that yield the various types of
indeterminate forms
Students will be able to find the general solution of a separable differential equation and a particular
solution given an initial condition.
Students will be able to solve exponential and logistic differential equations and use them in modeling
Students will be able to construct a slope field for a differential equation and interpret the significance of
various aspects of slope fields.
Unit 6
Students will be able to calculate the area between two curves and describe the significance
Students will be able to calculate the volumes of solids of revolution and describe the significance
Students will be able to calculate the volumes of solids using cross sections perpendicular to an axis and
describe the significance
Students will be able to find the length of a curve on a given interval
1. Definite integral as a limit of Riemann sums
Unit 7
2. Definite integral of the rate of change of a quantity over an interval interpreted as the change of
Students will be able to apply calculus techniques to parametric curves
Students will be able to demonstrate he relationship between rectangular and polar coordinates and
equations
Students will be able to determine the area bounded inside a polar curve
the quantity over the interval
3. Basic properties of definite integrals
B. Applications of integrals: Appropriate integrals are used in a variety of applications to model
physical, biological, or economic situations. Although only a sampling of applications can be
included in any specific course, students should be able to adapt their knowledge and techniques to
solve other similar application problems. Whatever applications are chosen, the emphasis is on
Unit 8
Students will be able to identify the different types of infinite series.
Students will be able to apply the Integral Test to test for convergence and divergence
Students will be able to apply the Comparison Tests to test for convergence and divergence
Students will be able to apply the Alternating Series Test to test for convergence and divergence
using the method of setting up an approximating Riemann sum and representing its limit as a
definite integral. To provide a common foundation, specific applications should include the area of
a region, the volume of a solid with known cross sections, the average value of a function, the
distance traveled by a particle along a line, the length of a curve, and accumulated change from a
rate of change.
C. Fundamental Theorem of Calculus
1. Use of the Fundamental Theorem to evaluate definite integrals
2. Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and
graphical analysis of functions so defined
D. Techniques of integration
1. Antiderivatives following directly from derivatives of basic functions
2. Antiderivatives by pattern recognition and u-substitution
E. Applications of antidifferentiation
1. Finding specific antiderivatives using initial conditions, including applications to motion along a
line
2. Solving separable differential equations and using them in modeling
F. Numerical approximations to definite integrals: Use of Riemann sums and trapezoidal sums
to approximate definite integrals of functions represented algebraically, graphically, and by tables
of values
Students will be able to interpret when a series is Absolutely Convergent using the Ratio and Root Tests
Students will be able to represent a function as a power series and recognize and interpret when a Power
Series is convergent or divergent
Students will be able to find and approximate the Taylor and Maclaurin Series for certain functions and use
them in various applications
AP Calculus BC
Unit 1 – What is the importance of a limit?
Content Outcomes
Students will understand and be able to determine
the limit of a function both numerically and
graphically
Essential Questions
What is a limit?
How is a limit found numerically and graphically?
Key Concepts
Notation of a limit
Definition of one sided limit
Definition of infinite limits
Standards Addressed
California: CA C: 1.0 , CA C: 2.0
Properties of limits
Direct substitution property
One sided limit theorem
Definition of greatest integer function
California: CA C: 1.0 , CA C: 2.0
Definition of continuity at x = c
Properties of continuity
Definition of continuity on an interval
Types of functions that are continuous in their
domains
Intermediate Value Theorem
Types of discontinuities
California: CA C: 1.0 , CA C: 2.0
Definition of limit at infinity
Definition of horizontal asymptote
Meaning of infinite limit at infinity
California: CA C: 1.0 , CA C: 2.0
Common Core:
When does a limit fail to exist?
What is an infinite limit and when do they occur?
Students will understand and be able to calculate
limits using the limit laws
What are the properties of infinite limits?
What are the properties of limits?
What are one sided limits?
Common Core:
What does indeterminate form mean?
Students will understand and be able to determine
the continuity of a function
What techniques can be used to find limits of
functions analytically?
What does it mean for a function to be continuous at
x = c?
How do we prove continuity at a single point?
What are the properties of continuity and why are
they true?
Common Core:
What functions are continuous everywhere in their
domain?
What is the purpose of the intermediate value
theorem?
Students will understand and be able to determine
limits at infinity-
How is a limit at infinity found?
What does a limit at infinity describe?
Common Core:
AP Calculus BC
Unit 2 – What is a tangent line and how does it relate to a function?
Content Outcomes
Students will understand and be able to determine
average rates of change on an interval
Essential Questions
What is a rate of change?
Key Concepts
Formula for average rate of change.
How do you find the average rate of change on a
given interval?
Students will understand the tangent line problem
and be able to calculate rates of change using the
difference quotient
What does the slope of a tangent line tell us?
Graphically, how can we make a secant line
between two points become a tangent line at a given
point?
Standards Addressed
California: CA C: 4.0 , CA C: 7.0
Common Core:
Definition of a secant line
Definition of a tangent line
Definition of derivative at a number x = c
Alternative definition of derivative at x = c
Definition of velocity and acceleration
California: CA C: 4.0 , CA C: 7.0
Formula for the derivative of f, f’(x)
Definition of differentiability
Differentiation notation
Ways a function can be non-differentiable
California: CA C: 4.0 , CA C: 7.0
Common Core:
What does a derivative describe?
How can we find the slope of a tangent line at a
given point using the formal definition of the
derivative?
What is the alternative form of the derivative?
Students will understand and be able to determine
the derivative of a function and view the result as a
slope.
What is the interpretation of a tangent line with
respect to various applications in the sciences?
How is differentiability related to continuity?
Where is a function not differentiable?
How is a higher ordered derivative found and what
are their meanings with respect to position?
AP Calculus BC
Unit 3 – How can derivatives be found for any function?
Common Core:
Content Outcomes
Students will be able to determine derivatives of
polynomials
Essential Questions
What is the power rule?
Key Concepts
Power Rule, Constant Multiple Rule, Sum and
Difference Rule, Derivative of e^x
What is the derivative of a sum, difference, and
constant multiple of functions?
Standards Addressed
California: CA C: 4.0 , CA C: 5.0 , CA C: 6.0 ,
CA C: 7.0
Common Core:
What is the derivative of an exponential function?
How do you write the equation of a tangent line?
Where does a tangent line have a given slope?
Where does a function have a horizontal tangent
line?
Students will be able to use the product rule and
quotient rule to determine the derivative of a
function
What is the product rule and when is it used?
Students will be able to determine derivatives of
trigonometric functions
How are the derivatives of sin x and cos x derived?
Product Rule, Quotient Rule
What is the quotient rule and when is it used?
Derivatives of Trigonometric functions
How can the derivatives of the remaining trig
functions be found using the quotient rule?
Students will be able to us the chain rule to
determine the derivative of a function
What is the chain rule and when should it be used?
How can u-substitution be used to differentiate a
composite function?
Students will understand and be able to apply
implicit differentiation to find the derivative of
functions defined implicitly for a given variable.
Students will be able to determine derivatives of
inverse functions, including exponential,
logarithmic, and inverse trigonometric functions
California: CA C: 4.0 , CA C: 5.0 , CA C: 6.0 ,
CA C: 7.0
When is an equation defined implicitly and
explicitly?
What steps are used to implicitly differentiate a
function?
How do you find the derivative of an inverse
function?
What is the relationship between the slope of the
original function and the slope of the inverse?
Chain Rule
Chain Rule using u-substitution
Derivative of au
Steps for implicit differentiation
Derivative of the inverse of f(x)
Steps for logarithmic differentiation
Common Core:
California: CA C: 4.0 , CA C: 5.0 , CA C: 6.0 ,
CA C: 7.0
Common Core:
California: CA C: 4.0 , CA C: 5.0 , CA C: 6.0 ,
CA C: 7.0
Common Core:
California: CA C: 4.0 , CA C: 5.0 , CA C: 6.0 ,
CA C: 7.0
Common Core:
Derivative of the inverse of f(x)
Steps for logarithmic differentiation
Derivatives of inverse trigonometric functions
Derivatives of logau, and lnu, where u is a function
of x
What is the derivative of au , logau, and lnu, where u
is a function of x?
How do we use logarithms to separate a function so
differentiation is easier (logarithmic
differentiation)?
How do we find the derivative of inverse
trigonometric functions using implicit
differentiation?
AP Calculus BC
Unit 4 – What are derivatives used for and how are they applied to real situations?
California: CA C: 4.0 , CA C: 5.0 , CA C: 6.0 ,
CA C: 7.0
Common Core:
Content Outcomes
Students will be able to demonstrate an
understanding of differentiation to solve application
problems involving rates of change in the sciences
Essential Questions
What is the relationship between position, velocity
and acceleration?
What other rates of change can be found using
derivatives?
Students will be able to demonstrate an
understanding of differentiation to solve application
problems involving related rates
What do we mean by related rates?
Students will be able to find the linear
approximation of a function and use differentials to
approximate function values
How can a linearization of f help to approximate a
function value?
Key Concepts
Formula for average rate of change
Relationship between position, velocity and
acceleration
Formula for the law of natural growth and decay
Standards Addressed
California: CA C: 3.0 , CA C: 8.0 , CA C: 9.0 ,
CA C: 11.0 , CA C: 12.0
Definition of related rates
Strategy for solving related rate problems
California: CA C: 3.0 , CA C: 8.0 , CA C: 9.0 ,
CA C: 11.0 , CA C: 12.0
Common Core:
How do you solve related rate problems?
Common Core:
Formula for linear approximation of f(x) at x = c
Definition and use of differentials
What is the meaning of the differentials dx and dy?
Students will be able to determine maximum and
minimum values of a function
What is a critical point?
What is the difference between absolute and relative
extrema?
California: CA C: 3.0 , CA C: 8.0 , CA C: 9.0 ,
CA C: 11.0 , CA C: 12.0
Common Core:
Definition of absolute maximum, absolute
minimum, relative maximum, relative minimum,
critical number
California: CA C: 3.0 , CA C: 8.0 , CA C: 9.0 ,
CA C: 11.0 , CA C: 12.0
Common Core:
Extreme Value Theorem
Where do extrema occur?
Steps to find absolute extrema on [a,b]
Students will be able to explain how the Mean
Value Theorem applies to various situations
What do the Mean Value Theorem and Rolle’s
Theorem say?
Rolle’s Theorem
Mean Value Theorem
Under what conditions do they apply?
California: CA C: 3.0 , CA C: 8.0 , CA C: 9.0 ,
CA C: 11.0 , CA C: 12.0
Common Core:
What applications does this have in the real world?
Students will be able to determine how derivatives
affect the shape of a graph
When does a function increase? Decrease?
Increasing/Decreasing Test, First Derivative Test,
Concavity Test, Second Derivative Test
Definition of concavity and inflection point
Guidelines for sketching a curve
What is the first derivative test?
California: CA C: 3.0 , CA C: 8.0 , CA C: 9.0 ,
CA C: 11.0 , CA C: 12.0
Common Core:
What is concavity and how does it relate to the first
derivative?
What is the second derivative test?
How does the slope of the graph give the graph of
the derivative?
Students will be able to demonstrate an
understanding of differentiation to solve application
problems involving optimization
How can one interpret f(x) or f’(x) from a graph?
How do we create a function from a word problem?
How do we maximize or minimize a quantity of a
real application using calculus techniques?
AP Calculus BC
California: CA C: 3.0 , CA C: 8.0 , CA C: 9.0 ,
CA C: 11.0 , CA C: 12.0
Common Core:
Unit 5 – How are antiderivatives found and what is their purpose?
Content Outcomes
Students will be able to find an antiderivative of a
function and use it to find the position of an object
using its velocity.
Essential Questions
What is an antiderivative?
Key Concepts
Definition of antiderivative and differential equation
Antidifferentiation formulas
What is a differential equation?
Standards Addressed
California: CA C: 13.0 , CA C: 14.0 , CA C:
15.0 , CA C: 16.0 , CA C: 17.0 , CA C: 21.0 ,
CA C: 27.0
How is a general solution and particular solution to
a differential equation found?
Common Core:
What are the rules for integration of common
functions?
Students will be able to find an approximation of
the area under a curve by using the left endpoint,
right endpoint, midpoint and trapezoidal rules.
What is the graphical meaning of the integral of a
function?
How can the area under a curve, bounded by the xaxis be found using left and right endpoints and
midpoints?
Graphical understanding of finding the sum of areas
of multiple rectanges
Definition of area of a region under a curve
Area formula using left endpoints, right endpoints,
midpoints, and trapezoidal rule
California: CA C: 13.0 , CA C: 14.0 , CA C:
15.0 , CA C: 16.0 , CA C: 17.0 , CA C: 21.0 ,
CA C: 27.0
Definition of definite integral
Integration notation
Interpretation of the definite integral as a net area
Integrability Theorems
Properties of integrals
California: CA C: 13.0 , CA C: 14.0 , CA C:
15.0 , CA C: 16.0 , CA C: 17.0 , CA C: 21.0 ,
CA C: 27.0
Fundamental Theorem of Calculus
Inverse nature of differentiation and integration
Definition of indefinite integral
Table of common indefinite integrals
Net Change Theorem
California: CA C: 13.0 , CA C: 14.0 , CA C:
15.0 , CA C: 16.0 , CA C: 17.0 , CA C: 21.0 ,
CA C: 27.0
Substitution Rule for integration
Integrals of trigonometric functions
Substitution rule for definitie integrals
Rules for integrals of symmetric functions
Rule for integration by parts
California: CA C: 13.0 , CA C: 14.0 , CA C:
15.0 , CA C: 16.0 , CA C: 17.0 , CA C: 21.0 ,
CA C: 27.0
Common Core:
What is the trapezoidal rule?
What happens if we use an infinite number of
rectangles to approximate the area under a curve?
Students will be able to describe the integral of a
function as the exact area under a curve between
two x-values. Students will then be able to find
both indefinite and definite integrals using the
properties and describe the difference between
them.
What is a definite integral?
What is the purpose of finding the norm of a
partition and how is it used to prove that the definite
integral from x = a to x = b describes the exact area
under the curve?
Common Core:
What are the properties of definite integrals?
Students will be able to use the Fundamental
Theorem of Calculus to find the area under a curve
and apply it to application problems involving net
change. Students will then be able to describe the
relationship between a derivative and an integral.
What is the Fundamental Theorem of Calculus and
how is it used to evaluate a definite integral?
What is the mean value theorem for integrals and
what is the meaning of the average value of a
function?
Common Core:
What is the purpose of the 2nd Fundamental
Theorem of Calculus?
What is the Net Change Theorem and what real
world applications does it allow us to solve?
How are total distance and displacement found?
Students will be able to apply various integration
techniques including pattern recognition, usubstitution, integration by parts, and partial
fraction decomposition
What is the substitution rule for integration?
What situations require integration to be done by usubstitution?
What situations require integration by parts to be
done?
Technique of decomposing a fraction
Common Core:
Definition of improper integrals with infinite
intervals
Definition of improper integrals with discontinuous
integrands
Definitions of convergence and divergence
California: CA C: 22.0 , CA C : 23.0
Types of Indeterminate Form
L’Hospital’s Rule
California: CA C: 22.0 , CA C : 23.0
What steps are used to integrate by u-substitution
and integration by parts?
How is a definite integral found by u-substitution
and integration by parts?
Students will be able to evaluate the two types of an
improper integral: Infinite interval and
discontinuous integrand.
How do we find an integral of a rational function by
partial fraction decomposition (linear factors only)?
What makes an integral improper and what are the
two types of improper integrals?
How do we evaluate an improper integrals of both
types?
Common Core:
How do we determine whether an improper integral
is convergent or divergent?
Students will be able to use L’Hospital’s Rule to
calculate limits of functions that yield the various
types of indeterminate forms
Students will be able to find the general solution of
a separable differential equation and a particular
solution given an initial condition.
What is the comparison theorem and how is it used
to determine convergence or divergence?
What are the various types of indeterminate forms?
How can we find limits of functions using
L’Hospital’s Rule?
How are separable differential equations solved and
what does their solution mean?
How do we verify that a general solution is a
solution to a differential equation?
Common Core:
Definition of general solution, particular solution
and initial condition
Form of a separable differential equation
Steps for solving differential equations
Euler’s Method
California: CA C: 13.0 , CA C: 14.0 , CA C:
15.0 , CA C: 16.0 , CA C: 17.0 , CA C: 21.0 ,
CA C: 27.0
The Logistic Model
The Law of Natural Growth
California: : CA C: 13.0 , CA C: 14.0 , CA C:
15.0 , CA C: 16.0 , CA C: 17.0 , CA C: 21.0 ,
CA C: 27.0
Common Core:
How do we find a numerical solution of a
differential equation using Euler’s method?
Students will be able to solve exponential and
logistic differential equations and use them in
modeling
How do we solve a logistic differential equation and
what is its purpose?
How can we derive the law of natural growth and
solve natural growth and decay applications?
Students will be able to construct a slope field for a
differential equation and interpret the significance
of various aspects of slope fields.
What is a slope field and what does the graph
signify?
How do we graph a slope field and it’s
corresponding solution through a particular point?
Common Core:
Definition of a slope field
Construct slope field given differential equation
Draw a solution to a differential equation, given an
initial condition
California: CA C: 13.0 , CA C: 14.0 , CA C:
15.0 , CA C: 16.0 , CA C: 17.0 , CA C: 21.0 ,
CA C: 27.0
Common Core:
AP Calculus BC
Unit 6 – What are the applications of antiderivatives?
Content Outcomes
Students will be able to calculate the area between
two curves and describe the significance
Essential Questions
How do we find the area between 2 non intersecting
or 2 intersecting curves?
Key Concepts
Rule for finding area between curves
Common Core:
How do we determine whether to integrate with
respect to x or y?
Students will be able to calculate the volumes of
solids of revolution and describe the significance
How do we find the volume of a solid generated by
rotating a region about a vertical or horizontal line?
Definition of solids of revolution
Formulas for disk and washer methods of finding
volumes of rotated solids
California: CA C: 16.0
Volume formulas of common solids
Definition of volume using the integral of area
Interpretation of volume as a sum of infinite areas
California: CA C: 16.0
Formula for Arc Length
California: CA C: 16.0
How is area under a curve related to volume of
rotated solids?
Students will be able to calculate the volumes of
solids using cross sections perpendicular to an axis
and describe the significance
When do we use the disk method as opposed to the
washer method for finding volumes of rotated
solids?
What are the functions for area of common shapes
with respect to their base, i.e. equilateral triangle,
isosceles triangle, semicircle, rectangle, square, etc?
Students will be able to find the length of a curve on
a given interval
How do we use these formulas to find the volume of
a solid where cross sections perpendicular to an
axis, for a given region, are a specified shape?
How do we find the length of the arc of a function
on [a , b]?
Standards Addressed
California: CA C: 16.0
Common Core:
Common Core:
Common Core:
How do we determine a function s(x) for the length
of a curve on [a , x]?
AP Calculus BC
Unit 7 – What is the calculus of parametric and polar equations?
Content Outcomes
Essential Questions
Key Concepts
Standards Addressed
Students will be able to apply calculus techniques to
parametric curves
What is a parametric equation and how do we graph
them?
The slope of the tangent line of a parametric curve
The concavity of a parametric curve
Arc length formula for parametric equations
California: CA C 6.0
Definition of Polar Coordinates
Conversion formulas between rectangular and polar
coordinates
Slope of tangent line formula for polar coordinates
California:
Formula for area inside polar curves
California:
How do you find the slope of the tangent line for a
set of parametric equations?
Common Core:
How do you use derivatives of parametric equations
to find the graph’s concavity?
Students will be able to demonstrate he relationship
between rectangular and polar coordinates and
equations
How do you find the arc length of a parametric
curve?
What are polar coordinates and how are they related
to rectangular coordinates?
What are the conversion formulas between
rectangular and polar coordinates?
Common Core:
How do you find the slope of the tangent line for
polar graphs?
Students will be able to determine the area bounded
inside a polar curve
How do you find the area inside a polar curve?
Common Core:
AP Calculus BC
Unit 8 – What are infinite series?
Content Outcomes
Essential Questions
Key Concepts
Standards Addressed
Students will be able to identify the different types
of infinite series.
What are infinite series and how are they considered
convergent or divergent?
What is the difference between a geometric,
telescoping, harmonic and alternating series?
Students will be able to apply the Integral Test to
test for convergence and divergence
How do we recognize the terms of a series as areas
of rectangles & their relationship to improper
integrals?
How and when do we test a series for convergence
and divergence using the integral test?
Definition of Infinite Series
Test for Convergence and Divergence
Definition of Geometric Series
Definition of Telescoping Series
Definition of Alternating Series
California: CA C 22.0 – 26.0
Integral Test
California: CA C 22.0 – 26.0
Common Core:
Common Core:
Students will be able to apply the Comparison
Tests to test for convergence and divergence
How and when do we test a series for convergence
and divergence using the direct comparison test?
Direct Comparison Test
Limit Comparison Test
California: CA C 22.0 – 26.0
Common Core:
How and when do we test a series for convergence
and divergence using the limit comparison test?
California: CA C 22.0 – 26.0
Students will be able to apply the Alternating
Series Test to test for convergence and divergence
How and when do we test a series for convergence
and divergence using the alternate series test?
Definition of an alternating series
Alternating Series Test
Students will be able to interpret when a series is
Absolutely Convergent using the Ratio and Root
Tests
What is the definition of absolute convergence?
Definition of absolute convergence
The Ratio Test
The Root Test
California: CA C 22.0 – 26.0
Definition of a Power Series
Definition of the interval of convergence of a power
series
Differentiation and Integration Theorem of Power
Series
California: CA C 22.0 – 26.0
Definition of Taylor Series
Definition of Maclaurin Series
Maclaurin series of ex , sin x, and cos x
California: CA C 22.0 – 26.0
Common Core:
Students will be able to represent a function as a
power series and recognize and interpret when a
Power Series is convergent or divergent
What are the Ratio and Root Tests and when do we
use them?
What is a power series?
How do you find the interval of convergence for a
power series?
Common Core:
Common Core:
How do we find the radius and interval of
convergence of power series?
Students will be able to find and approximate the
Taylor and Maclaurin Series for certain functions
and use them in various applications
How can we represent a function as a power series?
How do we find the Taylor polynomial
approximation with a graphical demonstration of
convergence?
How do we determine the Maclaurin series and the
general Taylor series centered at x = a?
How can we formally manipulate Taylor series and
apply shortcuts to computing Taylor series,
including substitution, differentiation,
antidifferentiation, and the formation of
new series from known series?
How do we determine the Lagrange error bound for
Taylor polynomials?
Common Core: