Download Advanced Placement Calculus AB

Course Outlines
Name of the course:
Advanced Placement Calculus AB
Course description:
This course is intended for students who have a thorough knowledge of college
preparatory mathematics, including algebra, geometry, trigonometry, analytic
geometry, and elementary functions. AP Calculus AB covers the traditional topics
of Calculus at a rapid pace and in a rigorous fashion. These topics include, but are
not limited to, functions and graphs, limits and continuity, differential calculus, and
integral calculus. Throughout the year, emphasis is placed on interpreting each
new concept graphically, numerically, analytically, and verbally. Assessments
typically include some problems involving a function defined only by a graph or
numerical data. The course is balanced between conceptual understanding,
acquisition of skills, and use of technology.
Graphing calculators play a key role as a tool of analysis throughout the course.
Each student has a TI-89 graphing calculator to use in class and at home. Our
school will lend one to students who cannot afford to purchase one. Graphing
calculators are used extensively during class to discover and reinforce concepts.
Assessments frequently have calculator and non-calculator sections that are
designed to ensure students are able to both complete algebraic computations of
calculus problems on their own, as well as make effective and efficient use of the
available technology.
Throughout the year, released open response questions from previous AP exams
are used extensively. They have several functions within the framework of the
course. They illustrate different approaches to a concept, and are given as
classwork after a new concept has been introduced. The questions can be used to
reinforce concepts, and are used as homework assignments that are graded using
the published scoring guides. They are also used as assessment tools when part of
a unit test. The problems provide excellent opportunities for students to practice
their written communication skills. Exemplary responses are always shown to
reinforce effective ways to justify answers and present a logical problem solving
process while using correct mathematical vocabulary and symbols. In addition,
students work in small groups to practice new concepts at least weekly. They find
the more intimate setting to be less threatening, and they participate equally and
actively in explaining problems to each other. This improves their oral
communication skills by giving them consistent, frequent practice in discussing
processes for solving problems among their peers.
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Essential Questions:
• When and why should we estimate?
• Is there a pattern?
• How does what we measure influence how we measure?
• How does how we measure influence what we measure or don’t
measure?
• What do good problem solvers do especially when they get
stuck?
• How precise should this solution be?
• What are the limits of this mathematical model and of
mathematical modeling in general?
Topics:
Students are assigned summer work that consists of a variety of problems
suggested in the College Board Vertical Teaming materials. These deal with the
concepts of rates of change and accumulation. Several problems call for the
graphical representation of a verbal statement, or the verbal interpretation of a
graph. Some problems involve distance traveled represented as area under the
graph of velocity and as a piecewise defined function. Students also find the area of
an irregular shape by summing trapezoidal slices.
Unit 1 – Chapter 0 – Preliminaries: A Brief Review of Prerequisite Knowledge
(3 weeks)
0.1 The Real Numbers and the Cartesian Plane
Interval Notation
0.2 Lines and Functions
Emphasis on graphic representation of a variety of functions – polynomial,
rational, exponential, logarithmic
0.3 Graphing Calculators
Exploration of extrema
Increasing function relates to positive slope; decreasing to negative
Inflection points – change in concavity
Vertical and horizontal asymptotes
0.4 Solving Equations
0.5 Trigonometric Functions
0.6 Exponential and Logarithmic Functions
Rules of exponents
Rules of Logarithms
0.7 Transformations of Functions
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0.8 Preview of Calculus
Unit 2 – Chapter 1 – Limits and Continuity (3 weeks)
1.1 The Concept of Limit
1.2 Computation of Limits
1.3 Continuity and its Consequences
Intermediate Value Theorem
Piecewise defined functions
Types of discontinuities – relate the graphical and algebraic representations
1.4 Limits Involving Infinity
Vertical, Horizontal, and Slant Asymptotes
Relate the graphical, numerical, and algebraic representations
Focus on showing graphical, numerical, and algebraic evidence when finding
limits
Student generated “Guide to Finding Limits”
Unit 3 – Chapter 2 Part I – Differentiation: Basic Concepts (2 weeks)
2.1 Tangent Lines and Velocity
Difference Quotient
2.2 The Derivative
Differentiability
Sketch graph of the derivative given the graph of a function
Sketch graph of the function given the graph of its derivative
2.3 Computation of Derivatives: The Power Rule
2.4 The Product and Quotient Rules
Numerical and algebraic representations
Unit 4 – Chapter 2 Part II – Differentiation: Computation of Derivatives of
Algebraic, Trigonometric, Exponential and Logarithmic Functions (3 weeks)
2.7 The Chain Rule
Numerical and algebraic representations
2.5 Derivatives of Trigonometric Functions
2.6 Derivatives of Exponential and Logarithmic Functions
2.8a Implicit Differentiation – first and second order derivatives
2.9 The Mean Value Theorem for Derivatives
Rolle’s Theorem
Using derivatives to find the limit of a difference quotient
2.8b Related Rates Problems – applications of implicit differentiation; use of
graphing calculator
Unit 5 – Chapter 3 – Applications of Differentiation (3 weeks)
3.1 Linear Approximations and L’Hopital’s Rule
3.3 Maximum and Minimum Values
Graphical and algebraic representations
3.4 Increasing and Decreasing Functions
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3.5 Concavity
Graphical and algebraic representations
Sketch graph of the second derivative given the graph of a function or its derivative
Sketch graph of the function or its first derivative given the graph of its second
derivative
3.6 Overview of Curve Sketching
3.7 Optimization Problems – application of maximum and minimum values; use of
graphing calculator
Unit 6 – Chapter 4 – Integration (3 weeks)
4.1 Antiderivatives
4.2 Sums and Sigma Notation
4.3 Area
Riemann sums – left endpoint, right endpoint and midpoint approximations
Trapezoidal approximations
4.4 The Definite Integral
Integral Mean Value Theorem; average value of a function
4.5 The Fundamental Theorem of Calculus
I. Evaluation
II. Antiderivative
4.6 Integration by Substitution
Unit 7 – Chapter 5 – Applications of the Definite Integral (3 weeks)
5.1 Area between Curves
5.2 Volumes
Solids with known cross-section
Solids of revolution – method of disks and method of washers
Use of graphing calculator
Unit 8 – Motion – Kinematics (2 weeks)
Position
Velocity – average and instantaneous
Acceleration – average and instantaneous
Graphical, numerical, verbal, and algebraic representations are integrated and
interrelated.
Unit 9 – Chapter 6 – Differential Equations (3 weeks)
6.1 Review of the Natural Logarithm function - rules of logarithms
6.2 Inverse functions and their derivatives
Numerical, graphic and algebraic representations
6.3 Review of exponential functions
6.4 Exponential Growth and Decay Problems
Newton’s Law of Cooling
6.5 Separable Differential Equations
6.7 Inverse Trigonometric Functions
6.8 Calculus of Inverse Trig Functions
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Slope Fields
Unit 10 – Review for Advanced Placement Exam (2 weeks)
Practice Multiple Choice exams – 1997, 1998, and 2003 exams; multiple choice
questions
from released exams 1969-1993
Open response questions in groups of three to practice timing
In addition to the questions available on the apcentral website, the publication
“Free
Response Questions and Solutions 1989-1997” are used.
Teacher created review sheets for specific topics are used extensively to
supplement the
above materials in a focused manner.
Assessment:
Chapter tests for Chapters 0 – 6
Advanced Placement released Open Response Problems aligning with each unit of
study
Facts and Formulas Quiz
Released Advanced Placement Multiple Choice questions
Learning Standards
MP.1
Make sense of problems and persevere in solving them.
MP.2
Reason abstractly and quantitatively.
MP.3.
Construct viable arguments and critique the reasoning of others.
MP.4
Model with mathematics.
MP.5
Use appropriate tools strategically.
MP.6
Attend to precision.
MP.7
Look for and make use of structure.
MP.8
Look for and express regularity in repeated reasoning.
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N.RN.1
Explain how the definition of the meaning of rational exponents follows from extending the
properties of integer exponents to those values, allowing for a notation for radicals in terms of
rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 =
5(1/3)3 to hold, so (51/3)3 must equal 5.
N.RN.2
Rewrite expressions involving radicals and rational exponents using the properties of exponents.
N.Q.1
Use units as a way to understand problems and to guide the solution of multi-step problems;
choose and interpret units consistently in formulas; choose and interpret the scale and the origin in
graphs and data displays.
N.Q.2
Define appropriate quantities for the purpose of descriptive modeling.
N.Q.3
Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
A.SSE.1
Interpret expressions that represent a quantity in terms of its context.★
a. Interpret parts of an expression, such as terms, factors, and coefficients.
b. Interpret complicated expressions by viewing one or more of their parts as a single
entity. For example, interpret P(1+r)n as the product of P and a factor not depending on
P.
A.SSE.3
Choose and produce an equivalent form of an expression to reveal and explain properties of the
quantity represented by the expression.★
a. Factor a quadratic expression to reveal the zeros of the function it defines.
b. Use the properties of exponents to transform expressions for exponential functions. For
example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the
approximate equivalent monthly interest rate if the annual rate is 15%.
3. Identify zeros of polynomials when suitable factorizations are available, and use the zeros to
construct a rough graph of the function defined by the polynomial.
A.APR.3
(+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a
positive integer n, where x and y are any numbers, with coefficients determined for example by
Pascal’s Triangle.1
A.APR.5
(+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a
positive integer n, where x and y are any numbers, with coefficients determined for example by
Pascal’s Triangle.1
A.APR.6
Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x),
where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of
b(x), using inspection, long division, or, for the more complicated examples, a computer algebra
system.
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A.APR.7
(+) Understand that rational expressions form a system analogous to the rational numbers, closed
under addition, subtraction, multiplication, and division by a nonzero rational expression; add,
subtract, multiply, and divide rational expressions.
A.CED.1
Create equations and inequalities in one variable and use them to solve problems. Include
equations arising from linear and quadratic functions, and simple rational and exponential
functions.
A.CED.2
Create equations in two or more variables to represent relationships between quantities; graph
equations on coordinate axes with labels and scales.
A.CED.3
Represent constraints by equations or inequalities, and by systems of equations and/or
inequalities, and interpret solutions as viable or nonviable options in a modeling context. For
example, represent inequalities describing nutritional and cost constraints on combinations of
different foods.
A.CED.4
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving
equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.
A.REI.1
Explain each step in solving a simple equation as following from the equality of numbers asserted
at the previous step, starting from the assumption that the original equation has a solution.
Construct a viable argument to justify a solution method.
A.REI.2
Solve simple rational and radical equations in one variable, and give examples showing how
extraneous solutions may arise.
letters.
A.REI.3
Solve linear equations and inequalities in one variable, including equations with coefficients
represented by letters.
A.REI.4
Solve quadratic equations in one variable.
a. Use the method of completing the square to transform any quadratic equation in x into an
equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic
formula from this form.
b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots,
completing the square, the quadratic formula and factoring, as appropriate to the initial
form of the equation. Recognize when the quadratic formula gives complex solutions and
write them as a ± bi for real numbers a and b.
A.REI.10
Understand that the graph of an equation in two variables is the set of all its solutions plotted in
the coordinate plane, often forming a curve (which could be a line).
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A.REI.11
Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x)
intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using
technology to graph the functions, make tables of values, or find successive approximations.
Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential,
and logarithmic functions.
F.IF.1
Understand that a function from one set (called the domain) to another set (called the range)
assigns to each element of the domain exactly one element of the range. If f is a function and x is
an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph
of f is the graph of the equation y = f(x).
F.IF.2
Use function notation, evaluate functions for inputs in their domains, and interpret statements that
use function notation in terms of a context.
F.IF.4
For a function that models a relationship between two quantities, interpret key features of graphs
and tables in terms of the quantities, and sketch graphs showing key features given a verbal
description of the relationship. Key features include: intercepts; intervals where the function is
increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end
behavior; and periodicity.
F.IF.5
Relate the domain of a function to its graph and, where applicable, to the quantitative relationship
it describes. For example, if the function h(n) gives the number of person-hours it takes to
assemble n engines in a factory, then the positive integers would be an appropriate domain for the
function.
F.IF.6
Calculate and interpret the average rate of change of a function (presented symbolically or as a
table) over a specified interval. Estimate the rate of change from a graph.★
F.IF.7
Graph functions expressed symbolically and show key features of the graph, by hand in simple
cases and using technology for more complicated cases.★
a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
b. Graph square root, cube root, and piecewise-defined functions, including step functions
and absolute value functions.
c. Graph polynomial functions, identifying zeros when suitable factorizations are available,
and showing end behavior.
d. (+) Graph rational functions, identifying zeros and asymptotes when suitable
factorizations are available, and showing end behavior.
e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and
trigonometric functions, showing period, midline, and amplitude.
F.IF.8
Write a function defined by an expression in different but equivalent forms to reveal and explain
different properties of the function.
a. Use the process of factoring and completing the square in a quadratic function to show
zeros, extreme values, and symmetry of the graph, and interpret these in terms of a
context.
b. Use the properties of exponents to interpret expressions for exponential functions. For
example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y =
(1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.
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F.IF.9
Compare properties of two functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one
quadratic function and an algebraic expression for another, say which has the larger maximum.
F.BF.1
Write a function that describes a relationship between two quantities.★
a. Determine an explicit expression, a recursive process, or steps for calculation from a
context.
b. Combine standard function types using arithmetic operations. For example, build a
function that models the temperature of a cooling body by adding a constant function to
a decaying exponential, and relate these functions to the model.
c. (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a
function of height, and h(t) is the height of a weather balloon as a function of time, then
T(h(t)) is the temperature at the location of the weather balloon as a function of time.
F.BF.3
Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific
values of k (both positive and negative); find the value of k given the graphs. Experiment with
cases and illustrate an explanation of the effects on the graph using technology. Include
recognizing even and odd functions from their graphs and algebraic expressions for them.
F.BF.4
. Find inverse functions.
a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and
write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ≠
1.
b. (+) Verify by composition that one function is the inverse of another.
c. (+) Read values of an inverse function from a graph or a table, given that the function
has an inverse.
d. (+) Produce an invertible function from a non-invertible function by restricting the
domain.
F.BF.5
(+) Understand the inverse relationship between exponents and logarithms and use this
relationship to solve problems involving logarithms and exponents
F.LE.1
Distinguish between situations that can be modeled with linear functions and with exponential
functions.
a. Prove that linear functions grow by equal differences over equal intervals, and that
exponential functions grow by equal factors over equal intervals.
b. Recognize situations in which one quantity changes at a constant rate per unit interval
relative to another.
c. Recognize situations in which a quantity grows or decays by a constant percent rate per
unit interval relative to another.
d.
F.LE.2
Construct linear and exponential functions, including arithmetic and geometric sequences, given a
graph, a description of a relationship, or two input-output pairs (include reading these from a
table).
F.LE.3
Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a
quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
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F.TF.1
Understand radian measure of an angle as the length of the arc on the unit circle subtended by the
angle.
F.TF.2.
Explain how the unit circle in the coordinate plane enables the extension of trigonometric
functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise
around the unit circle.
F.TF.3
(+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4
and π/6, and use the unit circle to express the values of sine, cosines, and tangent for x, π + x, and
2π – x in terms of their values for x, where x is any real number.
F.TF.4
(+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric
functions.
F.TF.5
Choose trigonometric functions to model periodic phenomena with specified amplitude,
frequency, and midline.
F.TF.6
(+) Understand that restricting a trigonometric function to a domain on which it is always
increasing or always decreasing allows its inverse to be constructed.
F.TF.7
(+) Use inverse functions to solve trigonometric equations that arise in modeling contexts;
evaluate the solutions using technology, and interpret them in terms of the context.
F.TF.8
Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ)
given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
F.TF.9
(+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to
solve problems
G.SRT.5
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in
geometric figures.
G.SRT.8
. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied
problems.
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Resources:
Primary Textbook
Smith, Robert T. and Roland B. Minton. Calculus. Second Edition. New York:
McGraw-Hill, 2002. 0-07-283093-X
Additional resources used by teacher:
Hughes-Hallett, Deborah, Andrew Gleason, William G. McCallum et al. Calculus
Single Variable 4th Edition. New York: John Wiley & Sons, 2005. ISBN 0-47148481-4
Albert, Benita, Raymond Cannon, Mark Howell, and Steven Olson. Teacher’s
Guide for AP Calculus to accompany Calculus Single Variable (Hughes-Hallett,
Gleason, McCallum) 4th Edition. New York: John Wiley & Sons, 2005.
Barton, Ray, John Brunsting, John Diehl et al. Preparing for the Calculus AP Exam
with Calculus: Graphical, Numerical, Algebraic. Boston: Pearson Addison-Wesley,
2006. ISBN 0-321-29265-0
Materials from “Mini Workshop in the Fundamental Theorem of Calculus” with
Caren Diefenderfer. College Board Online Events, 2006.
AP Calculus AB, AP Calculus BC Free-Response Questions and Solutions 19891997. College Entrance Examination Board, 2003.
Broadwin, Judith, George Lenchner, and Martin Rudolph. Solutions: AP Calculus
Problems Part II AB and BC 1987-2001. Bellmore, NY: Mathematical Olympiads
for Elementary and Middle Schools, 2001. ISBN 1-882144-09-0
AP Calculus Multiple Choice Question Collection 1969-1998. College Board,
2005.
800 Questions in Calculus: 25 Years of Continental Mathematics League Contests.
Andover, MA: Skylight Publishing, 2005. ISBN 0-9727055-4-6
Lipp, Allen. Little Books of Big Ideas Calculus – Visualizing Differential
Equations with
Slope Fields. Saddle Brook, NJ: Peoples Publishing Group, 2006. ISBN 1-41381322-4
Kahn, David S., Cracking the AP Calculus AB & BC Exams 2006-2007 Edition.
New York: Princeton Review, 2005. ISBN 0-375-76526-3
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