Download AP Calculus AB 2014

AP Calculus AB
2014 – 2015
Brief Description of Course
Calculus AB is a course in single-variable calculus that includes techniques and
applications of the derivative, techniques and applications of the definite integral, and the
Fundamental Theorem of Calculus. Algebraic, numerical, and graphical representations
are emphasized throughout the course.
Expectations
Students will be expected to work in collaborative small groups. Each student will be
teacher; each will be learner as they help each other master the material.
Emphasis will be placed on student’s communicating their mathematical thinking within
their groups, to the class as a whole, and on assessments. Students will be expected to
explain their processes and justify their results both orally and in written form, in clear,
concise, well-crafted sentences.
Objectives and Goals
Students should understand the meaning of the derivative in terms of rate of change and
local linear approximations.
Students should be able to work with functions represented graphically, numerically,
analytically, or verbally, and should understand the connections among these
representations.
Students should understand the meaning of the definite integral both as a limit of
Riemann sums and as a net accumulation of a rate of change; and understand the
relationship between the derivative and integral.
Students should be able to model problem situations with functions, differential
equations, or integrals, and communicate both orally and in written form.
Students should be able to represent differential equations with slope fields, solve
separable differential equations analytically, and solve differential equations using
numerical techniques such as Euler’s method.
Technology
Our study of calculus will include the continual use of a variety of technologies; chief
among them will be the graphing calculator. Urbandale High School requires students to
have a graphing calculator, with a recommendation of the TI-84 Plus, to use daily to
explore, discover and reinforce the concepts of calculus. Students may use the graphing
calculators on most but not all assessments.
Students will be able to use their calculator to:
 Graph a function within an arbitrary viewing window.
 Solve an equation graphically and numerically.
 Analyze and interpret their results.
 Compute the numerical derivative of a function.
 Compute definite integrals numerically.
 Conduct explorations.
Examples:

Students will use the graph and zoom feature to investigate lim
x®0
sinq
q
. They will
support their conclusion using the table feature.

After finding the derivative of a function students will graph f(x) and f´(x).
Students will then graph nDeriv(f(x)) to support their calculation of the derivative.
Timeline
This is a full year course (2 semesters or 36 weeks).
I.
II.
III.
IV.
V.
VI.
VII.
Prerequisites for Calculus (1 week) Completed as an independent study.
Limits and Continuity (5 weeks) One formative, one summative assessment.
Derivatives (6 weeks) Two formative, one summative assessment.
Applications of Derivatives (6 weeks) Two formative, one summative assessment.
The Definite Integral (6 weeks) Two formative, one summative assessment.
Differential Equations and Mathematical Modeling (5 weeks) Two formative, one
summative assessment.
Applications of Definite Integrals (6 weeks) Two formative, one summative
assessment.
Major Topics
I Prerequisites for Calculus
1.1 Lines
Increments; Slope of a Line; Parallel and Perpendicular Lines; Equations of
Lines; Applications
1.2 Functions and Graphs
Functions; Domains and Range; Viewing and Interpreting Graphs; Even and Odd
Functions; Symmetry; Functions Defined in Pieces; Absolute Value Functions;
Composite Functions
1.3 Exponential Functions
Exponential Growth and Decay; Applications; The Number e
1.4 Parametric Equations
Relations; Circles, Ellipses, Lines and Other Curves
1.5 Functions and Logarithms
One-to-One Functions; Inverses; Finding Inverses; Logarithmic Functions;
Properties of Logarithms; Applications
1.6 Trigonometric Functions
Radian Measure; Graph’s Periodicity; Even and Odd Trigonometric Functions;
Transformations of Trigonometric Graphs; Inverses
II Limits and Continuity
2.1 Rates of Change and Limit
Average and Instantaneous Speed; Definition of Limit; Properties of Limits; Onesided and Two-sided Limits; Sandwich Theorem
2.2 Limits Involving Infinity
Finite Limits as x±∞; Sandwich Theorem Revisited; Infinite Limits as
xa; End Behavior Models
2.3 Continuity
Continuity at a Point; Continuous Functions; Algebraic Combinations,
Composites; Intermediate Value Theorem for Continuous Functions
2.4 Rates of Change and Tangent Lines
Average Rates of Change; Tangent to a Curve; Slope of a Curve; Normal to a
Curve; Speed Revisited
III Derivatives
3.1 Derivative of a Function
Definition; Notation; Relationship Between the Graphs of f and f ´; Graphing the
Derivative from Data; One-sided Derivatives
3.2 Differentiability
How f ´(a) Might Fail to Exist; Differentiability Implies Local Linearity;
Derivatives on a Calculator; Differentiability Implies Continuity; Intermediate
Value Theorem for Derivatives
3.3 Rules for Differentiation
Positive Integer Powers; Multiples, Sums, and Differences; Products and
Quotients; Negative Integer Powers of x; Second and Higher Order Derivatives
3.4 Velocity and Other Rates of Change
Instantaneous Rates of Change; Motion along a Line; Sensitivity to Change;
Derivatives in Economics
3.5 Derivatives of Trigonometric Functions
Derivative of the Sine and Cosine Functions; Simple Harmonic Motion; Jerk;
Derivatives of Other Basic Trigonometric Functions
3.6 Chain Rule
Derivative of a Composite Function; Outside-Inside Rule; Repeated use of the
Chain Rule; Slopes of Parameterized Curves; Power Chain Rule
3.7 Implicit Differentiation
Implicitly Defined Functions; Tangents and Normal Lines; Derivatives of Higher
Order; Rational Powers of Differentiable Functions
3.8 Derivatives of Inverse Trigonometric Functions
Derivatives of Arcsine, Arccosine, Arctangent, Arc cotangent, Arc secant, and
Arc cosecant functions
3.9 Derivatives of Exponential and Logarithmic Functions
Derivative of ex; Derivative of ax; Derivative of ln x; Derivative of logax; Power
Rule for Arbitrary Real Powers
IV Applications of Derivatives
4.1 Extreme Values of Functions
Absolute (Global) Extreme Values; Local (Relative) Extreme Values; Locating
Extreme Values
4.2 Mean Value Theorem
Mean Value Theorem; Physical Interpretation Increasing and Decreasing
Functions; Other Consequences
4.3 Connecting f and f ´´with the graph of f
First Derivative Test; Concavity; Points of Inflection; Second Derivative Test;
Learning about Functions from Derivatives
4.4 Modeling and Optimization
Examples from Mathematics, Business and Industry; Economics Modeling;
Discrete Phenomena with Differentiable Functions
4.5 Linearization and Newton’s Method
Linear Approximation; Newton’s Method; Differentials Estimating; Change with
Differentials; Absolute, Relative and Percentage Change; Sensitivity to Change
4.6 Related Rates
Related Rate Equations Solution Strategy; Simulating Related Motion
V The Definite Integral
5.1 Estimating with Finite Sums
Distance Traveled; Rectangular Approximation Method (RAM); Volume of a
Sphere; Cardiac Output
5.2 Definite Integrals
Riemann Sums; Terminology and Notation of Integration; Definite Integral and
Area; Constant Functions; Integrals on a Calculator; Discontinuous Integrals
Functions
5.3 Definite Integrals and Antiderivatives
Properties of Definite Integrals; Average Value of a Function; Mean Value
Theorem for Definite Integrals; Connecting Differential and Integral Calculus
5.4 Fundamental Theorem of Calculus
Fundamental Theorem, Part I; Graphing the Function
ò
x
a
f (t)dt ; Fundamental
Theorem, Part II; Area Connection; Analyzing Antiderivatives Graphically
5.5 Trapezoidal Rule and Simpson’s Rule
Trapezoidal Approximations; Approximations by Simpson’s Rule; Error Analysis
VI Differential Equations and Mathematical Modeling
6.1 Slope Fields and Euler’s Method
Differential Equations; Slope Fields; Euler’s Method
6.2 Antidifferentiation by Substitution
Indefinite Integrals; Leibniz Notation and Antiderivatives; Substitution in
Indefinite Integrals; Substitution in Definite Integrals
6.3 Antidifferentiation by Parts
Product Rule in Integral Form; Solving for the Unknown; Integral Tabular
Integration; Inverse Trigonometric and Logarithmic Functions
6.4 Exponential Growth and Decay
Separable Differential Equations; Law of Exponential Change; Continuously
Compounded Interest; Radioactivity; Modeling Growth with Other Bases;
Newton’s Law of Cooling
6.5 Logistic Growth
How Populations Grow; Partial Fractions; The Logistic Differential Equation;
Logistic Growth Models
VII Applications of Definite Integrals
7.1 Integral As Net Change
Linear Motion Revisited; General Strategy Consumption Over Time; Net Change
from Data; Work
7.2 Areas in the Plane
Area Between Curves; Area Enclosed by Intersecting Curves; Boundaries with
Changing Functions; Integrating with Respect to y; Saving Time with Geometry
Formulas
7.3 Volumes
Volume As an Integral; Square Cross Sections; Circular Cross Sections;
Cylindrical Shells; Other Cross Sections
7.4 Lengths of Curves
A Sine Wave Length of Smooth Curve; Vertical Tangents, Corners and Cusps
7.5 Applications from Science and Statistics
Work Revisited; Fluid Force and Fluid Pressure; Normal Probabilities
Textbooks
Title: Calculus: Graphical, Numerical, Algebraic, AP Edition
Publisher: Addison Wesley Publishing Company
Published Date: 2006-02
Author: Franklin Demana
Second Author: Bert K. Waits
Third Edition 2007
Other Course Materials
Student Practice Workbook ISBN 0-13-201411-4
Barron’s, AP Calculus, 10th edition, Hockett, Bock
Pearson Education, AP Calculus, Barton, Brunsting, Diehl, Hill, Tyler, Wilson