COURSE DELIVERY AP Calculus AB is the study of limits, derivatives, definite and indefinite integrals, and the Fundamental Theorem of Calculus. Consistent with AP philosophy, concepts will be expressed and analyzed geometrically, numerically, analytically, and verbally.
Our online virtual schools offer students the opportunity to complete coursework for credit in grades k-12. The AP curriculum will be available to students online, in an asynchronous environment. Students proceed through the coursework at their own pace, so unit timeframes are given as approximate guidelines. Qualified teachers will evaluate student work and provide constructive feedback to better prepare students for the option of taking the AP Exam.
COURSE DESCRIPTION This is designed to be taught over a full high school academic year. It is possible to spend some time on elementary functions and still cover the Calculus AB curriculum within a year. However, if students are to be adequately prepared for the Calculus AB examination, most of the year must be devoted to topics in differential and integral calculus. These topics are the focus of the AP Exam.
AP Calculus AB is the study of limits, derivatives, definite and indefinite integrals, and the Fundamental Theorem of Calculus . Consistent with AP philosophy, concepts will be expressed and analyzed geometrically, numerically, analytically, and verbally. Additional information, including a topical outline and frequently asked questions is available at https://apstudent.collegeboard.org/apcourse/ap-calculus-ab
COURSE GOALS
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 pads 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.
OBJECTIVES These objectives are taken from the 2012 College Board Calculus Course Description.
I. Functions, Graphs, and Limits 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.
Limits of functions (including one-sided limits)
An intuitive understanding of the limiting process.
Calculating limits using algebra.
Estimating limits from graphs or tables of data.
Asymptotic and unbounded behavior
Understanding asymptotes in terms of graphical behavior.
Describing asymptotic behavior in terms of limits involving infinity.
Comparing relative magnitudes of functions and their rates of change (for example, contrasting exponential growth, polynomial growth, and logarithmic growth).
Continuity as a property of functions
An intuitive understanding of continuity. (The function values can be made as close as desired by taking sufficiently close values of the domain.)
Understanding continuity in terms of limits.
Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem).
Parametric, polar, and vector functions. The analysis of planar curves includes those given in parametric form, polar form, and vector form.
II. Derivatives Concept of the derivative
Derivative presented graphically, numerically, and analytically.
Derivative interpreted as an instantaneous rate of change.
Derivative defined as the limit of the difference quotient.
Relationship between differentiability and continuity.
Derivative at a point
Slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents.
Tangent line to a curve at a point and local linear approximation.
Instantaneous rate of change as the limit of average rate of change.
Approximate rate of change from graphs and tables of values.
Derivative as a function
Corresponding characteristics of graphs of ƒ and ƒ•.
Relationship between the increasing and decreasing behavior of ƒ and the sign ofƒ•.
The Mean Value Theorem and its geometric interpretation.
Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa.
Second derivatives
Corresponding characteristics of the graphs of ƒ, ƒ•, and ƒ •.
Relationship between the concavity of ƒ and the sign of ƒ •.
Points of inflection as places where concavity changes.
Assignment: Through exploration students will investigate critical points both numerically and graphically. Students will then verbalize their findings, describing when a point will be a maximum, minimum or point of inflection.
Applications of derivatives
Analysis of curves, including the notions of monotonicity and concavity.
Analysis of planar curves given in parametric form, polar form, and vector form, including velocity and acceleration.
Optimization, both absolute (global) and relative (local) extrema.
Modeling rates of change, including related rates problems.
Use of implicit differentiation to find the derivative of an inverse function.
Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration.
Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations.
Numerical solution of differential equations using Euler’s method.
L’Hospital’s Rule, including its use in determining limits and convergence of improper integrals and series.
Assignment: This assignment uses a graphical, numerical and analytic approach to help students to develop an understanding of the relationship between position, velocity and acceleration. Students will learn to reason from tables and graphs and to communicate the relationships using correct language and notation.
Computation of derivatives
Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions.
Derivative rules for sums, products, and quotients of functions.
Chain rule and implicit differentiation.
Derivatives of parametric, polar, and vector functions.
Assignment: Students will be able to: Parametric Equations:
Sketch the curve defined by the parametric equations and eliminate the parameter.
Find and evaluate them for a given value of t.
Write an equation for the tangent line to the curve for a given value of t.
Find the points of horizontal and vertical tangency.
Find the length of an arc of a curve given by parametric equations.
Vector Equations:
Find the velocity and acceleration vectors when given the position vector.
Given the components of the velocity vector and the position of the particle at a particular value oft, find the position at another value of t.
Given the components of the acceleration vector and the velocity of the particle at a particular value of t, find the velocity at another value of t.
Find the slope of the path of the particle for a given value of t.
Write an equation for the tangent line to the curve for a given value of t.
Find the values oft at which the line tangent to the path of the particle is horizontal or vertical.
Assignment: Reasoning from Tabular Data Students analyze data from a table. Students will use data to estimate rates or change, net changeand average value from values of functions given in the table. They will need to use a variety of methods to find derivatives and integrals as they relate to real world applications.
Assignment: Students Discover the First and Second Fundamental Theorems of Calculus through graphical exploration.
Assignment: Students discover and explore Second Fundamental Theorem of Calculus. Students then use theorem toanalyze the F(x) from the graph of the derivative of the function f.
III. Integrals Interpretations and properties of definite integrals
Definite integral as a limit of Riemann sums.
Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval:
Basic properties of definite integrals (examples include additivity and linearity).
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 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 finding the area of a region (including a region bounded by polar curves), 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 (including a curve given in parametric form), and accumulated change from a rate of change.
Fundamental Theorem of Calculus
Use of the Fundamental Theorem to evaluate definite integrals.
Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined.
Techniques of antidifferentiation
Antiderivatives following directly from derivatives of basic functions.
Antiderivatives by substitution of variables (including change of limits for definite integrals), parts, and simple partial fractions (nonrepeating linear factors only).
Improper integrals (as limits of definite integrals).
Applications of antidifferentiation
Finding specific antiderivatives using initial conditions, including applications to motion along a line.
Solving separable differential equations and using them in modeling (including the study of the equation y• = ky and exponential growth). + Solving logistic differential equations and using them in modeling.
Numerical approximations to definite integrals. Use of Riemann sums (using left, right, and midpoint evaluation points) and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values
ASSESSMENTS Students will be expected to complete sample problems in each unit as practice for unit tests. Unit tests include multiple-choice and free-response questions from AP Released Exams. Students will be asked to periodically meet with the instructor in the instructor’s chat room to review unit tests and discuss the scoring guidelines. In addition to practice problems and unit tests, students will be asked to complete activities requiring use of graphing calculators and written justifications of their solutions. Sample activities include estimating the limit from a graph and comparing to the numerical computation of the limit, performing differentiation and integration, calculate definite integrals and verify properties of functions. Grades will be determined by student participation in completing practice problems and graded activities as well as by unit test scores.
GRADING SYSTEM A 100% – 90% B 89% – 80% C 79% – 70% D 69% – 60% F 59% and below
MATERIALS Technology Requirement You will want to have a graphing calculator. I recommend the TI-84 We will use the calculator in a variety of ways including:
Conduct explorations.
Graph functions within arbitrary windows.
Solve equations numerically.
Analyze and interpret results.
Justify and explain results of graphs and equations.
1- Introduction & Review
Welcome to AP Calculus AB
Course Outline
Course Syllabus
Handout: Calculus 1
Handout: Calculus 2
MLA Formatting Word 2007
MLA Documentation Updates
MLA Citation
MLA Incorporating Sources
Calculus On The Web
Completing Math Assessments
Review — Stuff You Should Already Know
2- Limits & Continuity
An Introduction to Limits & Continuity
Quiz: Introduction to Limits & Continuity
Tangent Lines & Rates of Change
Handout: Tangent Lines & Rates of Change Worksheet
Assignment: Tangent Lines & Rates of Change
The Limit
Handout: The Limit Worksheet
Assignment: The Limit
One-Sided Limits
Handout: One-Sided Limits Worksheet
Assignment: One-Sided Limits
Computing Limits
Handout: Computing Limits Worksheet
Assignment: Computing Limits
Infinite Limits
Handout: Infinite Limits Worksheet
Assignment: Infinite Limits
Limits at Infinity
Handout: Limits at Infinity Worksheet
Assignment: Limits at Infinity
Continuity
Handout: Continuity Worksheet
Assignment: Continuity
Additional Resources
Handout: Limits & Continuity Exam
Assignment: Limits & Continuity Exam
3- Derivatives
Introduction to Derivatives
Definition & Interpretation of the Derivative
Handout: Definition & Interpretation of the Derivative Worksheet
Assignment: Definition & Interpretation of the Derivative
Differentiation Formulas
Handout: Differentiation Formulas Worksheet
Handout: Derivatives Exam 1
Assignment: Differentiation Formulas
Assignment: Derivatives Exam 1
Product & Quotient Rule
Handout: Product & Quotient Rule Worksheet
Assignment: Product & Quotient Rule
Derivatives of Trig Functions
Handout: Derivatives of Trig Functions Worksheet
Assignment: Derivatives of Trig Functions
Derivatives of Exponential & Logarithm Functions
Handout: Derivatives of Exponential & Logarithm Functions Worksheet
Assignment: Derivatives of Exponential & Logarithm Functions
Derivatives of Inverse Trig Functions
Derivatives of Hyperbolic Functions
Handout: Derivatives of Inverse & Hyperbolic Trig Functions Worksheet
Assignment: Derivatives of Inverse & Hyperbolic Trig Functions
The Chain Rule
Handout: The Chain Rule Worksheet
Handout: Derivatives Exam 2
Assignment: The Chain Rule
Assignment: Derivatives Exam 2
Implicit Differentiation
Handout: Implicit Differentiation Worksheet
Assignment: Implicit Differentiation
Related Rates
Handout: Related Rates Worksheet
Assignment: Related Rates
Higher Order Derivatives
Handout: Higher Order Derivatives Worksheet
Assignment: Higher Order Derivatives
Additional Resources
Handout: Derivatives Exam 3
Assignment: Derivatives Exam 3
4- Applications of Derivatives
What’s the Big Deal About Derivatives, Anyway?
Rates of Change
Critical Points & Extrema
Handout: Critical Points & Extrema Worksheet
Assignment: Critical Points & Extrema
The Shape of a Graph
Handout: The Shape of a Graph Worksheet
Assignment: The Shape of a Graph
The Mean Value Theorem
Handout: The Mean Value Theorem Worksheet
Assignment: The Mean Value Theorem
Optimization
Handout: Optimization Worksheet
Assignment: Optimization
L’Hopital’s Rule
Handout: L’Hosptial’s Rule Worksheet
Assignment: L’Hospital’s Rule
Linear Approximations
Handout: Linear Approximations Worksheet
Assignment: Linear Approximations
Differentials
Handout: Differentials Worksheet
Assignment: Differentials
Newton’s Method
Handout: Newton’s Method Worksheet
Assignment: Newton’s Method
Handout: Motion Part 1 Worksheet
Handout: Motion Part 2 Worksheet
Handout: Motion Part 3 Worksheet
Assignment: Motion Part 1
Assignment: Motion Part 2
Assignment: Motion Part 3
5- Integrals
Working Backwards
Indefinite Integrals
Handout: Indefinite Integrals Worksheet
Assignment: Indefinite Integrals
Computing Indefinite Integrals
Handout: Computing Indefinite Integrals Worksheet
Assignment: Computing Indefinite Integrals
Substitution Rule for Indefinite Integrals
Handout: Substitution Rule Worksheet
Handout: Integrals Exam 1
Assignment: Substitution Rule
Assignment: Integrals Exam 1
Area Problem
Handout: Area Problem Worksheet
Assignment: Area Problem
Definite Integrals
Handout: Definite Integrals Worksheet
Assignment: Definite Integrals
Computing Definite Integrals
Handout: Computing Definite Integrals Worksheet
Assignment: Computing Definite Integrals
Substitution Rule for Definite Integrals
Handout: Substitution Rule for Definite Integrals Worksheet
Assignment: Substitution Rule for Definite Integrals
Integration by Parts
Handout: Integration by Parts Worksheet
Assignment: Integration by Parts
Partial Fractions
Handout: Partial Fractions Worksheet
Assignment: Partial Fractions
Improper Integrals
Handout: Improper Integrals Worksheet
Assignment: Improper Integrals
Additional Resources
Handout: Motion Part 4 Worksheet
Handout: Motion Part 5 Worksheet
Assignment: Motion Part 4
Assignment: Motion Part 5
6- Applications of Integrals
What’s the Big Deal About Integrals, Anyway?
Area Between Curves
Handout: Area Between Curves Worksheet
Assignment: Area Between Curves
Volumes of Solids of Revolution (Part 1)
Handout: Method of Rings Worksheet
Assignment: Method of Rings
Volumes of Solids of Revolution (Part 2)
Handout: Method of Cylinders Worksheet
Assignment: Method of Cylinders
Volume
Work
Handout: Work Worksheet
Assignment: Work
Handout: Applications of Integrals Exam
Assignment: Applications of Integrals Exam
7- Review & Test Preparation
Exam Format & Scoring
Exam Strategies & Tips
Practice, Practice, Practice
Handout: Calculus AB Section I Part A
Handout: Calculus AB Section 1 Part B
Handout: Calculus AB Section II Part A
Handout: Calculus AB Section II Part B
Assignment: AP Calculus AB Practice – Section I Part A
Assignment: AP Calculus AB Practice – Section I Part B
Assignment: AP Calculus AB Practice – Section II Part A
Assignment: AP Calculus AB Practice – Section II Part B
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