interval notation, composite functions, Pythagorean Theorem, right triangle trigonometry products, factoring, finding zeros, evaluating functions at given points quiz Assessments: informal questioning end-of-unit quiz Interdisciplinary Connections: N/A Additional Resources: summer work packet College Board’s AP Calculus AB course description Subject: AP Calculus AB Grade: 10/11/12 Suggested Timeline: 20 – 42 minute periods Unit Unit 1 - Limits Unit Overview/Essential Understanding: The limit plays a major role in calculus, as it is used to define both the derivative and integral. In this unit students explore the concept of the limit and a variety of techniques for evaluating limits. Although L’Hospital’s Rule for evaluating limits is a required technique, this method is explained later within the course content, as it relies on students’ knowledge of the derivative. Unit Objectives: Upon completion of this unit, students will demonstrate an understanding of the following topics and/or be able to do the following: concept of the limit
A large conceptual leap is needed to steer students away from the idea that the limit and function values are not necessarily the same. Concepts/Content: Limits indeterminate forms limits involving infinity continuity piece-wise functions Intermediate Value Theorem Competencies/Skills: evaluate limits by graph evaluate limits by table evaluate limits with algebra test for continuity test for conditions of IVT Description of Activities: teacher led direct instruction question & answer check/review HW warm-up / exit ticket questions quiz problem set end-of-unit test specific example of activity thought experiment with visiting friend and the concept of limit Assessments: informal questioning quiz problem set end-of-unit test Interdisciplinary Connections: asymptotic behavior – a measurement of computer algorithm efficiency Additional Resources: Foerster textbook (Kendall Hunt) – sections: 2-2, 2-3, 2-4, 2-6 textbook supplemental materials (explorations) TI-89 graphing calculator, desmos.com selected MC/FR items from released AP exams College Board’s AP Calculus AB course description
rule derivatives of sine and cosine derivatives of the natural exponential and natural logarithmic functions calculate derivatives of composite functions apply derivatives to several applied problems teacher led direct instruction question & answer check/review HW warm-up / exit ticket questions quiz problem set end-of-unit test Assessments: informal questioning quiz problem set end-of-unit test Interdisciplinary Connections: physics application – harmonic motion Additional Resources: Foerster textbook (Kendall Hunt) – sections: 3-6, 3-7, 3-8, textbook supplemental materials (explorations) TI-89 graphing calculator, desmos.com selected MC/FR from released AP exams College AP Calculus AB course description 9
Unit Objectives: Upon completion of this unit, students will demonstrate an understanding of the following topics and/or be able to do the following: identify when to use implicit differentiation apply implicit differentiation simplify differentiated implicit relations and express these derivatives strictly in terms of x and y find 2nd order derivatives of implicit relations accurately translate English to appropriate calculus variables within the context of related rates problems solve problems involving related rates Focus Standards Addressed in this Unit: EK 2.2A2 – Key features of functions and their derivatives can be identified and related to their graphical, numerical, and analytical representations EK 2.1C5 – The chain rule is the basis for implicit differentiation EK 2.3C2 – The derivative can be used to solve related rates problems, that is, finding a rate at which one quantity is changing by relating it to other quantities whose rates of change are known EK 2.3D1 – The derivative can be used to express information about rates of change in applied contexts Important Standards Addressed in this Unit: EK 2.1A4 - For , notations for the derivative include and EK 2.3A1 – The unit for is the unit for f divided by the unit of x Misconceptions: Related problems tend be one of the toughest areas for students. Many students report that they simplify have difficulty determining what the problem is asking. Teaching a slow, methodical approach to solving problems helps. Concepts/Content: implicit differentiation related rates translation problems Competencies/Skills: differentiate implicit relations identify given rates identify unknown Description of Activities: teacher led direct instruction question & answer check/review HW warm-up / exit ticket questions quiz problem set end-of-unit test specific example of activity “blue water leaking from funnel demonstration” – goes along with the classic related rates problem involving water leaking from a cone-shaped container Assessments: informal questioning quiz
Assessments: informal questioning quiz problem set end-of-unit test Interdisciplinary Connections: Optimization problems can be applied to many practical examples (for example, finding the minimum amount of material needed to manufacture a can that can hold a specific volume). Additional Resources: Foerster textbook (Kendall Hunt) – sections: 8-2, 8-3, 5-5 textbook supplemental materials (explorations) TI-89 graphing calculator, desmos.com selected MC/FR items from released AP exams College Board’s AP Calculus AB course description Subject: AP Calculus AB Grade: 10/11/12 Suggested Timeline: 14 – 42 minute periods Unit Title: Unit 7 – Integral I Unit Overview/Essential Understanding: In this unit, students are introduced to the reverse process of differentiation – finding the antiderivative (indefinite integral). Students learn and practice several anti-differentiation rules. The definite integral is introduced as a limit of Riemann sums. Riemann sums are used as a way to not only define the definite integral, but approximate the definite integral. The unit culminates with an introduction to the first part of the Fundamental Theorem of Calculus (FTC) as a way to exactly evaluate definite integrals. Unit Objectives: Upon completion of this unit, students will demonstrate an understanding of the following topics and/or be able to do the following: find indefinite integrals using anti-differentiation rules calculate left, right, midpoint, upper, and lower Riemann sums to approximate definite integrals use the first part of the Fundamental Theorem of Calculus to evaluate definite integrals use properties of indefinite and definite integrals to evaluate integrals use algebraic manipulation and u-substitution to evaluate indefinite and definite integrals Focus Standards Addressed in this Unit: EK 3.1A1 – An anti-derivative of a function f is a function g whose derivative is f EK 3.1A2 – Differentiation rules provide the foundation for finding anti-derivatives EK 3.2A1 – A Riemann sum, requires a partition of an interval I, is the sum of products, each of which is a value of the function at a point in a subinterval multiplied by the length of that subinterval of the partition
N/A Foerster textbook (Kendall Hunt) – sections: 5-3, 5-4, 5-6, textbook supplemental materials (explorations) TI-89 graphing calculator, desmos.com selected MC/FR items from released AP exams College AP Calculus AB course description Subject: AP Calculus AB 10/11/12 Suggested Timeline: 14 – 42 minute periods Unit Title: Unit 8 – Integral II Unit Overview/Essential Understanding: In this unit, students learn some important applications of the definite integrals: physical area, physical volume, and accumulated change. Unit Objectives: Upon completion of this unit, students will demonstrate an understanding of the following topics and/or be able to do the following: determine whether a definite integral represents physical area calculate the physical area bound between 2 or more curves calculate volumes of revolution discs or washers calculate volume of a with known cross sections solve definite integral problems applied to other scenarios involving accumulated change Focus Standards Addressed in this Unit: EK 3.4A1 – A function defined as an integral represents an accumulation of a rate of change EK 3.4A2 – The definite of the rate of change of a quantity over an interval the net change of the quantity over that interval EK 3.4D1 – Areas of certain regions in the plane can be calculated with definite integrals EK 3.4D2 – Volumes of solids with known cross sections, including discs and washers, can be calculated with definite integrals EK 3.4E1 – The definite integral can be used to express information about accumulation and net change in many applied contexts Important Standards Addressed in this Unit: EK 3.3B2 – If f is continuous on the interval and F is an antiderivative of f, then Misconceptions: Students sometimes believe that there exists a different initial setup for finding volumes with integrals, when in fact the procedure is exactly the same as finding area with the definite integral. Students should be repeatedly asked to identify regions, draw
EK 1.1C2 – The limit of a function may be found by using algebraic manipulation, alternate forms of trigonometric functions, or the squeeze theorem Misconceptions: Some of the problems that students will work in this unit are very abstract due to the fact that they are defined by an integral. A slow, methodical approach is recommended when teaching this topic. The College Board has many examples that involve analytical and graphical approaches to functions defined by integrals. Concepts/Content: reciprocal functions functions defined by integrals properties of logarithms logarithmic differentiation 2nd part of the Fundamental Theorem of Calculus L’Hospital’s Rule Competencies/Skills: applying various differentiation rules applying various anti-derivative rules analyzing a function defined by an integral graphically applying the 2nd part of the FTC to find derivatives of functions defined by integrals apply L’Hospital’s Rule to evaluate limits Description of Activities: teacher led direct instruction question & answer check/review HW warm-up / exit ticket questions quiz problem set end-of-unit test specific example of activity “Why L’Hospital’s Rule Works” activity Assessments: informal questioning quiz problem set end-of-unit test Interdisciplinary Connections: N/A Additional Resources: Foerster textbook (Kendall Hunt) – sections: 6-2, 6-3, 6-4, textbook supplemental materials (explorations) TI-89 graphing calculator, desmos.com selected MC/FR from released AP exams College Board’s AP Calculus AB course description Calculus in Motion computer simulations
students will demonstrate mastery in all topics outlined in the College Board’s course description for AP Calculus AB through working many multiple choice and free response items Standards Addressed in this Unit: No new standards are introduced during this unit. All standards addressed in previous units will be reviewed as students work through the practice exams. Important Standards Addressed in this Unit: N/A Misconceptions: N/A Concepts/Content: AP Calculus AB topics as outlined in the College Board’s course description for AP Calculus Competencies/Skills: Work problems involving limits, derivatives, and integrals Description of Activities: assignment, review, and discussion of previously released exam questions Assessments: informal questioning formula / theorem quiz final exam Interdisciplinary Connections: N/A Additional Resources: Foerster textbook (Kendall Hunt) – sections: all relevant sections textbook supplemental materials (explorations) TI-89 graphing calculator, desmos.com selected MC/FR from released AP exams College Board’s AP Calculus AB course description