AP Calculus AB
2016 – 2017 Syllabus
Curricular Requirements
CR1a The course is structured around the enduring understandings within Big Idea 1: Limits. • See page 2
CR1b The course is structured around the enduring understandings within Big Idea 2: Derivatives. • See page 3
CR1c The course is structured around the enduring understandings within Big Idea 3: Integrals and the Fundamental Theorem of Calculus. • See page 4
CR2a The course provides opportunities for students to reason with definitions and theorems. • See page 5
CR2b The course provides opportunities for students to connect concepts and processes. • See page 5
CR2c The course provides opportunities for students to implement algebraic/computational processes. • See page 5
CR2d The course provides opportunities for students to engage with graphical, numerical, analytical, and verbal representations and demonstrate connections among them. • See page 5
CR2e The course provides opportunities for students to build notational fluency. • See page 5
CR2f The course provides opportunities for students to communicate mathematical ideas in words, both orally and in writing. • See page 6
CR3a Students have access to graphing calculators. • See page 6
CR3b Students have opportunities to use calculators to solve problems. • See page 6
CR3c Students have opportunities to use a graphing calculator to explore and interpret calculus concepts. • See page 6
CR4 Students and teachers have access to a college-level calculus textbook • See page 6
Course Overview
Advanced Placement (AP) Calculus AB is an enriched mathematics course designed to:
-Provide students the opportunity to master high level mathematical concepts and problem solving skills required in many science, technology and business fields;
-Prepare students for success in college level mathematics;
-Prepare students for success on the AP Calculus AB exam.
Students wanting to take AP Calculus AB must have successfully completed the prerequisite courses; Algebra I, Geometry, Algebra II, and Precalculus. Irving High School has an open enrollment policy that allows any student who has met the requirements to take an AP course. All students who take an AP course are required to take the AP exam. The costs for the AP exams are born by the district.
Students are formally assessed throughout the school year. Tests include problems selected by the instructor from a variety of sources, including previously released AP free response and multiple choice problems. All free response problems are graded using rubrics similar to those used on AP assessments. Formative assessments including homework and quizzes are used to measure student progress prior to testing. Students participate in outside study groups that provide support and encouragement throughout the year as part of their preparation for the AP exam.
As with any rigorous mathematics course, emphasis is given to not just the mathematical concepts themselves, but to the problem solving and critical thinking skills required to analyze and solve real world problems. Students will also be expected to write to demonstrate their understanding of mathematical concepts, as well as to improve their communication skills.
Course Outline
The course is a comprehensive study of calculus designed to meet all the goals outlined in the AP Calculus Course Description. A scope and sequence of the topics covered follows, along with a preliminary timeframe allotted for each unit.
First Semester
Unit 1 - Limits and Continuity –CR1a(2 weeks)
1. Introduction to Limits
a. graphically
b. numerically
2. Properties of Limits
3. Techniques for Evaluating Limits
4. Continuity and One-sided Limits
5. Intermediate Value Theorem
6. Infinite Limits and Vertical Asymptotes
7. Limits at Infinity and Horizontal Asymptotes
Unit 2 – Differentiation – CR1b(5-6 weeks)
1. Definition of the Derivative of a Function
2. Differentiability and Continuity
3. Theorems on the Derivative
a. Constant Rule
b. Power Rule
c. Scalar Product Rule
d. Sum and Difference Rule
e. Product Rule
f. Quotient Rule
4. Derivatives of Trigonometric Functions
5. Derivative of Exponential/Logarithmic Functions
6. The Chain Rule
7. Implicit Differentiation
8. Derivatives of Higher Order
9. Average Rate of Change
10. Position, Velocity, Acceleration
11. Motion in a Straight Line—Particle Problems
Unit 3 - Applications of the Derivative (6-7 weeks)
1. Related Rates
2. Extrema
a. Local (relative) Extrema
b. Absolute Extrema
3. Rolle’s Theorem and Mean Value Theorem
4. Analysis of Graphs
a. Increasing and Decreasing
b. Critical points
c. First Derivative Test
d. Points of Inflection
e. Second Derivative Test
f. Concavity
g. Relationship between f,f’,f’’
h. Summary of Curve Sketching
5. Approximating the Derivative using the Calculator
6. Optimization Problems
Semester Exam (2 days)
- AP Style Multiple Choice (day 1 40 minutes)
- 12 questions – no calculator allowed
- 5 questions – calculator allowed
- AP Style Free Response Questions (day 2 45 minutes)
- 2 questions – calculator allowed
- 1 question – no calculator allowed
Second Semester
Unit 4 – Integration –CR1c(9-10 weeks)
1. Antiderivatives
2. Indefinite Integration
3. The Area Under a Curve
a. Riemann Sums
i. Right Riemann Sums
ii. Left Riemann Sums
iii. Midpoint Riemann Sums
iv. The Definite Integral as a Limit of a Riemann Sum
b. Trapezoidal Rule
c. Definite integrals
4. The Fundamental Theorem of Calculus
5. The Second Fundamental Theorem of Calculus
6. Mean Value Theorem for Integrals/Average Value of a Function
7. Integration by Substitution
8. Integration of Even/Odd Functions
9. Integration of Trigonometric Functions
10. Integration of Inverse Trigonometric Functions
11. Integration of Logarithmic Functions
13. Differential Equations—Growth and Decay
14. Slope Fields
15. L’Hopital’s Rule
16. Numerical Integration - Graphing Calculator
Unit 5 - Applications of Integration (2 weeks)
1. Area Between Curves
2. Volume
a. Disk Method
b. Washer Method
c. Shell Method
d. Known Cross Sections
3. Motion in a Straight Line—Total Distance Traveled
Unit 6 - AP Examination Practice (3-4 weeks)
1. Discussion of AP Scoring Guidelines for AP Exam
2. Study of Released AP Examinations
3. Discussion of Current Question Trends/Types
4. Practice on Multiple Choice and Free-Response Questions
5. Completion of Two AP Practice Exams
Mathematical Practices
The following is a brief description of some of the activities included in the course.
- Reasoning with definitions and theorems –CR2a
In problems where students practice applying the results of key theorems (e.g., Intermediate Value Theorem, Mean Value Theorems, and/or L’Hopital’s Rule), students are required for each problem to demonstrate verbally and/or in writing that the hypotheses of the theorems are met in order to justify the use of the appropriate theorem. For example, in an in-class activity, students are given a worksheet that contains a set of functions on specified domains on which they must determine whether they can apply the Mean Value Theorem. There are cases where some of the problems do not meet the hypotheses in one or more ways.
- Connecting concepts and processes – CR2b
Students analyze position and velocity functions and their graphs. Students calculate distance travelled and the accumulation of velocity, and make connections between the relationship between the two functions and the processes of differentiation and integration.
- Implementing algebraic/computational processes – CR2c
Students are presented with a table of observations collected over time periods of different lengths (e.g., temperatures or stock prices). Students use Riemann sums to numerically approximate the average value of the readings over the given time period and interpret the meaning of that value.
- Connecting multiple representations – CR2d
Students collect data for the height of a meterstick being pulled away from a wall. Students present data numerically, graphically, analytically (in the form of a formula), and verbally (as a description in words of how the function behaves). Students develop equations which relate the variables distance from the wall and height of the meterstick, differentiate this equation with respect to a third variable (time), and then use the resulting equation to solve for specific scenarios.
- Building notational fluency –CR2e
Students are given a variety of growth and decay word problems where the rate of change of the dependent variable is proportional to the same variable (e.g., population growth, radioactive decay, continuously compounded interest, and/or Newton’s law of cooling). Students are asked to translate the problem situation into a differential equation using proper notation. Students show the steps in solving the differential equation, continuing to use proper notation for each step (e.g., when to keep or remove absolute value).
- Communicating –CR2f
Throughout the course, students are required to present solutions to homework problems both orally and on the board to the rest of the class. On assessments, students are expected to clearly communicate solutions in writing, provide explanations for solutions, and demonstrate that appropriate conditions have been met for application of various theorems.
- Solving problems with calculators –CR3b
Students use graphing calculators to solve integrals in problems requiring the evaluation of integrals involving complex functions representing rates of change. These problems sets are similar in scope to AP Free Response Questions.
- Exploring calculus concepts with technology –CR3c
Students approximate the slope of a tangent line to polynomial parent functions (quadratic, cubic, and quartic) at a specific point by finding the slope of a secant line where the second point of the secant line is extremely close to the point of tangency. Each student is given a different point to evaluate. Results from each student are accumulated, a scatter plot is sketched using graphing technology, and students determine the equation which best fits the scatter plot. After performing this analysis for all three functions, students then generalize a rule, thus discovering the power rule for differentiation.
Graphing Calculators and TechnologyCR3a
Graphing calculators are used on a daily basis to reinforce calculus concepts and interpret results. TI-84 Plus calculators are provided for use in school for classwork and assessments. Calculators may be checked out on a special needs basis overnight. While students are expected to be able to find solutions both with and without the use of a graphing calculator, the following calculator skills are emphasized:
-Plot the graph of a function with an arbitrary viewing window;
-Find zeros of a function (solve equations numerically);
-Numerically calculate the derivative of a function;
-Numerically calculate the value of a definite integral.
ReferencesCR4
1. Larson, Roland E., Robert P. Hostetler, and Bruce H. Edwards.
Calculus of a Single Variable. 8th ed. Houghton Mifflin, Boston, MA, 2006.
2. AP booklet with released free response questions.
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