Syllabus, 2015-2016
AP Calculus AB Syllabus, Bray, 2015-2016 Contact Information
Mr. Bray,
Rm. A8,
[email protected]
Materials
3-ring notebook Loose-leaf paper Graph paper
*Pencils/pens * TI-84 graphing calculator (required daily)
Classroom Rules
Be respectful.
* Be on time.
* Be prepared.
* Geometry
* Precalculus
*Be mindful of yourself and others.
Prerequisite
Algebra II
or
* College Algebra with Trigonometry
Textbook Finney, R.L., Demana, F.D., Waits, B.K., & Kennedy, D. (2012). Calculus: Graphical, numerical, algebraic, fourth edition, *AP edition. Boston, MA: Pearson. Course Description AP Calculus AB is designed to help students have an adequate knowledge of the calculus concepts through experiences and applications. Our study of Calculus is divided into two major topics: differential and integral Calculus. Differential Calculus enables us to calculate rates of change, find the slope of a curve, and calculate velocities and accelerations of moving bodies. Integral Calculus is used to find the area of an irregular region in a plane, and measure lengths of curves. Our task is to facilitate each student’s mechanics in order to develop his or her understanding of the theory of Calculus and the ability to use these ideas in applied Calculus. Objectives The student will be able to solve problems numerically, graphically, algebraically, and analytically by hands and with graphing calculator. They will learn how to use calculus to model and solve real-life problems. They will have a deep understanding of integrals, derivatives, and their practical applications. Students will gain enough knowledge to pass the AP Calculus AB Exam and get ready to succeed in college advanced math and science courses. Instructional Strategies I will introduce each unit numerically, graphically, analytically, and verbally through clear and simple illustrations (examples and non-examples). I will encourage students to work in small groups (2 or 3 people) in order to solve problems and support their answers clearly in both verbal and written ways. Students will have the opportunity to correct problems using specific terms and explanations required by the AP Calculus AB Exam. Students will be able to explore, discuss, and solve many old AP exams, especially the ones that require solutions and justifications. I will also use document camera, smart board, and PowerPoint presentations for lectures, projects, and class discussions. Students will have the opportunity to spend at least 15 days reviewing before they take the AP Exam.
AP Exam Date
MAY 5, 2016 AT 8:00 A.M.
Technology All students are required to use their own graphing calculator TI-83 or 84 because almost all AP topics involve the use of graphing calculator. Furthermore, I have a class set of graphing calculators TI-84 Plus. Students will analyze graphs (continuity, increasing, decreasing, etc...) by hands and with calculator; evaluate limits of functions graphically, numerically, and algebraically; evaluate integrals of functions numerically by hands and with graphing calculator; graphing many different functions; establish graphically the relationship between a function and its first and second derivatives. Graphing calculator will be used to find intersections between functions, and for complex computations. In other words, graphing calculator will help students visualize and verify reasonably their understanding, work, and conclusions about graphs, derivatives, and definite integrals, and polynomial approximations. Requirements and Procedures At the tardy bell, beginning of class, students take out homework, pencil, calculator, textbook and blank notebook paper. They start their daily given bell work, which will be checked and discussed daily. Students discuss a few of their homework questions using the white board or smart board or document camera. Every student is required to take note in Cornell Note Style for every single topic covered in class. All assignments need name, date, and period in the upper righthand corner of your paper. Homework is assigned almost every night and is discussed the following class. Homework is graded on a zero to five scales Credit will not be earned if students did not show and support work and write out the original problem. Show your work neatly! Show all work steps, even if you use a calculator. AP Calculus AB students should expect to spend 30-60 minutes nightly studying and completing assignments. Use examples worked in class and your book as guides. Assignments We will use many old AP exams for practice, especially those that require explanation and justifications of answers. There will be daily assignments (in groups or individually). Assignments are due to the end class period. It is the teacher’s choice when to grade and score a student’s work (But almost every assignment will be graded). For independent work, each student does all work in pencil through a step-bystep solution with clear justifications. Students will be encouraged to solve and verify their work in small groups. They will be frequently asked to express and support their solutions in both verbal and written ways. Cheating is an automatic zero and a discipline referral. No talking or communication during testing. Cheating is giving or receiving answers. It is the student’s responsibility to ensure that he/she understands how to do the problems and be able to show and explain how the problems are worked out correctly in proper writing. Any missing assignment will receive a grade zero. Grades Students earn points for their class notes (Cornell Notes), assignments, quizzes, project, and tests. At least two quizzes per week will be given either with or without the use of calculator. However, tests based on cumulative units covered will be given and divided into a calculator part and non-calculator part (similar format to AP Test). At the end of the session, a final comprehensive exam as well as small group projects will be given. The student’s grade is calculated by adding up the earned points then dividing by the total possible points giving a percentage which matches the following scale:
90-100% →A
80-89%→B
70-79%→C
60-69%→D
Tests:
50%
Class works:
25%
Bell work/Homework/Exit Slips:
25%
Final Exam:
20% for semester
0-59%→F
Attendance and Makeup It is extremely important that students attend class every day. “If you are absent, it is your responsibility to get the class notes (ask a classmate) and assignments, and make them up.” For any questions pertaining to absences, please see me EITHER BEFORE OR AFTER CLASS. All make-up assignments must be done in class (except homework). Course Topic Outline Analysis of Graphs (10 days - 1 test) Students will review graphical representations of various functions both manually and using the graphing calculator. They will note various behaviors in each graph. * Relations, functions, and their graphs * Identification of essential functions: constant, linear, quadratic, cubic, square root, cube root, absolute value, greatest integer, piece-wised, exponential, logarithmic, and trigonometric. * Transformation of functions * Using the graphing calculator to determine an appropriate viewing window * Using the graphing calculator to find roots * Domain, range, intercepts, and asymptotes * Composition of functions / * Writing equations of lines / * Inverses of functions
Limits and Rates of Change (15 days – 2 tests) Students will discover several methods of finding a limit of a function. They will investigate limits and their properties. * Tangent to a curve and velocity of an object-understanding limits intuitively * Evaluate the limit of a function numerically * Evaluate one-sided limits * Evaluate the limit of a function using limit laws * Evaluate the limit of a function analytically and graphically using graphing calculator.
* Determining when a limit does not exist * Continuity of a function * Discontinuity of a function * Limits involving infinity and the asymptotic relationship * 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 * Geometric understanding of graphs of continuous functions using the Intermediate Value Theorem and the Extreme Value Theorem
Derivatives (25 days – 3 tests) Students will discover the concept of the derivative graphically, numerically, and analytically. -Concept of the derivative * Presentation of the derivative graphically, numerically, and analytically * Interpretation of the derivative as a Rate of Change * Derivative defined as the limit of the difference quotient * Slopes, tangent lines, and derivatives * Relationship between differentiability and continuity * Differentiation Formulas-Constant Rule, Power Rule, Product Rule, Quotient Rule, and Chain Rule-of polynomial, rational, trigonometric, inverse trigonometric, exponential, and logarithmic functions * Implicit Differentiation * Logarithmic Differentiation -Derivative at a point * Slope of a curve at point- tangents, vertical tangents, and no tangents * Tangent line to a curve at a point and local linear approximation- Slopes, tangent lines, and derivatives * Instantaneous rate of change as the limit of the average rate of change * Approximate rate of change from graphs and table values -Derivative as a function * Corresponding characteristics of the graphs of f and f ’and f”
* Critical Values * Relationship between the increasing and decreasing behavior of f and the sign of f’ * Rolle’s Theorem / * The Mean Value Theorem and its geometric consequences * Translations of verbal descriptions into equations involving derivatives and vice versa -Second derivatives * Corresponding characteristics of the graphs of f, f’, and f’’ * Relationship between the concavity of f and the sign of f’’ * Inflection points as places where concavity changes - Applications Derivatives (20 days – 2tests) * Curve sketching involving derivatives and sign lines * The Newton’s Method * The first and second derivative tests * Analysis of curves, including notions of monotonicity and concavity * Analysis of planar curves given in parametric form, polar form, and vector form, including velocity and acceleration * Graphing Summary including roots, domain and range, asymptotes, intervals of increase and decrease, extrema, concavity, and inflection points * 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
Integrals (15 days – 2 tests) Students will understand the concept of integration and the definite integral as area under the curve, the accumulated change of a rate of change, and representation of the limit of an approximating Riemann sum, as well as a method of finding the area of a region, the volume of a solid, the average value of a function, and the distance traveled by a particle along a line. Students will use graphing calculator to explore reasonably their answers. -Interpretations and properties of definite integrals * Understand the concept of area under the curve using a Riemann sum over equal subintervals
* Calculate the definite integral as a limit of Riemann sums and Trapezoid sums. * Definite integrals and antiderivatives * Definite integral of the rate of change of a quantity over an interval interpreted as the change of quantity over the integral: * Use of the graphing calculator to compute definite integrals numerically * Basic properties of definite integrals including additivity and linearity -Applications of integrals (15 days – 2 tests) * Use of definite integrals to find the area under a curve * Use of definite integrals to find the area between two curves * Use of definite integrals to find the area of a region bounded by polar curves * Volumes of Solids of Revolution using Disks and Washers * Cylindrical Shells * Average Value of a function * Volumes of Solids with known cross sections * Distance traveled by a particle along a line * Arc Length of a curve
Fundamental Theorem of Calculus (10 days – 1 test) * Use of the Fundamental Theorem of Calculus to evaluate definite integrals * Use of the Fundamental Theorem of Calculus to represent a particular antiderivative, and the analytical and graphical analysis of such defined functions Techniques of antidifferentiation (25 days – 2 tests) * Antiderivatives following directly from derivatives of basic functions * Antiderivatives by substitution of variables (including changing limits for definite integrals, parts, and simple partial fractions) * L’Hopital Rule, Slope-Fields -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 such as with 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, upper, lower, and midpoint evaluation points to approximate definite integrals of functions represented algebraically, graphically, and by tables of values * Use of trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values Simpson’s rule
AP Calculus AB Review (at least 15 days)
Please Sign and Return in This Sheet to Mr. Bray Student’s Agreement: I read this syllabus and understand the content and expectations of this course.
______________________________
______________________________
______________________________
Print name
Signature
Date
Parent’s Agreement I read and discuss the content and expectations of this syllabus, and I will help my kid do his/her best.
______________________________
______________________________
______________________________
Parent/guardian
Signature
Date
e-mail ______________________________
Mr. Bray,
Rm. A8,
[email protected]
Materials
3-ring notebook Loose-leaf paper Graph paper
*Pencils/pens * TI-84 graphing calculator (required daily)
Classroom Rules
Be respectful.
* Be on time.
* Be prepared.
* Geometry
* Precalculus
*Be mindful of yourself and others.
Prerequisite
Algebra II
or
* College Algebra with Trigonometry
Textbook Finney, R.L., Demana, F.D., Waits, B.K., & Kennedy, D. (2012). Calculus: Graphical, numerical, algebraic, fourth edition, *AP edition. Boston, MA: Pearson. Course Description AP Calculus AB is designed to help students have an adequate knowledge of the calculus concepts through experiences and applications. Our study of Calculus is divided into two major topics: differential and integral Calculus. Differential Calculus enables us to calculate rates of change, find the slope of a curve, and calculate velocities and accelerations of moving bodies. Integral Calculus is used to find the area of an irregular region in a plane, and measure lengths of curves. Our task is to facilitate each student’s mechanics in order to develop his or her understanding of the theory of Calculus and the ability to use these ideas in applied Calculus. Objectives The student will be able to solve problems numerically, graphically, algebraically, and analytically by hands and with graphing calculator. They will learn how to use calculus to model and solve real-life problems. They will have a deep understanding of integrals, derivatives, and their practical applications. Students will gain enough knowledge to pass the AP Calculus AB Exam and get ready to succeed in college advanced math and science courses. Instructional Strategies I will introduce each unit numerically, graphically, analytically, and verbally through clear and simple illustrations (examples and non-examples). I will encourage students to work in small groups (2 or 3 people) in order to solve problems and support their answers clearly in both verbal and written ways. Students will have the opportunity to correct problems using specific terms and explanations required by the AP Calculus AB Exam. Students will be able to explore, discuss, and solve many old AP exams, especially the ones that require solutions and justifications. I will also use document camera, smart board, and PowerPoint presentations for lectures, projects, and class discussions. Students will have the opportunity to spend at least 15 days reviewing before they take the AP Exam.
AP Exam Date
MAY 5, 2016 AT 8:00 A.M.
Technology All students are required to use their own graphing calculator TI-83 or 84 because almost all AP topics involve the use of graphing calculator. Furthermore, I have a class set of graphing calculators TI-84 Plus. Students will analyze graphs (continuity, increasing, decreasing, etc...) by hands and with calculator; evaluate limits of functions graphically, numerically, and algebraically; evaluate integrals of functions numerically by hands and with graphing calculator; graphing many different functions; establish graphically the relationship between a function and its first and second derivatives. Graphing calculator will be used to find intersections between functions, and for complex computations. In other words, graphing calculator will help students visualize and verify reasonably their understanding, work, and conclusions about graphs, derivatives, and definite integrals, and polynomial approximations. Requirements and Procedures At the tardy bell, beginning of class, students take out homework, pencil, calculator, textbook and blank notebook paper. They start their daily given bell work, which will be checked and discussed daily. Students discuss a few of their homework questions using the white board or smart board or document camera. Every student is required to take note in Cornell Note Style for every single topic covered in class. All assignments need name, date, and period in the upper righthand corner of your paper. Homework is assigned almost every night and is discussed the following class. Homework is graded on a zero to five scales Credit will not be earned if students did not show and support work and write out the original problem. Show your work neatly! Show all work steps, even if you use a calculator. AP Calculus AB students should expect to spend 30-60 minutes nightly studying and completing assignments. Use examples worked in class and your book as guides. Assignments We will use many old AP exams for practice, especially those that require explanation and justifications of answers. There will be daily assignments (in groups or individually). Assignments are due to the end class period. It is the teacher’s choice when to grade and score a student’s work (But almost every assignment will be graded). For independent work, each student does all work in pencil through a step-bystep solution with clear justifications. Students will be encouraged to solve and verify their work in small groups. They will be frequently asked to express and support their solutions in both verbal and written ways. Cheating is an automatic zero and a discipline referral. No talking or communication during testing. Cheating is giving or receiving answers. It is the student’s responsibility to ensure that he/she understands how to do the problems and be able to show and explain how the problems are worked out correctly in proper writing. Any missing assignment will receive a grade zero. Grades Students earn points for their class notes (Cornell Notes), assignments, quizzes, project, and tests. At least two quizzes per week will be given either with or without the use of calculator. However, tests based on cumulative units covered will be given and divided into a calculator part and non-calculator part (similar format to AP Test). At the end of the session, a final comprehensive exam as well as small group projects will be given. The student’s grade is calculated by adding up the earned points then dividing by the total possible points giving a percentage which matches the following scale:
90-100% →A
80-89%→B
70-79%→C
60-69%→D
Tests:
50%
Class works:
25%
Bell work/Homework/Exit Slips:
25%
Final Exam:
20% for semester
0-59%→F
Attendance and Makeup It is extremely important that students attend class every day. “If you are absent, it is your responsibility to get the class notes (ask a classmate) and assignments, and make them up.” For any questions pertaining to absences, please see me EITHER BEFORE OR AFTER CLASS. All make-up assignments must be done in class (except homework). Course Topic Outline Analysis of Graphs (10 days - 1 test) Students will review graphical representations of various functions both manually and using the graphing calculator. They will note various behaviors in each graph. * Relations, functions, and their graphs * Identification of essential functions: constant, linear, quadratic, cubic, square root, cube root, absolute value, greatest integer, piece-wised, exponential, logarithmic, and trigonometric. * Transformation of functions * Using the graphing calculator to determine an appropriate viewing window * Using the graphing calculator to find roots * Domain, range, intercepts, and asymptotes * Composition of functions / * Writing equations of lines / * Inverses of functions
Limits and Rates of Change (15 days – 2 tests) Students will discover several methods of finding a limit of a function. They will investigate limits and their properties. * Tangent to a curve and velocity of an object-understanding limits intuitively * Evaluate the limit of a function numerically * Evaluate one-sided limits * Evaluate the limit of a function using limit laws * Evaluate the limit of a function analytically and graphically using graphing calculator.
* Determining when a limit does not exist * Continuity of a function * Discontinuity of a function * Limits involving infinity and the asymptotic relationship * 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 * Geometric understanding of graphs of continuous functions using the Intermediate Value Theorem and the Extreme Value Theorem
Derivatives (25 days – 3 tests) Students will discover the concept of the derivative graphically, numerically, and analytically. -Concept of the derivative * Presentation of the derivative graphically, numerically, and analytically * Interpretation of the derivative as a Rate of Change * Derivative defined as the limit of the difference quotient * Slopes, tangent lines, and derivatives * Relationship between differentiability and continuity * Differentiation Formulas-Constant Rule, Power Rule, Product Rule, Quotient Rule, and Chain Rule-of polynomial, rational, trigonometric, inverse trigonometric, exponential, and logarithmic functions * Implicit Differentiation * Logarithmic Differentiation -Derivative at a point * Slope of a curve at point- tangents, vertical tangents, and no tangents * Tangent line to a curve at a point and local linear approximation- Slopes, tangent lines, and derivatives * Instantaneous rate of change as the limit of the average rate of change * Approximate rate of change from graphs and table values -Derivative as a function * Corresponding characteristics of the graphs of f and f ’and f”
* Critical Values * Relationship between the increasing and decreasing behavior of f and the sign of f’ * Rolle’s Theorem / * The Mean Value Theorem and its geometric consequences * Translations of verbal descriptions into equations involving derivatives and vice versa -Second derivatives * Corresponding characteristics of the graphs of f, f’, and f’’ * Relationship between the concavity of f and the sign of f’’ * Inflection points as places where concavity changes - Applications Derivatives (20 days – 2tests) * Curve sketching involving derivatives and sign lines * The Newton’s Method * The first and second derivative tests * Analysis of curves, including notions of monotonicity and concavity * Analysis of planar curves given in parametric form, polar form, and vector form, including velocity and acceleration * Graphing Summary including roots, domain and range, asymptotes, intervals of increase and decrease, extrema, concavity, and inflection points * 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
Integrals (15 days – 2 tests) Students will understand the concept of integration and the definite integral as area under the curve, the accumulated change of a rate of change, and representation of the limit of an approximating Riemann sum, as well as a method of finding the area of a region, the volume of a solid, the average value of a function, and the distance traveled by a particle along a line. Students will use graphing calculator to explore reasonably their answers. -Interpretations and properties of definite integrals * Understand the concept of area under the curve using a Riemann sum over equal subintervals
* Calculate the definite integral as a limit of Riemann sums and Trapezoid sums. * Definite integrals and antiderivatives * Definite integral of the rate of change of a quantity over an interval interpreted as the change of quantity over the integral: * Use of the graphing calculator to compute definite integrals numerically * Basic properties of definite integrals including additivity and linearity -Applications of integrals (15 days – 2 tests) * Use of definite integrals to find the area under a curve * Use of definite integrals to find the area between two curves * Use of definite integrals to find the area of a region bounded by polar curves * Volumes of Solids of Revolution using Disks and Washers * Cylindrical Shells * Average Value of a function * Volumes of Solids with known cross sections * Distance traveled by a particle along a line * Arc Length of a curve
Fundamental Theorem of Calculus (10 days – 1 test) * Use of the Fundamental Theorem of Calculus to evaluate definite integrals * Use of the Fundamental Theorem of Calculus to represent a particular antiderivative, and the analytical and graphical analysis of such defined functions Techniques of antidifferentiation (25 days – 2 tests) * Antiderivatives following directly from derivatives of basic functions * Antiderivatives by substitution of variables (including changing limits for definite integrals, parts, and simple partial fractions) * L’Hopital Rule, Slope-Fields -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 such as with 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, upper, lower, and midpoint evaluation points to approximate definite integrals of functions represented algebraically, graphically, and by tables of values * Use of trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values Simpson’s rule
AP Calculus AB Review (at least 15 days)
Please Sign and Return in This Sheet to Mr. Bray Student’s Agreement: I read this syllabus and understand the content and expectations of this course.
______________________________
______________________________
______________________________
Print name
Signature
Date
Parent’s Agreement I read and discuss the content and expectations of this syllabus, and I will help my kid do his/her best.
______________________________
______________________________
______________________________
Parent/guardian
Signature
Date
e-mail ______________________________
