Title                                                                  Teacher                             

Calculus                                                            Mr. James Uy

                                                                          

Duration                                                          Contact

One trimester                                                   Email: james.uy@maryknollschool.org

                                                                           Voice Mail: 952-7281

                                                                           Room: 213, High School Campus

 

 

Course Description: Calculus is a compilation of ideas that provides a way of viewing and analyzing the physical world.  This course challenges students to learn the elements of and manipulate the calculus.  They receive a solid introduction to calculus and become proficient enough to understand the use of calculus in solving many seemingly complex problems that face engineers, chemists and many other professionals using mathematics.  This is required study for anyone studying engineering, any "hard" science (chemistry, physics, biology), and dozens of other fields in college.  Students will use a graphics calculator and other resources to solve problems with the calculus.  It is fast paced and requires the student to think and be creative.  Many of the concepts are difficult to comprehend initially.  Determination and diligence will help students overcome the initial confusion as this course becomes the basis for any serious study in mathematics, engineering or science.  In May each student is encouraged take the AP Calculus exam (whether or not taking the AP Calculus Prep elective) that will test their ability and, if they are successful and the college agrees, will count for college credit.

 

 

Expected School Learning Results:  This course supports the following parts of the Maryknoll School Expected Learning Results:

 

Habits of the mind – To become a creative and critical thinker and problem solver; Accepting responsibility for learning and personal choices; Valuing learning as a life‑long process; Seizing the challenges of the future with optimism.

 

Habits of the heart – To practice moral values and understand the gospel as expressed in the teaching of the Roman Catholic Church; respecting self and others.

 

Habits of the community – To contribute time, energy and talents to improve the quality of life in our school, community, and nation, while striving for a world of peace and justice; Demonstrating positive social relations and participating in effective collaboration.

 

 

 

Course Objective:  Students will learn the basics of the four concepts of calculus: limits, derivatives, antiderivatives, and definite integrals.  They will learn and communicate these concepts graphically, symbolically, numerically, and verbally.

 

 

Topical Course Outline: 

 

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.  (Close values of the domain lead to close values of the range.)

·        Understanding continuity in terms of limits.

·        Geometric understanding of graphs of continuous functions  (Intermediate Value Theorem and Extreme Value Theorem).

 

II.         Derivatives

 

Concept of the derivative.

·        Derivative presented geometrically, 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  f  and .

·        Relationship between the increasing and decreasing behavior of  f and the sign of .

·        The Mean Value Theorem and its geometric consequences.

·        Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa.

 

Second derivatives.

·        Corresponding characteristics of graphs of  f , , and .

·        Relationship between the concavity of  f and the sign of .

·        The Mean Value Theorem and its geometric consequences.

·        Points of inflection as places where concavity changes.

 

Applications of derivatives.

·        Analysis of curves, including the notions of monotonicity and concavity.

·        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.

 

Computations of derivatives.

·        Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions.

·        Basic rules for the derivative of sums, products, and quotients of functions.

·        Chain rule and implicit differentiation.

 

III.       Integrals

 

Interpretations and properties of definite integrals.

·        Computation of Riemann sums using left, right, and midpoint evaluation points.

·        Definite integral as a limit of Riemann sums over equal subdivisions.

·        Definite integral of the rate of change of a quantity over an integral 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, social, 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 applications problems.  Whatever applications are chosen the emphasis is on using the integral of a rate of change to give accumulated change or 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, the volume of a solid with known cross sections, the average value of a function, and the distance traveled by a particle along a line.

 

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).

 

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.  In particular, studying the equation  and exponential growth.

 

Numerical approximations to definite integrals.  Use of Riemann and trapezoidal sums to approximate definite integrals of functions represented algebraically, geometrically, and by tables of values.

 

 

            Ä  Late Assignments:  Assignments handed in late will be reduced in grade based on the amount of time that the assignment is late.

 

            Ä  Grading Policy:

                        Æ  Letter grades are based on the following schedule:

 

Letter Grade	Percentage Grade	Comment
A+	97-100	Outstanding Achievement
A	93-96
A-	90-92
B+	87-89	Average to Above Average Achievement
B	83-86
B-	80-82
C+	77-79	Satisfactory Achievement
C	73-76
C-	70-72
D+	67-69	Passing Achievement
D	63-66
D-	60-62
F	0-59	Unsatisfactory Achievement

 

            Æ  The final grade for the course is based on three items.  Quizzes and Chapter Tests are 60% of the grade. Homework is 20% of the grade. Trimester exam is 20% of the grade.

 

Assignments:

            Ä  Missed Assignments:  Students are responsible for any worked missed due to an absence of one or two days.  Anyone absent for a longer period must contact me to arrange a time to make up the instruction missed.

            Ä  Homework:  Homework will be assigned daily and is due according to the course syllabus.

            Ä  Quizzes:  Quizzes are administered daily.  They will be approximately 10 minutes long given at the beginning of the class.  Missed quizzes are not made up.

            Ä  Tests:  Tests may be comprehensive.  Missed tests will be made up as soon as possible.  Students absent the test day only, will be administered the test the day they return to school.  For students absent on any days prior to the test a determination will be made by me when the test will be given.  Unless otherwise instructed students should expect to take the test the day they return to school.  Students with a planned absence the day of the test must coordinate with me before the test as to when the test will be administered.

            Ä  Trimester Exam:  There will be an exam at the end of the trimester covering the trimester’s  material.

            Ä  Classroom Participation:  Students are expected to participate in all class discussions.  During the class students will be required to solve problems at the board and explain how the answer was or was not derived.

 

 

Course Schedule:  See course syllabus.

 

 

Material Requirements:

            Ä  Required:

                        Æ  Text:  Calculus Concepts and Applications by Paul A. Foerster, Key Curriculum Press.

                        Æ  Graphics calculator, TI-82, TI-83, TI-84 or TI-89.

                        Æ  Three ring binder and notebook paper for homework.

                        Æ  Pencil, wooden or mechanical.

                        Æ  Eraser

                        Æ  Straight edge

 

 

 

Class Decorum:

 

Ä     Food:  No food will be allowed in the classroom.

 

Ä  Student Behavior:  Lively discussions are encouraged.  Respect for other student’s opinions and opportunity to speak will be adhered to. There is to be one person speaking at a time.  I will act as the facilitator.  Students are to be on time and have the requisite materials with them.  Class time is not for retrieving forgotten items nor is it time for personal grooming.  These activities are to take place outside of the classroom.  I recommend that students have more than one pencil when they come to class.  I will not provide any materials during class.  Present in class means that the student is physically in the room with all the necessary equipment, in the proper dress and ready to start the lesson.  In all other matters the school policy on behavior will be in effect.

Daily Classroom Procedures: 

            Ä  Preparation:  Students are to bring to class their text, calculus, blank paper and any other item I previously requested be brought for that period.

            Ä  Daily Routine:  Class starts with a 10 minute quiz (except on days of tests).  Depending on the difficulty of the previous night’s material there will be a short lecture on the subject or we will look at some homework problems, or students will be given problems to work on the board. 

            Ä  Absences:  Students are to follow the school procedures for absences.  I will be available for instruction after school when previously arranged for by the student.

            Ä  Tardiness:  Students are to follow the school procedures for tardiness.

 

 

Additional Instruction:  I will be available during lunch and after the regular school day in room 213 to any student who needs and requests time for additional instruction.  If a student is in danger of failing I may require that the student come after school for further instruction.  Students attending or failing to attend a session I require will have this reflected in their class participation grade.

 

 

Student Ethics:  Above all I expect students to act ethically and with moral fortitude.  Cheating in any manner is not tolerated.  Any student caught cheating will receive a 0 grade for the test or quiz, and I will contact that student's parent(s) by phone.  For repeat cases I will request a conference with the student's parents to discuss the student's behavior.