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Bound Brook Public School District High School Calculus Curriculum
2007-2008
Dr. Edward Hoffman, Superintendent
2006-2007 Board of Education Members
Martin Gleason, President
Steve Clouser, Vice – President
Peter Allen – South Bound Brook Representative
Hal Dietrich
Terrence Hoben
Robyn Ann Jeskie
Carol Ann Koupiaris
Robert Murray
Kenneth Sella
Rae Siebel
Carole Deddy, Board Secretary
Administration
Mr. Dan Gallagher, Principal
Mr. Mario Bernardo, Vice Principal Mr. Robert Nixon, Vice Principal
Ms. Dianne Ianniello, Director of Pupil Services
Curriculum Revision by: Pamela J. Norgalis, Math Department Chairwoman
Mission Statement
Bound Brook High School is a supportive multicultural community that provides an innovative and academically challenging educational program while offering variety of extra-curricular and social opportunities that encourage life long learning and citizenship.
Vision Statement
The vision of the Bound Brook Public School community is to provide a comprehensive educational environment that will:
• Develop tolerant citizens • Prepare graduates for their educational and vocational choices in life
• Develop life long learners • Allow students to be users of technology • Develop finders and users of data • Provide educational opportunities both within and outside the classroom • Challenge students educationally • Provide a positive learning environment • Make students aware of their strengths and weaknesses • Recognize student successes.
Belief Statement
We believe that the Bound Brook community will provide a supportive environment for academic and personal growth that will:
• Foster independence, self-reliance, and self-worth • Prepare students for a diverse and ever-changing society. • Encourage the development of programs that promote good character in the school community. • Enable everyone to feel physically, emotionally, and intellectually safe (free to verbally express opinions and ideas). • Value all for their unique qualities. • Encourage all to pursue their individual goals in a challenging, supportive, and safe environment. • Provide a positive learning environment where mutual respect and opportunity exist for the exchange of ideas among teachers, students,
parents, and community members. • Deliver an instructional program that encompasses a variety of learning styles, interests, and levels of readiness for all students in all
disciplines • Demonstrate honesty, integrity, and trustworthiness in academic pursuits and social interactions. • Respect all people and cultures • Encourage participation in one’s community as a social, civic, and personal responsibility. • Promote learning as a life-long process.
Academic Goals
Learning Goals
1. Students are able to use basic communication and mathematics skills for purposes and situations they will encounter throughout their lives.
2. Students shall develop their abilities to apply core concepts and principles from mathematics, the sciences, the arts, the humanities, social
studies, practical living studies, and vocational studies to what they will encounter throughout their lives.
3. Students shall develop their abilities to become self-sufficient individuals.
4. Students shall develop their abilities to become responsible members of a family, work group, or community, including demonstrating
effectiveness in community service.
5. Students shall develop their abilities to think and solve problems in school situations and in a variety of situations they will encounter in life.
6. Students shall develop their abilities to connect and integrate experiences and new knowledge from all subject matter fields with what they have
previously learned and build on past learning experiences to acquire new information through various media sources.
Academic Expectations
• Students will use reference tools such as dictionaries, almanacs, encyclopedias, and computer reference programs and research tools
such as interviews and surveys to find the information they need to meet specific demands, explore interests, or solve specific problems.
• Students will make sense of the variety of materials they read, observe, and hear.
• Students will use mathematical concepts and procedures to communicate, reason, and solve problems.
• Students will organize and classify information through an understanding of terms defined in this course
• Students will use appropriate conventions and styles in their written work to communicate ideas and information to different audiences and
for different purposes.
• Students’ oral communication will incorporate appropriate forms, conventions, and styles to communicate ideas and information to
different audiences and different purposes.
• Students will use of technology to collect, organize, and communicate information and ideas.
• Students will understand scientific ways of thinking and working and use those methods to solve real-life problems.
• Students will identify, analyze, and use patterns such as cycles and trends to understand past and present events and predict possible
future events.
• Students will identify and analyze systems and understand how their components work together or affect each other.
• Students will use and scientific models and scales to explain the organization and functioning of living and non-living entities and predict
other characteristics that might be observed.
• Students will understand that under certain conditions nature tends to remain the same or move toward a balance.
• Students will understand how living and nonliving things change over time and the factors that influence the changes.
• Students will understand number concepts and use numbers appropriately and accurately.
• Students will understand various mathematical procedures and use them appropriately and accurately.
• Students will understand space and dimensionality concepts and use them appropriately and accurately.
• Students will understand measurement concepts and use measurement appropriately and accurately.
• Students will understand mathematical change concepts and use them appropriately and accurately.
• Students will understand mathematical structure concepts including the properties and logic of various mathematical systems.
• Students will understand probability and use statistics appropriately.
• Students will understand the democratic principles of justice, equality, responsibility, and freedom and apply them to real-life situations.
• Students will accurately describe various forms of government and analyze issues that relate to the rights and responsibilities of citizens in
a democracy.
• Students will observe, analyze, and interpret human behaviors, social groupings, and institutions to better understand people and the
relationships among individuals and among groups.
• Students will interact effectively and work cooperatively with the many ethnic and cultural groups of our nation and world.
• Students will understand economic principles and are able to make economic decisions that have consequences in daily living.
• Students will understand, analyze, and interpret historical events, conditions, trends, and issues to develop historical perspective.
• Students will recognize and understand the relationship between people and geography and apply their knowledge in real-life situations.
• Students will present works of art convey a point of view.
• Students will analyze and reflect on their own and others' artistic products and performances using accepted standards.
• Students will gain knowledge of major works of art, music, and literature and appreciate creativity and the contributions of the arts and
humanities.
• In the products they make and the performances they present, students will show that they understand how time, place, and society
influence the Arts and Humanities such as languages, literature, and history.
• Students will demonstrate skills that promote individual well-being and healthy family relationships.
• Students will evaluate consumer products and services and make effective consumer decisions.
• Students will demonstrate the knowledge and skills they need to remain physically healthy and to accept responsibility for their own
physical well-being.
• Students will demonstrate strategies for becoming and remaining mentally and emotionally healthy.
• Students will demonstrate the skills to evaluate and use services and expectation resources available in their community.
• Students will perform physical movement skills effectively in a variety of settings.
• Students will demonstrate knowledge and skills that promote physical activity and involvement in physical activity throughout their lives.
• Students will use strategies for choosing and preparing for a career.
• Students will demonstrate skills and work habits that lead to success in future schooling and work.
• Students will demonstrate skills such as interviewing, writing resumes, and completing applications that are needed to be accepted into
college or other postsecondary training or to get a job.
• Students will use critical thinking skills such as analyzing, prioritizing, categorizing, evaluating, and comparing to solve a variety of
problems in real-life situations.
• Students will use creative thinking skills to develop or invent novel, constructive ideas or products.
• Students will organize information to develop or change their understanding of a concept.
• Students will use a decision-making process to make informed decisions.
• Students will use problem-solving processes to develop solutions to complex problems.
• Students will connect knowledge and experiences from different subject areas.
• Students will use scaffolding to acquire new knowledge, develop new skills, or interpret new experiences.
• Students will expand their understanding by making connections to new paradigms, skills, and experiences COURSE PHILOSOPHY Calculus is both a product of the Industrial Revolution and a tool which enables technological changes to occur. Calculus is the study of continuous change which occurs in small increments. It has its own set of very logical principals which flow from one to the other.
It is both logical and well-organized, and is a total system for studying change. It is the tool of science and technology, of engineering, business, industrial processes and design. It is the source of the tools which enable us to quantify and categorize. It allows students to think clearly, to select a course of action and to verify the results. COURSE DESCRIPTION AP Calculus is an honors course designed for the above average math student. It contains topics studied in a typical introductory college level course. The course content and curriculum is guided by the College Board’s description of their ideal AP AB course. The instructor should have attended a summer institute run by the College Board. Students may, if they wish, take the AB Calculus Advanced Placement Exam in early May. If students scores a 3 or better, they will receive AP credit for the course on their transcripts. If students do not take the exam, the high school will list honors calculus on their transcripts. The use of graphing calculators is an integral part of the course. Course Prerequisites A recommendation from student’s math teacher. Successful completion of Honors pre-calculus , attaining an 85 average, or successful completion of Academic pre-calculus or IMP III, attaining a 90% average Course Resources Larson, Hostetler and Edwards. Calculus of a Single Variable. Houghton Mifflin. 1998 Lederman. Multiple Chioce & Free Response Questions in Preparation for th AP Calculus (AB) ExaminationTI 84 Plus calculator from Texas Instrument Eighth edition. D&S Marketing Systems Inc. 2003 TI 84 instruction Manual from Texas Instrument
Standards: NJCCCS/CEEB Requirements
Students who are entering Calculus have, as a prerequisite, passed the HSPA in, preferably, the advanced proficient category. Therefore they have theoretically mastered the NJCCCS in mathematics. For the AP AB Calculus course, the instructor needs to be very cognizant of the requirements of the College Board, the organization which designs the AP exams. There is an excellent website which they run, http://apcentral.collegeboard.com This website will give the instructor several sample syllabi for the AB course. Thes syllabi may be downloaded and printed and serve as a helpful supplement to the AP teacher. They also publish a manual each year, for the current sum of $15.00, which serves as a guide to the AP teacher. There are numerous day and half day conferences held during the fall for AP teachers who have completed the one week summer course in AP Calculus. Whoever teaches this course should have completed one of these summer institutes. The school will be inundated in the spring with mail from many local sites, such as University of Delaware. Assessment/Testing Rubric
Superior Proficient Non-Proficient
Honors 40% Traditional 50% Written 10% Other
50% Traditional 40% Written 10% Other
70% Traditional 20% Written 10% Other
College Prep
50% Traditional 40% Written 10% Other
60% Traditional 30% Written 10% Other
80% Traditional 10% Written 10% Other
Academic 60% Traditional 70% Traditional 90% Traditional
30% Written 10% Other
20% Written 10% Other
5% Written 5% Other
Note: ‘Authentic Assessments’ will be categorized under either the “Traditional” or “Other” Categories
Workload Distribution for Honors Calculus Rubric Homework: 30-40 minutes per day Supplemental Work-Worksheets, multi-day projects, cooperative efforts, note taking Multi-Media projects-proficiency using the graphing calculator as a tool, use of the library to research newer topics and history of calculus and its creators Reading-10 minutes for each new section to read and take notes on it. (1-2times/wk)
Instructional Methods
The instructional methods used in calculus are similar in some ways to all good instruction; however, they are on a higher level. Students are expected to utilize the instructor as a coach, while the student does most of the wok. Students are encouraged to interact with each other, to the point of working on problems together. As in all good instruction, there is daily feedback to assess student learning, as well as quizzes for checking assimilation of short term knowledge, and test which cover months of work. There are projects for the students to do which is calculus related, but fun for the students. There are always opportunities for students to relearn material which may have confused them
Instructional Outcomes
Participation in the AP Calculus course may mean one of three outcomes for the student:
1. If the student takes the AP exam and receives a 3 or higher, that students will be given advanced placement credits, and placement in mathematics
2. If the student takes his or her college placement exam, and receives the requisite grade as determined by the college, that
student may receive advanced placement in a math course, but no additional college credit.
3. If the student is placed in an introductory college calculus class, that student will usually receive a very high grade, with a minimum of effort, in his or her freshman.
Subject : Calculus
Content/Topics College Standards Key Skills Enduring Understandings Essential Questions Assessments
Support Activities/Experiences
Precalculus Functions (Summer Assign)
The nature of functions, odd and even functions
What is a function? An even or an odd function?
Worksheet on Functions; test class discussion
on precalculus Time= Limits
Finding limits of various types of functions
Concept of a limit,Graphs may be used; in polynomials: lim f(x)as x c=f©
How do we find the limit of polynomial functions?
practice problems
boundaries and/or limits on actual entities such as room occupants
Evaluating limits; limits which fail to exist
Some functions have limits; others have no boundaries
How do we determine if a limit exists or not?
drawing diagrams
limitless entities, such as, possibly, the universe
Limits can be evaluated. How do we evaluate limits?
quiz on limits (1.1-1.2)
Using the calulator as an aid
Limits can be evaluated algebraically
How do we use algebraic techniques? sample problems
recall division of rational functions, rationalization techniques
Limit Strategies Limits can be evaluated using other functions
How do we use the "Squeeze Theorum? sample problems
Limit requirements and continuity are related but different How do we use limits to evaluate
continuity? drawing diagrams
Students find two functions which will squeeze a third in an interval; Students evaluate each other's diagrams
There are three criteria which need to be met in order for a function to be cont.
How can we determine if a function is continuous? Quiz 1.3
students discern that all polynomials are continuous
The Intermediate Value Theorum requires continuity, and helps us find zeros
How do we use the IVT to demonstrate that a function has a zero within a given interval?
practice problems
Informal "proof" via diagrams
Continuity Functions can "go to" zero, infinity, or a specific number
How do particular functions behave as x approaches infinity?
group problems, quiz 1.4, writing assignment, test on limits
students create a summary of the concept of limits; the best one is chosen and reproduced for class to study
time=5 days End Und p. 1
Limits
Content/ Enduring Essential Questions Assessments Connected
Topics Understanding Co-Curricular Support Key Skills Activities / Experiences
Tangent Lines and Slopes
As a secant line becomes shorter, one approaches the tangent line
How did calculus help find the slope of a line tangent to a curve at a single point?
Board problems; written sheet for students to study
Students draw actual secant lines and have them become shorter until they are tangents.
The definition of a derivative
Develop the 4 step process, both in general and at a point.
What is the limit of f(x +delta x) -f(x)/delta x as delta x approaches 0?
worksheet on f(x+delta x); practice both single point and general method
Have students try to make generalizations to simplify process
Differentaibility and Continuity
Develop the shortcut known as the power rule
What is the connection between differentiability and continuity?
practice on both 4 step process, checking with shortcut
What other shortcuts have you learned in math? (Syn div)
The derivative of Sine and Cosine
How can we use the 4 step process to find the derivatives of trig functions
Use the 4 step process on sin x, then practice
Have students perform a similar exercise using the cosine function
The four step process and trig identities are used here
Second and Third Derivatives
The deriv of a function is a function and may be applied to the real world
What is the meaning of a second or third derivative?
discussion , referral to physics, practice
Develop velocity as the deriv of position and accel as the deriv of velocity
Standards Guiding
Dropping and Throwing Things up in the air
Velocity, and position all stem from the force of gravity, which is -32 ft/sec/sec
What is the relationship between position, velocity and accelaeration?
Two worksheets on "physics" problems
Have students who have taken physics relate how they do similar problems without the aid of calc.
Products and Quotients
There are rules for taking the derivatives of products and quotients
How can we take the derivative of a product or a quotient?
Practice problems, big quiz on taking derivatives
Have student do problems with and without the rules to verify the results are same.
National Content/ Enduring Essential Questions Assessments Connected Standards Guiding Topics Understanding Co-Curricular Support Program: AP Curricul- Key Skills Activities / Experiences Lum Derivatives (cont) Trig Functions One can use the prod &
quot rules, & trig relats to obtain the derivs of the 4 remaining trig functions
How can we take the derivs of tan x, cot x, sec x,& csc x?
practice, memorization drills Have students review sin(a+b) from trig in order to determine derivatives
Intro to applications of the Derivative
Derivatives are useful How can we use this derivative?
worksheet on applications; writing-Three word definition for the derivative?
Have students suggest other ways we can use the derivative, based on its definition
Simplifying Derivatives
Algebraic skills are needed to simplify, especially factoring
How can we make these complex answerssimpler?
Worksheet of products and quotients which require simplifying
Practice on factoring out fractional exponents
The Chain Rule Composite functions
require the use of a special rule, the Chain Rule
How do we take the derivative of a composite function?
quiz on product and quotient rule, four worksheets on the chain rule
Review composite functions from precalc
Implicit If both X and Y affect What is implicit quiz on chain rule, Discuss simplifying and
Differentiation the curve as dependent variables, we need to use implicit differentiation
differentiation, why do we need it, and how do we do it?
worksheets on implicit differentiation
using original function to express y' and y"
Derivatives Related Rates Time can be the independent variable in many situations
How do you use implicit differentiation when time is the independent variable?
Discussions, sample problems take-home project
Students try to find situations where time is the variable, and set it in a mathematical context
End Und p.3
National Content/ Enduring Essential Questions Assessments Connected Standards Guiding Topics Understanding Co-Curricular Support
Program: AP Curricul- Key Skills Activities / Experiences Lum Using the Derivative Extremes in a
closed interval Horizontal tangent lines can indicate the existence of maxes and mins in a curve
How can we use the first derivative to assist in graphing a curve?
Quiz on related rates; interpreting graphs to see if an absolute max or min may be determined
Have students discuss the implications of having an open versus a closed interval
Rolle’s Theorum If f(a) is one sign and
f(b) is the opposite sign, and if the curve is continuous, then there is a zero in(a,b)
How does Rolle’s Theorum help us and how do we use it?
Board problems verified by graphing calculator work
Discussion of functions such as profit and the need to know the break-even point
The Mean Value Theorum
If f(a) is not equal to f(b), but the curve is continuous, at some point f’x = the slope of the secant line
How does one use this new � heorem/
Practice problems aa discussion on why this � heorem would be of value to anyone
The First Derivative Test
F’x dives the graph into increasing and decreasing areas. F’x=0 lets us find maxes and mins, and helps us draw the curve accurately
What facts about the curve does the first derivative divulge?
Quiz on Rolle’s thm and Mean Value Thm; practice sheets to work on cooperatively and check with the gc
Write: “Why do you still need to use calculus techniques when you have a graphing calculator?”
The Second Derivative Test
F’’x allows us to find points of inflection, where the concavity changes
What does “changing the concavity” mean and imply?
Have students graph the derivative of a derivative and look for trends
Write: “What value is there to knowing where the concavity changes?”
The first and second derivative test together
Between the two tests, we can draw the graph of a function quite accurately
How do we put these two tests together to graph accurately?
Plenty of practice, inc. 2 worksheets
Watch a video where the skilss are pu together and slowly explained
End Und p. 4
National Content/ Enduring Essential Questions Assessments Connected Standards Guiding Topics Understanding Co-Curricular Support Program: AP Curricul- Key Skills Activities / Experiences Lum Using Derivatives Behaviour at
infinity-Horizontal Some functions have horizontal or slant
How does one determine if a function has a horizontal
Quiz on first and 2d derivative test.
Write, “What is the significance of a horizontal
and Slant Assymptotes
� symptote� as x goes to infinity
� symptote? A Slant � symptote?
� symptote in a real function?”
Curve Sketching All the points learned to
date allow us to draw a very accurate picture of a function
How does one apply what we have learned thus far to various polynomial functions?
Project using symmetry, intercepts, maxes, mins, zeros, inc, dec, pts. Of inflection & assymtotes
Look for curves in journals and newspapers that display some of these characteristics
Optimization The analysis of a
function allows us to find ways to maximize or minimize variables
What would one wish to optimize, to minimize? How is calculus used in business?
Have students define marginal revenue, marginal profit and other business terms
examine real situation-profit, revenue. Show video on subject
Newton’s Method There are methods to
determine where a function crosses the x axis
What did Newton figure out? Students will work on problems than check with gc.
Internet- brief paragraph on newton’s contributions to math
Differentials Take a differential and
understand how it is different from a derivative
What is a differential? How is it used to understand error in measurement?
Students will do a little measurement experiment
Studentys will relate their experiment to applications in business. We’ll discuss the Japanese vs American business model
End Und P. 5
National Content/ Enduring Essential Questions Assessments Connected Standards Guiding Topics Understanding Co-Curricular Support Program: AP Curricul- Key Skills Activities / Experiences Lum Integration Antiderivatives Integration is the
reverse process from differentiation
What is an antiderivative or an integral?
Work from known derivative functions backward to the appropriate integral
Write, “Why is the +C important?”; “What is meant by a family of integrals?”
Series and Find the nth term of a How do we find if a series Worksheet Recall work from pre-
Sequence, Sigma Notation
sequence, the sum of a series
has a finite sum? calculus on the subject
Upper and Lower Sums, Area
We can take the area of irregular shapes by using inscribed and circumscribed rectangles
How can rectangles help us take the area of an irregular object?
Practice on drawing and calculating inscribed and circumscribed rectangles
Discuss what happens as we increase the number of rectangles we are using
Reiman Sums and the Definite Integral
Introduce Definite Integrals as the area under the curve
What happens when we use an infinite number of rectangles in taking an irregular area?
Group Practice Find an area using first 4, then 8, then 16 rectangles and compare results
The Fundametal � heorem of Calculus; Average value of f(x)
An accumulation function can be correlated to the area under the curve
What is this theorem and how do we use it?
Board Problems; quiz on previous section
Discuss,”Why do you think this theorem is called the fundamental theorem of Calculus?
Pattern Recog., Definite Integral, Change of Variable
The chain rule can work in reverse, but one must have du/dx
How do we work backwards to recognize a function from its derivative?
Practice chgani rule going forward to understand how it works in reverse
Discuss changing the limits of integration once one determines u
The Trapezoidal& Simpson’s Rule and Error Analysis
Trapezoidal&Simpson’s Rule are pre-Calc,&are used to calculate error
Demonstrate Trap rule, simson’s, and compare to inner and outer rectangles
How do we use integration to calculate degree of error?
Discvuss how we can pick the degree of error we can tolerate and work back to
possible values Enduring Understanding p. 6
Course Title: Calculus
National Content/ Enduring Essential Questions Assessments Connected Standards Guiding Topics Understanding Co-Curricular Support Program: AP Curricul- Key Skills Activities / Experiences Lum Logarithmic and Trigonometric Functions
Natural logs, e, differentiating ln u
Logs are exponents, follow their rules, have � erive� , & help take � erive� of funcs
What’s the � erive of ln u? Why do we need logs? How do they assist us in simplifying problems?
Worksheet on logs and exps; quiz on previous work, sample problems
Discussion of history of logs and exps, why we needwhen we have gcs, how they are useful.
Integrating u’/u Introduce ln x as the
area under the curve of What does ln u represent? How do we determine u and
quiz on differentiating ln u; exercises with overhead
When would we use this new integral?
1/t from 1 to x. u’? Inverse functions Some functions have
inverses with special charateristics
How do you determine if a function has an inverse and how do you find it?
Quiz on integrating u’/u; worksheet on inverses
Write, “What is the relationship between the domains and ranges of a function and its inverse?
Ex, concept an development
ex is its own derivative and hence its own integral
How can a function be its own derivative? Why is it its own derivative?
Board problems, practice problems, quiz on inverses
How is the gc going to help with the integrals?
Bases other than e We can work in others
bases, such as 10, binary, 16.
The gc has baes e and base 10; how do we calculate in base 4 or 6 etc?
review sheet and quiz on e and ln u; skill drill on other bases
Discussion of base 2 (binary) and base 16 and their applications
Growth and Decay Many natural processes
follow growth according to base e.
How are e and ln u important in the real world?
Problems using e such as radioactive decay, Dopler and Richter scale, decibel levels
Change in the level of intensity of one unit produces a tenfold increase in the object
Differential Equations-Sep of Variables, Slope Fields
Use Cex as soln of y’ + y=0, use a slope field, solve a differential equation
What is a differential equation? What is a slope field?
Draw slope fields and interpret others, quiz on applications of e
Internet research- What is a slope field and how do we use them?
End Und p. 7
Course Title: Calculus
National Content/ Enduring Essential Questions Assessments Connected Standards Guiding Topics Understanding Co-Curricular Support Program: AP Curricul- Key Skills Activities / Experiences Lum Logarithmic and Exponential Functions
Inverse Trig Functions
Trig functions will have an inverse if limit � omain & range; they have derivs
What are the inverses of the six trig functions?
Create review of integration to date; quiz on differentials
What types of problems require the use of inverse trig functions and how does the gc help?
Summary of integration
The derivatives of the inverse trig functions have antiderivatives
How do I use completing the square to assist in this process?
Test on all of integration to date
Discussion-we now have the basics, from hereon we learn applications and techniques of integration
National Content/ Enduring Essential Questions Assessments Connected Standards Guiding Topics Understanding Co-Curricular Support Program: AP Curricul- Key Skills Activities / Experiences Lum Applications of Intergration
Area between two curves
It’s an of application of definition of int
How can we find the area between 2 functions?
Practice problems Have students create a problem
Applications of Integration Discs, Washers & Solids with known X sections
We can use calculus to find the volume of irregular objects
How do we use calculus to find the volume of an object?
Practice with shapes and rotating them
work with party decorations, cheese slices, baloney todevelop theory
Arc length and Surface of Revolution
Use calculus to find the arc length of a curve and the surface area of a volume
How can we find lengh and/or surface area if we don’t have straight sides?
Practice problems, quiz Work with lengths of string for arc length
Work done by a variablw force
work is the product of the function for the force x the displacement
What is work, and how do we find it if the force is not constant?
Problems with real situations, trying to determine the force function
Review � otential vs kinetic energy, constant vs variable force
At this point, all the material required for the AP exam has been covered. Depending on how many students are taking the exam, it is hoped thata few days of intensive review can fit in here. All along, AP questions should have been thrown into the problem mix, using released exams, a review book, and the AP Central website
The Shell method, comparison to disc
Certain situations lend themselves to s different method, the shell method, esp. if rotating about y axis
What’s the shell method? How do I decide which method to use?
Worksheet on both methods with student choice, class generalizations, individual project on all methods
Discussion of how a pearl is formed as an example of how the shell method works
Course Title: Calculus
National Content Topics/Key
Skills Enduring Understandings
Essential Questions Assessments Connected Co-curricular Support Activites/Experiences
Standards Guiding Program: AP Curricul- Lum Techniques of Integration L'Hopital's Rule We can take the limit of
a quotient of functions that are not algebraic
What happens when algebraic techniques fail to allow us to take limits of functions?
Practice determining what is an indeterminate form and how to manipulate to get one
Compilation and Assimilation of Differentiation
There are 22 rules for differentiation and that is all.
Now that I can Integrate, why do I need to differentiate
worksheet on 50 derivatives, test on derivatives, test on applications of derivatives
Compilation and Assimilation of Integration
There are some rules know for integration, but there are many more.
What about all the integrals that I still cannot do? What do I do with them?
worksheet on 70 integrals, test on integrals, test on applications of integrals
Integration by Parts A large portion of
calculus is learning techniques of Integration
How do I determine which is the function and which is the derivative?
worksheet, cooperative learning, practice in picking u and v'
Subject/Math /Course/Calculus Content / Skill Assessment
1.What is a function? What are even or odd functions? Unit:The Cartesian Plane and Functions 2. Review chapter P Days to Complete:5
What are the concepts which we need from precalculus?
sample problems, pretest and test on Chapter P
1.1 What is calculus?1.2 Use graph, diagrams to find the limits of polynomial functions
1.3 Evaluate limits w. algebra; use rationalization techniques, squeeze theorum
Unit:Limits
1.4 Develop concept of continuity;elicit concept that limit requirements and continuity requirements are not the same. Develop three criteria for continuity of a function.Be able to distinguish a removable vs a non-removable discontinuity. Use continuity and the Intermediate Value Therum to check for zeros of a function
Days to Complete:12
How do we find and use limits?
1.5 Examine the behavior of functions as they approach infinity; categorize as going to zero, a specific number of infinity
quiz 1.1-1.2, sample problems, quiz 1.3, practice problems, group problems, writing assignment, quiz 1.4, test on chapter 1.
Unit: Derivatives
How and why do we differentiate?
2.1 Begin with a secant line, and let it get smaller until it becomes a tangent line. Develop the four step process to find
Read and Notes,worksheet on f(x + delta x), homework, board work, worksheets on
derivatives, using both the general form and the point form. Then develop the pattern for the shortcut
2.2 Use the shortcut to find derivatives of polynomials; then use the definition of a derivative to find the deriv of sine and cosine. Find the derivative of a derivative, and apply it to the real world usin position, velocity and acceleration.
2.3 Study the proof of the product and quotient rules and apply them. Use the product and quotient rules to develop the derviatives of tan, cotan, sec, csc. Demonstrate the use of the derivative in a variety of situation. Practice factoring out common factors
2.4 Introduce the Chain Rule, its theory and uses
2.5 Introduce Implicit Differentiation, how and when and why to use it
Days to Complete:33
2.6 Study related rates with diffentiation with respect to time, and apply to many real situations
distance, velocity, and acceleration, falling objects, products and quotients, applications of derivatives, 4 worksheets on the chain rule, related rates problems; quizes 2.1-2.2, 2.3, 2.4; review of 2.1-2.5, test on 2.1-2.5, quiz on 2.6
3.1 Find the maxima and minima of a closed interval 3.2 Use Rolle's theorum and the Intermediate Value Theorum
3.3 Use the first derivative to find critical numbers and where functions are increasing or decreasing
3.4 Find the second derivative of a function and use it to determine concavity; put the first and second derivative tests together to completely analyze and graph a function.
3.5 Find vertical, horizontal and slant asymptotes for rational functions
3.6 Put all skills together to graph functions 3.7 Use derivatives to optimize functions 3.8 Use Newton's method to find x intercepts 3.9 Find differentials and use them to analyze error 3.10 Apply calculus to business situations
Unit:Applications of Derviatives/Days to Complete-23
What are the ways in which we use the derviative?
Read & Notes, homework, group problems, worksheets on 1st and 2d derivatives and graphing,video, quizes 3.1-3.2, 3.3-3.4, 3.5, test on chapter, graphing project
4.1 Take Antiderivatives, find specific member of a family.
4.2 Discuss sequence and Series in order to find Upper and Lower Sums and Areas
Unit:Integrals
What are Integrals and how do we find them?//Days to Complete-
4.3 Develop Reiman sums and the Definite Integral,
Read and Notes, various worksheets, including 50 Integrals, Quizzes 4.1, 4.2, 4.3, 4.4 and two day chapter test. Midterm included in this time period
properties of Integrals and their evaluation
4.4 Use the first and Second Fundamental theorums of Calculus, find average value of functions
4.5 Develop use of u and du, change of variavle, changing the limits of Integration
Days to Complete:30
4.6 Use the Trapezoidal Rule, Simpsons Rule and perform Error Analysis
5.1 The Natural Logarithmic Function and Differentiation 5.1 The Natural Logarithmic Function and Integration 5.3 Inverse Functions 5.4 Exponential Functions: Diffentiation and Integration 5.5 Bases Other than "e" and Applications 5.6 Growth and Decay 5.7 Differential Equations 5.8 Inverse Trigonometric Functions and Differentiation
Logarithmic and Trigonometric Functions
5.9 Inverse Trigonometric Functions and Integration Days to Complete:37
How do we use integrals?
6.1 The Area between Two curves
6.2 Volume: Disks, Washers, and Solids with Known Cross Sections
6:3 Volume: The Shell Method and Comparisons with Disks 6.4 Arc Length
Unit:Applications of Integration
6.5 Surface of Revolution Days to Complete:20
What are various techniques of integration?
6.6 Work 7.1 Summary of Derivatives 7.2 Summary of Integrals 7.3 Integration by Parts
Unit: Techniques of Integration
7.4 L'Hopital's Rule Days to Complete:
What techniques can we use to assist in integration?