Course Overview
Ms. Ornella Bascuñán
The main objective of this course is to follow the guidelines of AP Calculus and have students appreciate the beauty of Calculus. Students will receive a strong foundation that will give them the tools to succeed in future mathematics courses.
Textbook: James Stewart “Single Variable Essential Calculus, Early Transcendentals”, edition 2007
Goals
Students should be able to: • Work with functions represented in a variety of ways numerical, graphical, analytical or verbal and being able to see the connection between these representations. • Understand the meaning of derivatives as rates of change and local linear approximation. • Understand the meaning of definite integral as net accumulation of change and Riemann sums. • Understand the Fundamental Theorem of Calculus and see the connection between derivatives and the definite integral. • Use technology to help solve problems, verify conclusions and interpret results. • Explain and communicate results orally and in written sentences, making sure the answers to their problems are reasonable and include units of measurement. • Model physical situations with functions, integrals or differential equations.
Course Planner
Chapters 1.1 1.2 1.3 1.4 1.5 1.6 2.1 2.2 2.3 2.4 2.5 2.6 2.7 3.1 3.2 3.3 3.4 3.5 3.6 4.1 Topics Functions and their representation A catalog of essential Functions The Limit of a function Calculating Limits using Algebra and estimating limits from graphs and tables. Continuity Limits involving infinity. Intermediate Value Theorem and Extreme Value Theorem. Derivatives and Rate of Change The Derivative as a Function Basic Differentiation Formulas The Product and Quotient rule The Chain rule Implicit Differentiation Related Rates Exponential Functions Inverse Functions and Logarithms Derivatives of Logarithmic Functions Exponential Growth and Decay Inverse Trigonometric Functions Indeterminate forms Maximum and Minimum Values Timeline in days 4 2 3 7 4 4 4 2 3 3 3 3 3 2 3 2 3 2 2 4
4.2 4.3 4.4 4.5 Chapters 4.6 5.1 5.2 5.3 5.4
The Mean Value Theorem. The Extreme Value Theorem. Derivatives and the shapes of graphs Curve sketching Optimization Problems Topics Antiderivatives Areas and distances. Riemann sums The Definite Integral Evaluating Definite integrals The Fundamental Theorem of Calculus Average value of a Function Mean Value Theorem The Substitution rule Areas between curves Volumes of Solids of Revolution Applications of Integral Calculus to Physics and Engineering Differential Equations. Slope fields
2 3 3 4 Timeline 3 5 2 4 3 2 1 3 4 5 4 5
5.5 7.1 7.2 7.5 7.6
Chapter 1 • Discusses the basic ideas concerning functions, their graphs (which with the use of graphing calculators are easy to produce) and the different ways of representing functions. • We look at the different types of functions and use them as models of real-world phenomena. • Introduces the concept of Limits in an intuitive way by observing the graph of functions and the corresponding tables associated with it. • Finding Limits by using Properties. • Describes asymptotic behavior in terms of Limits approaching infinity. • Analyzes and applies The Squeeze Theorem. • Continuity is introduced first in an intuitive way to follow the understanding in terms of Limits
Chapter 2 • Presents a special type of Limit, called a Derivative that we use when we want to find the slope of the tangent line, a velocity, or any instantaneous rate of change. • Tangent line to a curve at a point and local linear approximation. • Derivatives are presented numerically, graphically and analytically. • Derivatives are defined as the limit of the difference quotient. • Differentiable functions are defined. Also, how can a function fail to be differentiable? • Basic Differentiation formulas for derivatives of sums, products and quotients of functions. • Derivatives of power, logarithmic, exponential, trigonometric, and inverse trigonometric functions. • Implicit Differentiation and Chain Rule. • Related rates. Interpretation of the derivative as a rate of change under different contexts, like speed, velocity, acceleration. • Equation involving derivatives. Word problems are translated into equations and vice versa. Chapter 3 • The common theme that links exponential, logarithmic, and inverse trigonometric functions is that they occur as pairs of inverse functions. • Properties of these functions are investigated, and their derivatives computed. • Exponential growth and decay in biology, physics, chemistry, and other sciences are studied. • Indeterminate forms and L’Hospital rule are discussed( even though is not a requirement for AP Calculus AB )
Chapter 4 • Applications of differentiation in greater depth. Here we learn how derivatives affect the shape of the graph of a function. • How derivates help us to locate maximum and minimum value of a function.(global and relative) • Analysis of curves, monocity and concavity. • Special characteristics of graphs of f, f’ and f’’ Increasing/Decreasing functions and the relationship with the graph of f’ and f’’ .Concavity. • The Extreme value Theorem. • Fermat’s Theorem. • Finding Critical numbers and points of Inflection. • Rolle’s Theorem. • The Mean Value Theorem... • Guidelines for Curve Sketching. • Optimization problems. • First Derivative test for Absolute Extreme Values. • Antiderivatives. Creating tables of antidifferentiation formulas. • Rectilinear motion. Using antiderivatives for acceleration, velocity and position problems.
Chapter 5 • Area and distance problems are used to formulate the idea of the Definite Integral. • Calculating areas under the curve by using Riemann sums. • Use of Riemann sums (using right, left, midpoints as evaluation points. Use of trapezoidal sums) to approximate definite integrals of functions. • Properties of the Definite Integral. • Evaluating Definite Integrals. • Indefinite Integrals. Tables of Indefinite Integrals. • Net change as the integral of a rate of change. • Finding distances and displacement. • The Fundamental Theorem as a connection between differential and integral Calculus. • Use of the Fundamental Theorem to evaluate definite integrals and to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined. • Average value of a function. • Integrals of symmetric functions. • Applications of integrals to model physical, biological, or economical situations.
Chapter 7 • • • • • • • • Finding the area between curves. Finding the volume of a solid of revolution (disc, shell, and cross-sections). Use of integral calculus to calculate work, force due to water pressure, and centers of mass. Initial conditions for differential equations. Separable Differential Equations. Modeling population growth Direction fields or slope fields. Sketching slope fields for given differential equations.
After the AP Exam • • • Integration by parts Partial Fractions Newton’s Method
Technology. Teacher and students use TI 84 Plus Silver Edition and TI 89 Titanium calculators. PowerPoint presentations for most of the lessons.
Group Work Activities These group activities are taken from the Instructor Guide for Stewart’s Calculus Concepts and Contexts 3E.There is one for every section.( I included some examples on the hard copy of this document ). An example of one of these group activities is: “For each of the given integrals in Problem 1-3, first sketch the corresponding area, and then approximate the area using right and left endpoint approximations and the Trapezoidal rule, all with n=4. From your sketch alone, determine if each approximation is an overestimate, an underestimate, or if there is not enough information to tell. Share your results with your group and make sure every member of the group agrees with the conclusions. A paper shall be written with the group conclusions explaining with details the results and given to me. Transparencies will be provided to present and discuss your results with the rest of the class.” Students work in groups of 3 or 4 , they talk about the methods they use to solve the problems and write a single paper with the results, making sure they use correct terminology and every step is justified . The solutions to the problems are then presented to the whole class.
Classroom Management Plan
Commitment to excellence in everything we do: academics, activities and citizenship. Single Variable Essential Calculus, Early Transcendental, first Text
edition,James Stewart, Thomson/Brooks/Cole,2007
Attendance
You are expected to be in class every day. If you miss any assignment while you were absent, it is your responsibility to make it up in a reasonable amount of time (e.g., 2 days for 1 absence). If absent during a Test, you should report to this room to take it on the day of your return at 7:15 a.m. Your absence has to be justified. You will be considered Tardy 2 minutes after the start of each class period. Homework has to be done on a daily basis and it will require a minimum of 1 hour. The more you practice, the better you get at it. Collected homework will count as a Quiz grade.
Tardy Homework
Tests/ Home There are 3 to 4 tests each marking period. They count as 25% of your final grade. Assignments
Home Assignments are graded on the number of problems completed. A student earns full credit for showing work on all the problems. Students, who do not try all the problems, receive partial credit. Home Assignments count for 25% of your final grade.
Quizzes
Quizzes account for 25% of the final grade. Quizzes are given daily. They are similar to the multiple-choice questions that are on the AP Exam. The questions are usually related to the topics being taught, but may include review topics. If a student misses a quiz, (with a justified absence), that grade is not included in the average. Being
late or cutting class will get you a 0.
Quarterly Assessment Extra Help Discipline
This exam is 25% of your final grade and it will cover material from the entire quarter. It may also contain material from past quarters. I am available before school at 7:30 am every day, and also after school by appointment 1.First Offense: A remainder that unacceptable behavior has occurred 2. Second Offense: Repeated behavior will mean an Office Card. 3. Third Offense: A call will be made home to parents for intervention. **No food or drinks are allowed in the classroom!!
You will be expected to complete academic work both independently and cooperatively in a productive manner.