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Calculus AB Instructor: M Challender Text Book: Calculus, concepts and applications. Paul A. Foerster, 1998 Key Curriculum Press. We will cover almost all the material in the first ten chapters of the text in the sequence outlined in the attached syllabus. Broadly speaking, the topics will be functions, graphs and limits. The study of continuity as a property of functions, the concept of the derivative, application of the derivative and computation of basic derivatives is studied. Basic integrals and their application to differential equations and the fundamental theorem of calculus will be taught as well as numerical methods of integration. We will concentrate on learning the material in preparation to successfully challenge the College Board Calculus AB exam. Homework will be one part of your grade; quizzes will make up the second part, and the final is the third. You must pass all the components of the course to earn a letter grade of C or better. There is no remediation of the material in this course. The grading scale for collected homework assignments is on a 4-point scale. Work turned in late will earn no more than half credit. All assigned work must be completed in order to earn a passing grade. It is expected that the homework be completed correctly. Collaboration on homework is encouraged but remember that understanding how someone else does a problem does not necessarily mean that you will be able to reproduce the solution. The purpose of my reading the homework assignments is to better prepare you to write complete solutions to problems. Quizzes over the homework material will be graded using a rubric style system in preparation for the Calculus AB exam. Some of the quiz questions will be open ended and require many steps similar to the homework problems. There will also be multiple- choice questions requiring you to recall previously learned concepts. Your understanding of the homework material will be determined by the quiz questions that are very similar to the homework problems. You are expected to pass the quizzes on a regular basis. The final exam will be very similar to the College Board Calculus AB exam. You will know in advance what the questions will cover. We will prepare for this exam over the course of several class meetings. One way to master math is to do many problems. You should plan to spend about two or more hours of work out side of class for every hour in class. The time spent by each individual will vary. Problem assignments for each class day are listed in the syllabus. Certain parts of assignments will be collected. I suggest you do all the problems. Have assignments prepared for collection the first class meeting of the week following the week after they are assigned. If the material is presenting difficulties, ask questions. People are encouraged to work together. You can learn from each other, however group learning may only make you feel you understand. I will be available at 6:45 A.M. during the week for help and by appointment. This time is set aside to answer your questions; you should come prepared to get help. Apart from homework that will be collected as outlined above you will be doing problems in class sometimes with collaborators, please bring your book. There will be quizzes of about 10-30 minutes duration almost daily to help us assess how you are coming with the current material. Be prepared for the quizzes they will be over the most recently covered material. I will inform you of the material that will be covered on the next quiz, your input is valued. If you feel you need more practice in a certain area we will practice these concepts. On quizzes and exams collaborations cease, of course, and people do their own work. The material covered in this course is the Calculus material outlined in the College Board Calculus AB syllabus. I am confident that we can reach a reasonable level of mastery. It will take commitment from you. Since this is an advanced placement class, you are expected to work at the college level. If you do not understand a concept it is your responsibility to see to it that you do what it takes to learn the material. There is no remediation of assessments. Continued failure of assessments will result in a grade lower than you may be used to. An additional grade point is offered for this course. Students earning a B or better in the course should be able to score a 4 or 5 on the AB exam. Students earning a C or lower are encouraged to challenge the AB exam, but will not earn the additional grade point if their score is below a 3 on the AB exam. I will not be surprised if a student earning below a B passes the AB exam with a respectable score. Students earning a C are meeting the minimum level of expectation for the course. The purpose of this course is to produce highly prepared students for a college environment. Students successfully completing this course with a B or better will have demonstrated mastery the high school mathematics curriculum. All students will have an understanding of the effort that is required of them to meet the expectations of a very challenging curriculum. The knowledge about themselves and methods they use to acquire knowledge will help them meet the challenges ahead. The following is an outline of the problems you are to complete to meet the minimum expectations for the course. You are encouraged to work ahead, come to class prepared to ask questions regarding your concerns about particular concepts. Chapter 1 1.1 The concept of instantaneous rate. The idea of a limit, average rate, instantaneous rate of change and the derivative are introduced. Problem 1.1.1 requires the use of a graphing calculator in radian mode. Problems 1.1.1 and 1.1.2. 1.2 Rate of change is studied by using equations, graphs and tables. The meaning of derivative and the definition of limit are learned by approximating rate of change from graphs and tables. Problem 1.2.25 illustrates how to use the TI83 to solve problems related to rates of change in f(x). Problems 1.2.1 to 1.2.27. 1.3 The idea of integral as area under a curve. The meaning of definite integral is shown. Problems 1.3.1 to 1.3.13. 1.4 Estimating the value of a definite integral. The area under a curve is found using the trapezoid rule. Problem 1.4.7 requires that the student write a program to estimate a definite integral using the trapezoid rule. Problems 1.4.1 to 1.4.5 odd and 1.4.9 to 1.4.17 odd. 1.5 The formal definition of a limit. Indeterminate forms, step discontinuity, removable discontinuity are introduced. Problem 1.5.19 uses the Ti 83 to experiment with the limit of sin(x)/x as x0. Problems 1.5.1 to 1.5.19 odd. 1.6 Calculus journal. A selection of problems serves as examples to key concepts. Problems 1.2.14, 1.3.6, 1.3.10, 1.4.2, 1.5.12, 1.5.14, 1.5.16. 1.7 Chapter review. Problems R1 to R6 all. Chapter 2 2.1 Numerical approach to the definition of a limit. The formal definition of a limit is continued using an intuitive approach and introducing delta and epsilon values. Problems 2.1.1 to 2.1.3. 2.2 Graphical and algebraic approach to the definition of limit. The values of delta () and epsilon () and the limit L are calculated as well as estimated from tables and graphs. Problem 2.2.7 to 2.2.11 will need the table feature of the TI83 to confirm delta values for a given epsilon value. Problems 2.2.1 to 2.2.13 odd. 2.3 The limit theorems. The properties of limits are used in proofs, and explained why they are true. Indeterminate forms and undefined limits are discussed. Problems 2.3.1 to 2.3.25 odd. 2.4 Continuity. The definition of continuity, step discontinuity, one-sided limits, cusp, and asymptotes is studied. Problems 2.4.1 to 2.4.69 odd. 2.5 Limits involving infinity. Definition of limit as x goes to infinity and as limit goes to infinity. Problem 2.5.11 experiments with the limit as applied to an integral using the trapezoid program to conjecture a limit. Problems 2.5.1 to 2.5.13 odd, 2.5.12, and 2.5.14. 2.6 Intermediate value theorem and its consequences. The notion of continuous functions is reinforced and the idea of extreme values is exercised. Problems 2.6.1 to 2.6.14 all. 2.7 Chapter review Problems 2.7.R0 to 2.7.R6 all. Chapter 3 3.1 Graphical interpretation of the derivative. The connection to the slope of the tangent line in connection to the instantaneous rate of change is explored. Problems 3.1.1 to 3.1.9 all. 3.2 Difference quotient and one definition of derivative. The definition of the derivative as x goes to c is used. The idea of local linearity is explored using the TI83 zoom feature. Problems 3.2.1 to 3.2.20 all 3.3 Derivative functions. Symmetric difference quotient, forward and backward difference quotients, and the idea that x is a tolerance. Problems 3.3.5 and 3.3.6 require students to make a conjecture about the derivative of sin(x) based on the graph of the numerical derivatives graph. Problems 3.3.1 to 3.3.9 odd, 3.2.12 to 3.3.13 all. 3.4 Derivative of the power function and another definition of the derivative. The use of h0 is used, derivative of a power function, derivative of a sum, a constant times a function and a constant function are covered. Problems 3.4.1. to 3.4.29 odd. 3.5 Displacement, velocity and acceleration. The connection between acceleration, velocity and displacement are established. The idea of speed, and relative versus absolute values is covered. Problems 3.5.1 to 3.5.12 all. 3.6 Introduction composite functions using sine and cosine. The concept of inside function and outside function are established. Problems 3.6.1 to 3.6.8 all. 3.7 Derivative of a composite function; the chain rule. The chain rule as it applies to trigonometric and polynomial functions is learned. In problems 3.7.3 to 3.7.22 students use the numerical derivative feature of the TI83 to support their conclusions. Problems 3.7.1 to 3.7.26 all. 3.8 Proof of the derivatives of sine and cosine. The squeeze theorem is shown, sinusoidal function applications are explored using the TI83 calculator. Problems 3.8.1 to 3.8.13 odd. 3.9 Anti-derivatives and indefinite integrals. The constant of integration is learned and the use of initial conditions to find a particular equation is shown. Problems 3.9.1 to 3.9.25 all. 3.10 Chapter review and journal entry. Problems 3.10.R0 to 3.10.R9 all. Chapter 4 4.1 Introduction to the chain rule, product rule, and quotient rule. Problems 4.1.1 to 4.1.6. 4.2 Derivative of the product of two functions. The formula for the derivative of a product is proven. Problems 4.2.1 to 4.2.21 odd require that the student check their answers by comparing the graphs of the calculated derivative and the numerical derivative. Problems 4.2.1 to 4.2.35 odd. 4.3 Derivative of a quotient of two functions. The formula for the derivative of a quotient is shown. Problems 4.3.1 to 4.3.25 use the TI83 is used to confirm solutions. Problems 4.3.1 to 4.3.31 odd. 4.4 Derivative of the trigonometric functions other than sine and cosine. The derivative of the six trigonometric functions is shown. Problems 4.4.1 to 4.4.43 odd. 4.5 Derivative of the inverse trigonometric functions. The arctangent, arccosine, etc. are reviewed. Implicit differentiation is introduced. Problem 4.5.29 is key to discovering the derivative of the inverse of a function. Problems 4.5.1 to 4.5.24 all, 4.5.25, 4.5.26, 4.5.29. 4.6 Differentiability and continuity. The idea of differentiability is covered, including differentiability at a point and on an interval. Differentiability implies continuity is studied. Problems 4.6.1 to 4.6.38 all. 4.7 Derivative of parametric equations is not covered. 4.8 Graphs and derivatives of implicit relations. Differentiation of implicit relations is covered in detail. Problems 4.8.1 to 4.8.27 all. 4.9 Chapter review and journal write. Problems 4.9.R0 to 4.9.R8 all. Chapter 5 5.1 A definite integral problem. This section shows other ways to find the area under a curve. Problem 5.1.4 requires the students to use the TI83 to estimate a definite integral using a midpoint sum. Problems 5.1.1 to 5.1.7 all. 5.2 Review of antiderivatives. The meaning of antideivative, indefinite integral and the need for a constant of integration is covered. Problems 5.2.1 to 5.2.17 odd. 5.3 Linear approximations and the differentials. Linearization of a function is used find approximations for functions. Problems 5.3.1 to 5.3.39 odd. 5.4 Formal definition of antiderivative and indefinite integral. The integral sign ( ) is introduced, and the integral sign becomes an operation. Problems 5.4.1 to 47 odd. 5.5 Riemann sums and the definition of definite integral. Definition of Riemann sums are introduced along with an extensive vocabulary list. Partitions, lower, midpoint, and upper sums are covered. The idea of integrability is shown, and the integral as a limit of a Riemann sum is covered. Problem 5.5.10 requires the student to write a program that will be used to estimate integrals using the upper, lower, and midpoint Riemann sums. Problems 5.5.1 to 5.5.12 all. 5.6 The mean value theorem and Rolle’s theorem. The mean value theorem is proven algebraically, Rolle’s theorem and the mean value theorem are used in a variety of questions. Problem 5.6.33 uses the TI83 it illustrate the proof of Rolle’s mean value theorem. Problems 5.6.1 to 5.6.18 all, 5.6.19 to 5.6.33 odd. 5.7 Some special Riemann sums. The idea that a Riemann sum can be used to find and exact value of a definite integral is discovered. Problems 5.7.1 to 5.1.12 all. 5.8 The fundamental theorem of Calculus. The fundamental theorem is proven using an intuitive approach, by applying the Riemann sum and the squeeze theorem. Problems 5.8.1. to 5.8.9 all. 5.9 Definite integral properties and practices. Evaluating the definite integral using the fundamental theorem is covered. The properties of integral are established, such as positive and negative integrands, reversal of the limits on integration, sum of integrands, etc. Problems 5.9.1 to 5.9.37 odd. 5.10 A way to apply the definite integral. A variety of questions exercise writing Riemann sums and evaluating definite integrals using the fundamental theorem. Problems 5.10.1 to 5.10.7 all. 5.11 Numerical integration by Simpson’s rule and a graphing calculator. Simpson’s rule is derived and applied to questions involving tables of values. Students write a program for the TI83 calculator to evaluate integrals by Simpson’s rule. Problems 5.11.1 to 5.11.13 odd. 5.12 Chapter review and journal write. Problems 5.12.R0 to 5.12.R12 all. Chapter 6 6.1 Integral of the reciprocal function, a population growth problem. The foundations for question involving direct proportion, and separating variables is introduced. Problems 6.1.1 to 6.1.7 all. 6.2 Antiderivative of the reciprocal function. The 1/x dx = ln x + C is introduced in a discovery lesson aided by the TI 83 calculators graphing capabilities. Problems 6.2.1 to 6.2.12 all. 6.3 Natural logarithms, and another form of the fundamental theorem. The derivative of an integral is investigated. The definition on the natural log is illustrated. Problems 6.3.1 to 6.3.57 odd, 6.3.58, and 6.3.60. 6.4 ln x really is a logarithmic function. The uniqueness theorem for derivatives is used to establish the properties of logarithms. Problems 6.4.1 to 6.4 13 all. 6.5 Derivatives of exponential functions and logarithmic functions. The technique of logarithmic differentiation is established. Problems 6.5.1 to 6.5 37 odd, 6.5.38. 6.6 The number e and the derivative of base b logarithm functions. The definition of e is covered, along with the change of base property, and the interchange of limits within continuous functions. Problems 6.6.1 to 6.6.21, and 6.6.20. 6.7 The natural exponential function and the inverse of ln. The derivative and the integral on f(x) = ex is shown. Problem 6.7.63 uses the TI 83 to discover L’Hospital’s rule. Problems 6.7.1 to 6.7.63 odd. 6.8 Limits of indeterminate forms: I’Hospital’s rule. The application of l’Hospital’s rule is covered and a variety of algebraic manipulations are exercised. The indeterminate forms are covered. Problems 6.8.1 to 6.8.29 odd, 6.8.31 to 6.8.35 all. 6.9 Derivative and integral practice for transcendental functions. The focus is to master integrals ad derivatives of exponential functions, and the integrals of tangent, cotangent, secant and cosecant. Problems 6.9.1 to 6.9.90 all. 6.10 Chapter review and journal write. Problems 6.10.R1 to 6.10.R9 all. Chapter 7 7.1 Exponential growth and decay. The concept of a differential equation is introduced. The direct proportion property of exponential functions is explored. All the problems in this chapter require the TI83 to interpret the students’ results. Problems 7.1.1 to 7.1.6 all. 7.2 Direct proportion property of exponential functions. General solutions to differential equations is covered, separation of variables and initial conditions are used to find the particular solution to a differential equation. Problems 7.2.1 to 7.2.9 all. 7.3 Other differential equations for real-world applications. Differential equations are further explored using rates of change that are directly proportional to functions. More complicated problem scenarios are used. Problems 7.3.1, 7.3.4, 7.3.5, 7.3.7 plus a selection of problems at a later date. 7.4 Graphical solution of differential equation using slope fields. Given a slope field the graph of the approximate particular solution is determined. The conjecture is confirmed algebraically when possible. Slope fields, and logistic equations are introduced. Problems 7.4.1 to 7.4.7 odd, 7.4.10. 7.5 Numerical solutions of differential equations by Euler’s method. This topic is not a part of the AB curriculum and will be covered at the end of the course. 7.6 Predator prey population problems. This is not a part of the AB curriculum and will be covered at the end of the course. 7.7Chapter review and journal write. Problems 7.7.1 to 7.7.4 all. Chapter 8 8.1 Cubic functions and their derivatives. The connection between functions and their derivatives is introduced. Concavity, points of inflection and the second derivative are discussed in a discovery lesson. Problems 8.1.1 to 8.1.5 all. 8.2 Critical point and point of inflection. Critical points as they apply to local and global extremes are discussed. Points of inflection and cusps are also learned. The idea of infinite versus undefined is presented. The notation for the first and second derivatives is presented. Students will become familiar with the different notations and their uses. Problems 8.2.1 to 8.2.41 odd. 8.3 Maximum and minimum in plane and solid figures. The idea of extremes is applied to plane and solid figures. Problems 8.3.1 to 8.3.31 odd, additional problems are suggested if student needs further practice. 8.4 Area of a plane region. The area of regions bounded by two curves is studied and the application of the definite integral and its properties is reviewed. Students use the numerical integration feature of the TI83 to estimate the area of a region bounded by curves. Problems 8.4.1 to 8.4.24 all. 8.5 Volume of a solid by slicing. The methods for finding the volumes of solids by cross sectional area and the disk and washer methods of revolution are covered. Students will use the TI83 calculator’s numerical integration feature to determine the validity of their results. Problems 8.5.1 to 8.5.27 odd. 8.6 Volume of a solid of revolution by cylindrical shells. The method for finding the volumes of solids using shells is covered. Problems 8.6.1 to 8.6.18 all. 8.10 Chapter review and journal write. Problems 8.10.R0 to 8.10.R6 all. Chapter 9 (This chapter is not part of the AB curriculum and will be done at the end of the course.) 9.1 Introduction to integration of a product of two functions. This is an introduction to integration by parts. A numerical solution to an integral is used to discover the technique of integration by parts. Problems 9.1.1 to 9.1.7 all. 9.2 Integration by parts-A way to integrate products. The technique of integration by parts is covered. Problems 9.2.1 to 9.2.10 all. 9.3 Repeated integration by parts. A method of integration by parts is introduced. Different reoccurring patterns are explored, making the original integral repeat, use of trigonometric properties, regrouping factors between steps. Problems 9.3.1 to 9.3.49 odd. 9.4 Reduction formulas and computer software. An iterative processes is used the powers of trigonometric functions. Problems 9.4.1 to 9.4.23 odd. 9.5 Integration of special powers of trigonometric functions. Trigonometric identities and “u” substitution techniques are exercised in this section. Problems 9.5.1 to 9.5.33 odd. 9.6 Integration by trigonometric substitution. Techniques that utilize trigonometric substitution are used, reinforcing the techniques learned in section 4.5. Problems 9.6.1 to 9.6.29 odd. 9.7 Integration of rational functions by partial fractions. A few simple methods of breaking rational functions into partial fractions are covered. Problems 9.7.1 to 9.7.19 odd. 9.8 Integrals of inverse trigonometric functions. Integration techniques for the inverse trigonometric functions are covered. Problems 9.8.1 to 9.8.6 all. 9.9 Calculus of hyperbolic and inverse hyperbolic functions. This topic is not part of the AB curriculum. 9.10 Improper integrals. The ideas of convergence and divergence are used to determine if a solution to an improper integral exists. Problems 9.10.1 to9.10.25 odd. 9.11 Miscellaneous integrals and derivatives. A selection of problems that will serve to review techniques of integration and derivatives in a random order. Problems 9.11.1 to 9.11.102 every other odd. 9.13 Chapter review and journal entry. Problems 9.13.R0 to 9.13.R11 all. Chapter 10 10.1 Introduction to distance and displacement for motion along a straight line. Given a velocity function distance traveled and displacement are determined. Through out the chapter the TI83 calculator will be used to graph and analysis problems. Students will use the numerical integration and derivative features of the TI83 in a routine manner while problem solving. Problems 10.1.1 to 10.1.5 all. 10.2 Distance, displacement, and acceleration for linear motion. Given velocity or acceleration the distance traveled and the displacement are determined. Problems 10.2.1 to 10.2.15 odd. 10.3 Average value problems in motion and else where. The concept of average velocity of a function and average velocity are discussed. Problems 10.3.1 to 10.3.15 odd. 10.4 Related rates. The relationship between different variables and the rates at which they are changing with respect to each other is covered. Problems cover changing volume of a variety of solids as well as linear rates. Problems 10.4.1 to 10.4.21 odd. 10.5 Minimal path problems. Exercises involve setting up functions that can be used to find minimum values. Problems 10.5.1 to 10.5.13 odd. 10.6 Maximum and minimum problems in motion and elsewhere. A variety of problems involving related rates and maximum volume and minimize surface area are solved. Global minimum and maximum are also included. Problems 10.6.1 to 10.6.11 odd. 10.8 Chapter review and journal write. Problems 10.8.R0 to 10.8.R5 all. To review for the Calculus AB exam, students will work all open-ended questions from the 2001 to 2007 exams. The review of basic concepts will be done using the Barron’s AB Calculus review.