Derivatives - Mater Academy Charter Middle / High School



IntroductionThe course will cover all topics associated with Functions and their Graphs, Limits of a Function, Derivatives and Integrals, in accordance with the AP Calculus BC Course Description.AP Calculus BC is organized as a full-year course that includes all topics taught in Calculus AB (AP Calculus BC is an extension of the previous Calculus AB course) plus supplementary topics such as Parametric, Polar, and Vector Functions, Equations Involving Derivatives, several Integration Techniques like Integration by Parts, Simple Partial Fractions, Improper Integrals, Polynomial Approximations and Series among others. Technology The AP Calculus BC lessons will enforce the use of graphing calculators as an essential goal of the course.C5— The course teaches students how to use graphing calculators to help solve problems, experiment, interpret results, and support conclusions.In addition, the following resources may be helpful for those students looking for online tutoring or for those who are interested in consulting references regarding the AP Exam: . (for free online tutoring)calculus-calculus/fr.html (to find questions and solutions from previous AP Exams) (this site allows the students to explore excellent visual presentations, and it is a great graphing tool) (an interactive way to learn about derivatives)Primary Textbook Stewart, James. “Calculus”. Books/Cole Cengage Learning. 6th Edition. 2008Alternative TextbooksFinney, Ross L., Franklin Demana, Bert Waits, and Daniel Kennedy. “Calculus: Graphical, Numerical, and Algebraic”. Reading, Mass.: Addison-Wesley, 2007.“Cracking the AP Calculus AB & BC Exams”, 2010 Edition / Princeton Review.Course Outline (Organized based on the College Board’s Course Description)Functions, Graphs and Limits ( 5 weeks)C2— The course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; Integrals; and Polynomial Approximations and Series as delineated in the Calculus BC Topic Outline in the AP Calculus Course Description. Analysis of graphs. (2 weeks) Limits of functions (3 weeks) An intuitive understanding of the limiting process. Language of limits, including notation and one-sided limits. Calculating limits using algebra. Properties of limits. Estimating limits from graphs or tables of data. Estimating limits numerically and graphically.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 Continuity as a property of functions. An intuitive understanding of continuity. Understanding continuity in terms of limits. Types of discontinuities. Geometric understanding of graphs of continuous functions. Intermediate Value and Extreme Value TheoremParametric, polar and vector functions. Derivatives ( 5 weeks)C2— The course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; Integrals; and Polynomial Approximations and Series as delineated in the Calculus BC Topic Outline in the AP Calculus Course Description. Concept of the derivative. Derivative presented graphically, 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 ? and ??. Relationship between the increasing and decreasing behavior of ? and the sign of ??. The Mean Value Theorem and its geometric interpretation. Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa.Second derivatives. Corresponding characteristics of the graphs of ?, ?? and ? ?. Relationship between the concavity of ? and the sign of ? ?. Points of inflection as places where concavity changes.Applications of derivatives Analysis of curves, including the notions of monotonicity and concavity.Analysis of planar curves given in parametric form, polar form and vector form, including velocity and acceleration.Optimization, both absolute (global) and relative (local) extreme.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 diverse 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.Numerical solution of differential equations using Euler’s method.L’Hospital’s Rule, including its use in determining limits and convergence of improper integrals and putation of derivativesKnowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric and inverse trigonometric functions.Derivative rules for sums, products and quotients of functions.Chain rule and implicit differentiation.Derivatives of parametric, polar and vector functions.Integrals (10 weeks)C2— The course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; Integrals; and Polynomial Approximations and Series as delineated in the Calculus BC Topic Outline in the AP Calculus Course Description. Interpretations and properties of definite integrals. Summation notation. Definite integral as a limit of Riemann sums. Basic properties of definite integrals. Linearity properties of definite integrals. Applications of integrals. A variety of applications to model physical, biological or economic situations. Finding the area of a region (including a region bounded by polar curves), the volume of a solid with known cross sections, the average value of a function, the distance traveled by a particle along a line, the length of a curve (including a curve given in parametric form), and accumulated change from a rate of change.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), parts, and simple partial fractions (nonrepeating linear factors only). Improper integrals (as limits of 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 (including the study of 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 and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically and by tables of values.Polynomial Approximations and Series (10 weeks)C2— The course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; Integrals; and Polynomial Approximations and Series as delineated in the Calculus BC Topic Outline in the AP Calculus Course Description. Concept of series. A series is defined as a sequence of partial sums, and convergence is defined in terms of the limit of the sequence of partial sums. Technology can be used to explore convergence and divergence.Series of constantsMotivating examples, including decimal expansion.Geometric series with applications.The harmonic series.Alternating series with error bound.Terms of series as areas of rectangles and their relationship to improper integrals, including the integral test and its use in testing the convergence of p-series.The ratio test for convergence and paring series to test for convergence or divergence.Taylor seriesTaylor polynomial approximation with graphical demonstration of convergence (for example, viewing graphs of various Taylor polynomials of the sine function approximating the sine curve).Maclaurin series and the general Taylor series centered at x = a.Maclaurin series for the functions ex , sin x, cos x, and 1/1-xFormal manipulation of Taylor series and shortcuts to computing Taylor series, including substitution, differentiation, antidifferentiation and the formation of new series from known series.Functions defined by power series.Radius and interval of convergence of power series.Lagrange error bound for Taylor polynomials.Review for AP Exam and Final Exam The course leaves about 2 weeks to teach the most difficult topics. The remaining time will be used to review the main topics before the AP exam.Teacher StrategiesStudents are expected to follow the syllabus systematically, completing all assignments with adequate time to be prepared for the AP exam. Students will receive materials and course topics on the first day of class.The teacher will coach the progress of the students and will work on every point on the course planner to achieve the goals of doing well on the AP exam.Each topic will be presented in different ways: numerically, geometrically, symbolically, and verbally in order to teach students the connection among these representations. The teacher and the students are going to spend two (2) weeks at the beginning of the school year to review a variety of Pre-Calculus topics, basically functions and their graphs, in order to familiarize students with the basic functions and be able to represent functions in a variety of ways (graphically, numerically, analytically, and verbally). Students should identify the connection among these representations.C3— The course provides students with the opportunity to work with functions represented in a variety of ways - graphically, numerically, analytically, and verbally- and emphasizes the connections among these representations.Students will be instructed to use a graphing calculator to help them in solving problems from the real world. Students who are taking calculus BC have already been trained to use the graphing calculator. In any case, the teacher will spend additional time (by means of after school tutoring) during the first weeks of the school year, training students in the use of the TI-83 calculator, and addressing concerns about its usage. Students will be provided with a graphing calculator every class. Students will use the graphing calculators to:Investigate limits of functionsExplore continuity of a function Confirm and discover characteristics of graphs of functions and their derivatives (e.g. ,Extreme Values, Inflection Points, and Concavity)Determine the asymptotic behavior of a function.Perform numerical integrationShow Riemann sumsCompute partial sumsUse Euler’s methodsShow a slope fieldDraw a solution curve on a slope fieldTo sketch implicitly defined functions C5— The course teaches students how to use graphing calculators to help solve problems, experiment, interpret results, and support conclusions.Students will justify responses and support their conclusions as a typical practice in class. To develop student’s communication skills, the course includes a diversity of teaching strategies to encourage students to expand their vocabulary and explanation skills.C4— The course teaches students how to communicate mathematics and explain solutions to problems both verbally and in written sentences.Students will be evaluated by different methods. From middle of October throughout the rest of the school year, students will work on questions from AP released exams at the beginning of every class, as “the question of the day”). Questions will be graded.Every week students will have a quiz containing 5-10 multiple-choice questions, and a test after the conclusion of every topic. Each student will work in building an AP Calculus BC Portfolio. As a final point, students will be required to take both, a Mid-Term and a Final Exams in AP Exam format. Students should make conclusions and justify their responses as a part of the class routine. Assignments and assessments are conceived to persuade students to express their ideas in carefully written sentences to support conclusions when soling problems. Presentations and discussions are been including to improve the student’s oral communication skills. Major Assessments and AssignmentsStudents will be evaluated by using different methods. Different assessments and assignments will be used in class in order to expand the student’s understanding of the concepts of calculus and their applications, and to develop the student’s skill in recognizing a variety of techniques for solving calculus problems. In addition, these assignments emphasize a multi-representational approach to calculus and the use of diverse manners to express concepts, problems, and results, including graphically, numerically, analytically, and verbally, and the correlation among these representations. Power Point presentations, banners, and posters could be built by students to show their results; as well portfolios should be created to organize information in written manner. Students will work in groups in solving real-world problems. Consequently, they should participate in discussions of notorious calculus topics and explain their solutions and conclusions to the rest of the class orally. The following activities represent some examples of the most important assignments of the course:Activity #1: Discussion about how derivatives affect the shape of a graph. (Working in groups)Points to stress:The use of the first derivative to determine whether a function is increasing or decreasing.The first and second derivative test for local maxima and minimaThe use of the second derivative to determine concavity and points of inflections.The geometric description of concavity, and the relationship between concavity and the behavior of the first derivative.All students will answer the following question by using both, a graphical display and by written sentences to justify their response: Why is it true that if f is concave upward, then all tangent lines to f lie bellow the curve? Students will discuss in group in order to formulate the team conclusion and then, each team will show their conclusions to the rest of the class. In addition, each group will receive a graph of f’ in order to sketch the function f from the graph of its first derivative. After all the teams have had the chance to reconstruct a function from the derivative, they will show and explain their answers to rest of the class. Students also will work in sketch the graph of the second derivative of f (f”). This activity enriches the students’ communication skills and teaches them how to work in groups in order to prepare for the discussion.Activity # 2 “Graphing with Calculus and Calculators” (Working in groups)Points to stress:The interaction between graphical methods and the analytical aspects of calculus.The use of graphing calculators for estimation of local extrema and inflection points, contrasted with the use of calculus for precise computation of such points. The need for special care when using graphing technology.The use of graphing calculators as a tool to explore families of curves.Working in groups, students will examine the functions f(x) = (X+1)5 sin (X-3) from x=-3 to x=0 (At first glance, the maximum appears to be at x= - 1). Some maxima or minima may be very small compared to the scale of the draw graphs. Calculus helps us determine an appropriate viewing scale. One of the focal points of this activity will be the proper use of the graphing calculator and the correct way to enter the information to graph a function and look for its interesting points. Activity #3 “Application of Differentiation” (Working in groups). Working in groups students will investigate about application of differentiation in a variety of problems, and they will show their conclusion to rest of the class. Each group should make a written report and use manipulative tools to facilitate student’s comprehension. The following list represent some of the problems will be used for this activity:“The Shape of a Can”. Students will investigate the most economical shape for a can. They must interpret this mean that the volume V of a cylinder is given and we need to find the height h and radius r that minimize the cost of the metal to make the can. “Creating a Pyramid” . Given a sphere with radius r, find the height h of a pyramid of minimum volume whose base is a square and whose base and triangular faces are all tangent to the sphere. Students will use the formula V= 1/3 Ah, where A is the area of the base.“Snowball”. Assume that a snowball melts so that its volume decreases at a rate proportional to its surface area. If it takes three hours for the snowball to decrease the half its original volume, how much longer will it take for the snowball to melt completely?“Speed of a Bullet Train” A high-speed bullet train accelerates and decelerates at the rate of 4ft/s2. Its maximum cruising speed is 90 mi/h. What is the maximum distance the train travel if it accelerates from rest until it reaches its cruising speed and then runs at that speed for 15 minutes? Suppose the train starts from rest and must come to a complete stop in 15 minutes. What is the maximum distance it can travel under these conditions?Each team will create a portfolio including a summary of the main concepts and vocabulary involving in the solution of the problems. Also there are going to “make a picture” to help the team in explain methods and concepts behind their answers. The activity will be a contribution to expand the student’s organizational skills and their abilities in communicating mathematics and explain solutions to problems verbally and in written sentences. Activity # 4 “Newton, Leibniz, and the Invention of Calculus” (Individual Project and Class Discussion) Student will read about the contribution of these men in one or more of the given references and write a report on one of the following three topics. Students could include biographical details but basically the report should show a description of their methods and notations. The Role of Newton in the Development of Calculus The Role of Leibniz in the Development of CalculusThe controversy between the Followers of Newton and Leibniz over Priority in the Invention of Calculus. After students complete their individual reports, the class will be organized in three different teams. Each team will be in charge of a topic in order to formulate ideas to support the presentation and discussion in class. Activity #5 “The Origins of L’Hospital’s Rule”( Writing Project)Students will write a report on the historical and mathematical origins of L’Hospital’s Rule, including biographical details of two mathematicians involving in this mathematical discovery and a description of the statement of this rule.Activity #6 “Complementary Coffee Cups” . (Application of Integrals/ The use of a graphing calculator will be required)Students will work in the following individual project:Suppose you have a choice of two coffee cups, one that bends outward and one inward, and you notice that they have the same height and the shapes fit together snugly. You wonder which cup holds more coffee. Of course you could fill one with water and pour it into the other one but, being a calculus student, you decide on a more mathematical approach. Ignoring the handles, you observe that both cups are surfaces of revolution, so you can think of the coffee as a volume of revolution. Suppose the cups have height h, cup A is formed by rotating the curve x=f(y) about the y-axis, and cup B is formed by rotating the same curve about the line X=K. Find the value of K such that the two cups hold the same amount of coffee. Based in your own measurements and observations, suggest a value for h and an equation for x= f(y) and calculate the amount of coffee that each cup holds. Activity #6 “Investigating about the Fundamental Theorem of Calculus” (Reading Activity and Discussion)“Functions defined By Integrals” By Ray Cannon “Exploring the FTC from numerical and graphical points of view” By Mark Howell“Using the Fundamental Theorem of Calculus in a variety of AP Questions” By Larry RiddleThese articles will help students to understand and assess the Fundamental Theorems of Calculus (FTC). These materials give excellent examples of concepts and focus on some special calculus topic and its connection to the FTC. This activity enriches the students’ communication skills and teaches them how to work in groups in order to prepare for the discussion.Activity #7 “A Rose is a Rose is a Rose”( Group Activity Exploration/ Using a graphing calculator to experiment)This activity introduces students to the polar curves described by mathematicians as rose curves. Students will graph r =2 sin (nθ) for various positive integer values of n until they can state a rule for how n determines the number of petals in the rose. Also students will find the total area enclosed by the petals of r =2 sin (2θ), and they will repeat the process for different values of n. Students will make a conjecture about the area of the rose generated by r =2 sin (nθ) for an arbitrary positive integer n. At the end of the activity students will be able to write the general rule for the area of the rose generated by r =2 sin (nθ) for any positive integer n. Through of this activity students could interpret the effect of varying a parameter on a family of functions by using the graphing calculator. C5— The course teaches students how to use graphing calculators to help solve problems, experiment, interpret results, and support conclusions.Additionally students will receive sets of questions in AP format for each topic to help them prepare for the AP exam. They will be required to take a Mid-Term in the AP Exam format. As a part of the students’ preparation to accomplish the class, they should complete selected questions from the section “Chapter Review” which appear at the end of each chapter in their textbook. ................
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