ࡱ> 241Sb O1jbjb{x{x $P)66666664Lh8 JLLLLLLLLLLLLL,SNRPxL-6 xL.!66 L.!.!.! 6 6 JL.!JT66666 JL.!.!*JD66JL, q :nK&JLL0LK;Q.!;QLJL.!6X XAP Calculus AB Course Syllabus Course Outline: Teaching Strategies: I strongly feel that students best learn mathematics, and specifically Calculus, when they are allowed to come up with an idea, rule, or theorem on their own. I try to introduce the majority of calculus topics by creating investigations for my students. These investigations frequently require the use of a graphing calculator and connecting graphical, numerical, and analytical representations of a problem. I then ask them to verbally communicate what conclusions they think they have found. Once the students are able to grasp a new concept or rule on their own, I then take the time to introduce the proper mathematical vocabulary and notation so that they can apply the concept in the future. I find this very beneficial to the students because when they approach problems in the future they are less likely to feel that they need to have a rule memorized, they know that they can figure out the problem using other methods. Textbook: Calculus with Analytic Geometry, 7th Edition by Larson, Hostetler, and Edwards published by Houghton Mifflin Company 2002 Required Calculator: Although students may choose any graphing utility that performs the required functions for this class, I encourage students to own TI-83 or TI-84 Plus editions because I use a TI-84 SmartView on my SmartBoard. I also have several calculators that students may borrow if they leave their calculator at home or cannot afford a calculator. Students need to have a graphing calculator on a daily basis because most of my lessons use graphing calculators to discover and explore new concepts or reinforce previous concepts. Students are allowed to use graphing calculators on a part of every assessment (they are all broken into a calculator and a no calculator sections). Unit 1: Limits and Continuity (3-4 weeks) Contrasting Pre Calculus and Calculus Finding Limits Graphically and Numerically Finding Limits Analytically Limit notation and definition of a limit Properties of limits Two-sided limits One-sided limits Continuity Relation to the limit at a point Common discontinuities Infinite limits Limits at infinity Intermediate Value Theorem ** To introduce limits where students must reduce and rationalize a function I give them limits of rational functions that can be simplified (i.e.  EMBED Equation.3 ). I then ask them to evaluate the function and describe why this happens. They then evaluate the function at multiple points on both sides of the function using the ASK function on their graphing calculators. They must verbally describe what they think this means in terms of the limit of the function. Then they are asked to graph the function using their calculator, and again describe that this means in terms of the limit of the function. Finally, they are asked to analytically tie together the numerical and graphical representations of the limit of the function and communicate their findings to the class. This allows them to see why dividing out works and is a valid technique. It also allows them to have multiple ways of finding a limit when they are unsure of what to do. Unit 2: Differentiation (4-5 weeks) Definition of the derivative Differentiability and how it relates to continuity Derivative as a rate of change Local linearity Tangent lines and their equations Numeric derivatives using graphing calculators Basic Differentiation Rules Properties of derivatives Derivatives of polynomials Product Rules Quotient Rule Derivatives of trigonometric functions Chain Rule Second Derivatives Applications to velocity and acceleration Implicit Differentiation Related Rates Problems ** An activity I like to do in this unit is an exploration of the chain rule where students work in groups to come up with the chain rule on their own. At this point they have learned the definition of the derivative of trigonometric functions. They start by graphing  EMBED Equation.3  in Y1 and in Y2 they put  EMBED Equation.3  which I explain is an approximation of the derivative at each point (I show them why numerically this works). They are then able to hypothesize what the rule for the derivative is, and check their answer by plugging in their guess into Y3. I have them start with a function they already know so that they make sure they understand all of the steps. I then have them repeat the process for  EMBED Equation.3  (which again they know how to do). They then are asked to find the derivative of  EMBED Equation.3 ,  EMBED Equation.3 , and  EMBED Equation.3 . Each time they have to make a hypothesis, check their answer, and if they are not correct come up with a different guess until they find the correct answer. I find that working in groups they are really allowed to communicate their guesses with their peers, explain why their thinking is, and help each other get to the correct answer in the end. By the end they are able to verbally communicate the chain rule to me, and I in return give them the written version of the chain rule. Unit 3: Applications of Differentiation (5 weeks) Extrema Local or relative extrema Global or absolute extrema Critical numbers Rolles Theorem and the Mean Value Theorem The First Derivative Test Increasing/Decreasing Identify maximums and minimums analytically and graphically The second Derivative test Concavity Identify points of inflection analytically and graphically Curve Sketching Use graphs of f, f', or f'' to predict the behavior of related graphs (increasing, decreasing, maximums, minimums, concavity, or points of inflection) Optimization Problems ** I like to have students discover the rules/properties of the first derivative test using their prior knowledge and their graphing calculators. I first ask students to find the critical points of  EMBED Equation.3 . I then ask students to find the derivative of at x = -1, .5, 2. I then give them a table with the intervals (-", 0), (0, 1), (1, "), the test values they just found, the sign of f (x), and ask them to use their graphing calculator to determine where the function is increasing or decreasing. This allows them to make the connection that the sign of the derivative dictates where a function is increasing or degreasing. They are also then asked to draw conclusions about what happens at critical points where the derivative changes sign from positive to negative, or negative to positive. They realize that these signs determine whether a critical point is a local minimum or maximum. To check for understanding I ask a few students to verbally communicate their findings and their reasonings they have for the conclusions they drew. Unit 4: Integration (5 weeks) Area under a curve Approximate the area under the curve using summation of rectangles (inscribed and others) Riemann Sums (left-hand, right-hand, mid-point) graphically and analytically using a graphing calculator Trapezoidal Approximation graphically and analytically using a graphing calculator Definite Integrals Rules of integration Average Value Theorem Fundamental Theorem of Calculus First and second Integration by substitution ** To introduce the concept of Riemann Sums I first introduce the concept that an integral is the area under a curve. I then ask students to find the integral of numerous linear piece wise functions. They are able to do this simply breaking the graph into numerous rectangles, triangles, or trapezoids. I use functions that are both above and below the x-axis so I can discuss positive and negative area. To help students transition their current skills into Riemann Sums I ask them to fill in a table for a function (i.e.  EMBED Equation.3  with x: [-4, -3, -2, -1, 0, 1]) and I provide them with the graph. I then have them draw in the rectangles, on top of the graph, that they would use for a right-hand approximation (I explain what this means before they do it). They are then asked to use the table they put together, as well as the rectangles they drew, to approximate the area of the function. They are finally asked to find the actual integral of the function using their graphing calculators and hypothesize how they could make their approximation more accurate. This discussion leads to different Riemann Sums (left-hand and midpoint) as well as a Trapezoidal approximation, and an infinite number of rectangles and thus the limit definition of the integral. Unit 5: Logarithmic, Exponential and other Functions (5 weeks) Natural logarithmic functions Review of basic graphs and properties Derivatives Integrals Inverse Functions Review of the definition (both graphical and algebraic) Derivatives Exponential Functions Review of basic graphs and properties Derivatives Integrals Exponential and logarithmic functions with bases other than e Derivatives Integrals Inverse Trigonometric Functions Review of graphical relationships and definitions Derivatives Integrals Differential Equations Solve simple differential equations Growth and decay models Solving using separation of variables Slope Fields Unit 6: Applications of Integrals (2-3 weeks) Area between two curves Volumes of solids with known cross-sectional areas Volumes of revolution Disk method Washer method Summing rates of change Distance traveled by a particle ** An activity I like to do towards the end of the unit is an Application problem where students need to find the volume of a flowerpot. They are given a cross sectional sketch of the pot on graph paper and told that it is perfectly circular. To solve this problem they need to determine what function best represents the pot, determine what method they want to use to find the volume, and finally evaluate the integral. 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