Mrs. Traylor's Class



AP Calculus AB SyllabusBrief Description of CourseAP Calculus AB is designed to help students pass the AP Calculus AB exam and to provide them with a substantial mathematical background to be successful in Calculus I and II in college. Technology with the graphing calculator is used extensively to allow students to record and interpret data. Students investigate limits, derivatives, and definite integrals analytically, graphically, numerically, and verbally. Students will frequently work in small groups to practice the content and discuss their findings.Derivatives are interpreted as slope, rate of change, and local linearization. Definite integrals are interpreted as limits of Riemann sums, areas under or between curves, as well as volumes of solids, and net accumulation of a rate of change. Students explain their work and relate these ideas as to why a formula is being used. For example, they may write a paragraph or explain to the class how the derivative relates to slope and rate of change.Students are reviewed the algebraic and trigonometric skills the first week of school. Sample AP exam questions are woven into the curriculum. Released AP exams are used during the second semester to prepare students for the AP test.Required Materials:3-Ring Binder (2 to 3 inch width)PencilsFolder with brads (this folder is to be used for AP Calculus only. Students should keep all graded papers/tests/course work/etc. in this folder)Graphing Calculator (provided in class)Procedures:Late work: No late work will be accepted. Any assignment that is not turned in when due will be a zero. This does not include assignments that are late due to an absence (see student handbook for grading guidelines). If a student will be absent because of an extra-curricular activity, he/she should ask for his/her assignment in advance and turn it in on or before the day that it is due. Missed assignments due to extracurricular activities will not be accepted after the day the student returns.Make-up work: If a student is absent it is his/her responsibility to ask for make-up work before or after class. Restroom Breaks: Please go to the restroom before or after class. No passes will be given during class.No food or drink will be permitted in class.Cell phones and other electronic devices will be taken up immediately and turned in to the office. Such devices can be retrieved for a fine of $15.Tutorials: * (More tutorials will be held during the second semester in order to prepare for the AP exam)Tuesdays from 3:30 to 4:30Thursdays from 7:15 to 8:00FIRST SIX WEEKSReview algebraic, logarithmic, exponential, and trigonometric functions. (Six days)LIMITS AND CONTINUITY:Introduce average and instantaneous speed.Define limit formally using epsilon and delta.Estimate limits from graphs and tables of value.Find right-hand and left-hand limits and calculate limits algebraically.Evaluate limits involving infinity.Determine end behavior asymptotes using limits of infinity.Investigate continuity and different types of discontinuity.Define continuity using limit and function value.Apply the Intermediate Value Theorem.Find equations of tangent lines and normal linesContent and/or Skills Taught:Students analyze graphs to support the concept of limits. Using either a graph from the text or one they graph from their calculator, they can approach x sufficiently close to find an acceptable y-value that indicates the limit, as well as, both right-hand and left-hand limits. Using their calculators they also learn to zoom in to a particular area to find removable discontinuities. These limits are also found algebraically. Asymptotes are also easily graphed to show end behavior of different types of functions: polynomials, logarithmic, rational, exponential, and trigonometric.Sample Assessment: Calculate limits using algebra, graphs, and tables of data.A grapher can suggest what a limit is or when a limit does not exist. Work with your group to use a grapher to examine what happens to the values of the function f(x) = sin(1/x) as x approaches zero (each group must consist of 3-5 students). Investigate the limit by allowing x to be both positive and negative .1, .01, .001, and .0001. Compare your findings with your graph and discuss. Together, make a table of data. Describe the function f(x) = (sinx)/x in the same way. What conclusions can you make? Discuss with the class. SECOND SIX WEEKSThe Derivative:Evaluate the derivative defined as the limit of the difference quotient. Compare the graphs of f and f1.Find the slope and equation of tangent lines. Compare the average rate of change to the slope of secant lines.Find the instantaneous rate of change.Approximate rate of change from graphs and tables of values.Evaluate the derivative numerically using nDeriv.Use differentiation rules to evaluate derivatives.Find the velocity of a moving particle and distance traveled.Find speed and acceleration.Evaluate marginal cost and marginal revenue.Find the derivatives of the six trigonometric functions.Find the derivative using the chain rule.Find the derivative of implicit functions.Content and/or Skills Taught:Students find the slope of a tangent line using the limit of the difference quotient giving them a good workout with algebraic skills. They compare the slope of the secant line to the slope of the tangent line. Students analyze why differentiability implies continuity and local linearity. Students will discuss when a function does not have a derivative: at a corner, at a discontinuity, and at a vertical tangent. Also, if time allows, implicit functions are graphed in the calculator, showing tangent lines passing through unique points on the curve. Particle motion along a horizontal line is looked at from the graphing calculator. Parametric equations are used to better show when a particle changes direction, slows down, and speeds up. Students also find derivatives numerically, as well as graphically and algebraically.Sample Assessment: Model physical situations algebraically and graphically.A blast blows a heavy rock straight up with an initial velocity of 160 ft/sec. It reaches a height of h(t) = 160t2 ft after t seconds. Solve the following graphically and algebraically. Simulate the moving rock using parametric mode.Graph the rock’s height as a function of time using parametric mode.How high does the rock go?How long is the rock in the air?How high is the rock at 3 seconds and 7 seconds? What does this tell you?How fast is the rock going when it is 256 ft above the ground?Collaborate with your group (3-5 students) to make up your own problem. Each group will present an original problem to the class and explain the solution, including a graph.THIRD SIX WEEKSTHE DERIVATIVE: (continued)Evaluate derivatives of inverse trigonometric functions.Evaluate derivatives of exponential and logarithmic functions.APPLICATIONS OF DERIVATIVES:Find critical points analytically and graphically, relative to the Extreme Value Theorem.Find the local extrema or relative maximum and minimum.Find the global extrema or absolute maximum and minimum.Apply the Mean Value Theorem to average rate of change and instantaneous rate of change.Test for inflection point, concavity, increasing, and decreasing using f, f1, and f11.Test for increasing and decreasing of monotonic functions.Optimize problems in industry and business.Evaluate local linearization to curves.Estimate change with differentials.Set up and evaluate related rates of change.Content and/or Skills Taught:Students study the characteristics of graphs and their points of inflection, critical points, increasing and decreasing behavior. They must find the above algebraically and then use the graphing calculator to support their work. At this time, students can compare the graph of the function with its derivative, either by sketching the derivative by hand or using their graphing calculator. They also learn to analyze the graph of the derivative and then sketch the graph of the original function. The Mean Value Theorem and Rolle’s Theorem are studied and compared. Students must write and explain the difference between the two theorems to show their understanding. Students learn to set up optimization problems and solve graphically, numerically, and algebraically. Students calculate such problems as the pipeline construction from a refinery to an offshore rig through land and water, both algebraically and graphically. Local linearization to curves is graphically shown by zooming in to a point to show how the curve becomes locally straight at a particular point and use this idea to work with how accurately they should measure the radius of a sphere to calculate its surface area within a certain percentage of its true value. I also have students explain their answers to make sure they understand the concept of differentials. Students also learn where Newton’s formula originates and how to use their calculator with this formula. They must write their iterations until they successfully find the root.Sample Assessment: Understand the behavior of graphs of functions from their derivatives.Graph any cubic polynomial and its derivative in the same viewing screen. Find the portions of the domain over which the graph of y rises. What do you notice about the derivative? Do the same for the portion of the domain over which the graph of y falls. Make a conjecture that connects the idea of the increasing and decreasing behavior of a function f with the behavior of the derivative function. Now test for increasing and decreasing using f1 algebraically. Write the intervals in interval notation form once you find them. Test your conjecture with another function. Discuss your findings with your group (3-5 students).FOURTH SIX WEEKSTHE DEFINITE INTEGRALFind the distance traveled relating this to area under a curve.Use the Rectangular Approximation Method (Left, right, and midpoint RAMS).Interpret the definite integral as a limit of a Riemann sum.Evaluate definite integrals geometrically.Evaluate definite integral values numerically using fnInt.Apply the basic rules or properties of definite integrals.Find the average value of a function and average daily inventory.Use the Fundamental Theorem of Calculus, Part I and Part II.Find net area and total area under a curve using antiderivatives. Use the Trapezoidal Rule to approximate definite integrals.Apply slope fields to approximate solution curves for differential equations.Solve initial condition problems.Integrate functions by the substitution method.Content and/or Skills Taught:The Rectangular Approximation Method is studied using a graphing program that literally draws on the screen a right RAM, left RAM, and mid RAM, evaluating each one numerically. Students also approximate definite integrals of functions represented algebraically, graphically, and by tables of values. Students must explain in writing the concept of a limit of the Riemann sum, giving specific examples. Students learn antiderivative formulas and apply them to definite and indefinite integrals. The Fundamental Theorem of Calculus is introduced, with students later writing an explanation of this theorem giving specific example for both Part I and Part II. In finding the average value of a function, students are asked to draw the function and prove the area under the curbe is the same as the rectangle they draw, after finding the average value, f(c), and the value of c. Students must find the area under a curve using the Trapezoidal Rule. They must sketch the curve, drawing the trapezoids. They compare this area graphically, numerically, and algebraically. Students construct slope fields for differential equations and solve initial condition problems.Sample Assessment: Use the Trapezoidal Rule to approximate definite integrals.Work with your group to find an appropriate area under the curve f(x)=sindx from x=0 to x=π using the Trapezoidal Rule. Make a drawing showing 4 subintervals, labeling the numerical x and y values. Then find the height of the trapezoids using h = (b-a)\n and plug into the formula. Do not use a calculator to find your answer. Leave your answer in terms of pi and a radical number. Find the area under the curve numerically using fnInt. Show what you entered into your calculator to get the numerical answer. Round your answer to three decimal places. Find the area algebraically, setting up the correct integral and evaluating the exact pare your answers. What conclusion can you make? Discuss with the class.FIFTH SIX WEEKSINTEGRATION (CONTINUED): Application of IntegralsSolve for y by separation of variables.Using dy/dt= ky, derive the Law of Exponential Change, y=y0ekt.Solve exponential growth problems.Solve exponential decay problems.Use integrals of velocity to yield net distance traveled,.Calculate the total distance traveled by a particle along a horizontal line. Find the areas between two curves and intersecting curves. Find the volumes of solids of revolution using:diskswasherscylindrical shellsContent and/or Skills Taught:Students investigate compounding interest continuously, relating it to limits; model growth with other bases, and work with half-life and carbon-14 dating. Calculating total distance traveled reinforces an integral as net change. Volumes of solids of revolution are set up as integrals. I have students compare the cylindrical shell method to the disk method to show that they can get the same answer for finding the volume bounded by the curve y = √x, the x-axis, and the line x=4, revolved about the x-axis.Sample Assessment: Work with your group to solve separable differential equations. Use them in modeling and study the equation y1=ky and exponential growth.Unchecked bacterial growth assumes that at any time t, the rate dy/dt at which the population is changing is proportional to the number y(t) of bacteria present. If the population’s original size is yo, this leads to the initial-value problem dy/dt=ky. Explain how to find y=yoekt by separation of variables and give an example of exponential growth to model bacterial growth. You will share your findings with the class.SIXTH SIX WEEKSINTEGRATION: (Continued)Find the volume of solids of revolution by slicing cross-sectional areas.TECHNIQUES OF INTEGRATION:Integrate by completing the square.Integrate by using trigonometric identities.Integrate by parts, including tabular integration.Evaluate partial fractions, all four cases.Content and/or Skills Taught:Students find volumes of solids by slicing and finding the cross-sectional area and then integrating the area in terms of x. This lesson is the last lesson before students take their AP exam in May. After the exam I teach integration by parts, integration by completing the square, and integration using trigonometric identities. Students also learn how to find partial fractions, all linear factors, none repeating; all linear factors, one or more repeating; at least one irreducible quadratic factor; and one or more repeating. These lessons are excellent fillers to end the school year and students find them easy and fun to do. Sample Assessment: Develop and appreciation of calculus as a coherent body of knowledge and as a human accomplishment.As you are nearing the completion of this AP Calculus course, write a composition discussing the knowledge you have learned and the appreciation of calculus that you have developed. Mention areas of calculus that have intrigued and fascinated you the most. Discuss the use of technology and how it has helped in developing your understanding of calculus as a human accomplishment. Textbooks/Course MaterialsFinney, Ross L., Franklin D. Demana, Bert K. Waits, and Daniel Kennedy. Calculus-Graphical, Numerical, Algebraic. Pearson Prentice Hall, 2007.Finney, Ross L., George B. Thomas, Bert K. Waits, and Daniel Kennedy. Calculus-Graphical, Numerical, Algebraic. Addison Wesley Publishing Company, Inc., 1995.CalculatorTI-83 Plus and TI-84 Plus ................
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