Curriculum Map. Discipline: Math Course: AP Calculus AB Teacher: Louis Beuschlein

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1 Curriculum Map Discipline: Math Course: AP Calculus AB Teacher: Louis Beuschlein August/September: State: 8.B.5, 8.C.5, 8.D.5 What is a limit? What is a derivative? What role do derivatives and limits play as a foundation for the calculus and in practical applications? Intuitive notion of a limit, limit theorems, finding limits, continuity, intermediate value theorem, trig limits Formal definition of a derivative, finding derivatives via definition, graphical interpretations, simple derivative theorems, finding derivatives of functions, equations of tangent lines, power rule, product rule. Students will apply the definition of the derivative. Students will use limit theorems to find limits of functions. Students will apply the definition of the derivative correctly to find derivatives of functions without resorting to derivative theorems. Students will take derivatives of polynomials via the sum and power rules. October: Frequent mini-quizzes Traditional quizzes State: 8.B.5, 8.C.5, 8.D.5 In what types of problems do the various differentiation rules apply? How can a function be transformed prior to differentiation in to apply a simpler differention rule?

2 How can derivatives be applied to solving motion problems? Quotient rule, chain rule, derivatives of trig functions, implicit differentiation, related rates problems. Combinatorics and probability. Students will use the power, quotient, sum, product and chain rules to find the derivatives of composite functions, and use these rules appropriately while differentiating implicitly. Students will set up and solve equations in related rates problems. Frequent mini-quizzes Traditional quizzes November: What information do the first and second derivatives of a function give one about the function itself? How can differentiation techniques be used in estimation problems? What information does calculus give us concerning graphs of functions? Absolute and relative extrema, critical values of a function, Rolle's theorem, mean value theorem, average vs. instantaneous rates of change, optimization problems (geometic, business, scientific) increasing/decreasing functions and first derivative, concavity and second derivative, inflection points, horizontal/vertical/oblique asymptotes, curve sketching (rational, polynomial, and trigonometric functions), first and second derivative tests (scroll) Antidifferentiation; fundamental theorems of calculus; integration techniques: power rule, u-substitution, long division, trig identities, radical conjugates. Proofs by inductions; summation formulae; area approximations: left/right endpoints, midpoints, trapezoid, Simpson; mean value theorem for integrals; symmetry (even and odd functions). Students will sketch curves of functions after identifying all asymptotes and intercepts and after using the first and second derivative to identify intervals over which function is increase/decreasing, concave up/down, extrema, and inflection points.

3 Students will use the above listed techniques to find antiderivatives for a wide variety of functions. Students will compute definite integrals by taking limits of Riemann sums, checking there work with the fundamental theorem of calculus. Daily 8th hour problem sessions December: What is an integral? How are integrals related to derivatives? What is the relationship between an integral and area? How can one apply numerical techniques to compute an integral without knowing the associated antiderivative? Integration techniques: complex u-substitution. Newton's method, linear approximations, error approximation Riemann sums; definite integrals. Students will integrate complex trigonometric, polynomial functions. Students will set up and solve differential equations that model a variety of phenomena in science, business, and population dynamics. Students will "linearize" functions, set up and solve optimization problems, and approximate zeros of functions via Newton's method (on paper and using recursive operations on a graphing calculator). Students will approximate the area under curves by hand and via calculators using all the methods listed above. Daily 8th hour problem sessions

4 January: What is a logarithm and how can a natural log be defined in terms of an integral? What is so special about the number e? What is a differential equation? How can one use differential equations to model real world problems? How does one deal with exponential and logarithmic functions in derivatives and integrals. Integral definition of the natural logarithm, derivations of log properties, inverse functions, the calculus definition of the number e, logs and exponentials of other bases. Integration techniques: exponentials, natural logarithms First order, linear differential equations with constant coefficients; exponential population growth; Newton's law of cooling; compound interest; logistic growth model; radioactive decay. Students will derive various properties of exponential and logarithmic functions. Students will integrate complex trigonometric, polynomial, exponential, and logarithmic functions. Students will set up and solve differential equations that model a variety of phenomena in science, business, and population dynamics. February: What role do inverse trigonometric and hyperbolic functions play in calculus? How can one approximate solutions to differential equations numerically?

5 What is a slope field and how can it be used to find solutions to differential equations? Using inverse trig functions to integrate Slope fields and Euler's method for approximating solutions to differential equations in the form dy/dx = f(x, y). Students will identify integrals that involve inverse trig, make the appropriate substitutions, integrate, and convert back to the original variable of integration. Students will draw slope fields and graphical solutions to differential equations. They will also interpret the meaning of an initial condition. March: How can integrals be used to find areas of complex figures? What are the practical applications of finding such areas? What is an improper integral and under what circumstances do they arise? Areas between curves Integrals involving x as a function of y Integrals involving a variety of integration techniques Students will integrate a wide range of functions, require a broad spectrum of techniques. Students will graph two curves, find their intersections, set up an integral for the area between the curves, and compute the area. April:

6 How can integrals be used to find volumes of complex figures? What are the practical applications of finding such volumes? What is about certain functions that lend themselves naturally to one method but not another? Volumes of revolution: disc, washer, and shell methods. Volumes of geometric solids via cross-sections and integration; Cavalieri's principle. Derivations of volumes of cone, pyramid, sphere, etc. via above techniques. Students will explain the difference between the disc, washer, and shell methods. They will also determine which method is preferred in particular cases and explain why only one method will work in certain cases. May/June: How is calculus useful in science, business, and other fields? What is the relationship between derivatives and integrals? Review and AP practice exams Presentations of projects Students will demonstrate the ability to make use of all the main concepts acquired throughout the school year. Practice AP exams for homework