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

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1 Curriculum Map Discipline: Math Course: AP Calculus BC 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? In what types of problems do the various differentiation rules apply? Formal limit definition, epsilon-delta proofs, 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, quotient rule, chain rule, derivatives of trig functions, implicit differentiation, related rates problems Students will apply the definition of the derivative and prove limits via epsilon-delta. Students will use limit theorems to find limits of complex functions. Students will apply the definition of the derivative correctly to find derivatives of functions without resorting to derivative theorems. Students will use the power, quotient, sum, product and chain rules to find the derivatives of complex composite functions, and use these rules appropriately while differentiating implicitly. Students will set up and solve equations in related rates problems. October: Traditional quizzes and computer-based quizzes.

2 State: 8.B.5, 8.C.5, 8.D.5 How can a function be transformed prior to differentiation in to apply a simpler differention rule? How can derivatives be applied to solving motion problems? 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? combinatorics and probability absolute and relative extrema, critical values of a function, Rolle's Theorem, mean value theorem, average vs. instantaneous rates of change, optimization problems (geometric, 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) Newton's method, linear approximations, error approximation 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. 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). Computer-based quizzes and computer-based quizzes. November: What is an integral? How are integrals related to derivatives? What is the relationship between an integral and area?

3 How can one apply numerical techniques to compute an integral without knowing the associated antiderivative? 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). Riemann sums; definite integrals. Students will approximate the area under curves by hand and via calculators using all the methods listed above. 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 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: complex u-substitution, 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.

4 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. Daily 8th hour problem sessions January: What role do inverse trigonometric and hyperbolic functions play in calculus? How are hyperbolic functions related to parametizations of hyperbolas? How can one approximate solutions to differential equations numerically? What is a slope field and how can it be used to find solutions to differential equations? Using inverse trig functions to integrate; trigonometric substitutions in integrals Hyperbolic functions: definitions and graphs; comparisons with circular functions; parameterizations for hyperbolas; hyperbolic identities and comparisons to trig identities; conversion to equivalent natural log functions; using them as substitutions in integrals. Slope fields and Euler's method for approximating solutions to differential equations in the form dy/dx = f(x, y). Students will identify integrals for which trigonometric substitution is appropriate, make the substitutions, integrate, and convert back to the original variable of integration. Students will prove basic hyperbolic identities. Students will draw slope fields and graphical solutions to differential equations. They will also interpret the meaning of an initial condition.

5 February: How can integrals be used to find areas and volumes of complex figures? What are the practical applications of finding such areas and volumes? What is about certain functions that lend themselves naturally to one method but not another? Areas between curves 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. Arc length of a curve; surface area swept out by the graph of a function rotated about various axes. Physics applications: work problems; pressure/force problems; center of mass of laminae of uniform thickness/density whose borders are defined in terms of two given functions. More techniques of integration: substitution techniques; integration by parts; trig integrals; more trig substitution; partial fractions Students will set up and solve (via techniques of integration or numerical integration/calculator) integrals associated with area, volume, arc length, and surface area problems, as well as a variety of physics applications. 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. Students will integrate a wide range of functions, require a broad spectrum of techniques. March:

6 What is an improper integral and under what circumstances do they arise? What is the difference between a sequence and a series? What does it mean for a sequence or series to converge? How can it be proven that a sequence or series converges? L'Hospital's rule, improper integrals. Sequences, monotonic and bounded sequences, finite and infinite series, convergence and divergence of infinite series and sequences, geometric series, infinite series convergence theorems and the proofs: nth term test, integral test, direct comparison test, limit comparison, alternating series test (& remainder theorem), ratio test, root test; absolute/conditional convergence, Taylor and MacClaurin series, power series derivations, radius/interval of convergence. Students will set up and evaluate integrals to find areas under curves despite discontinuities in the graphs. They will also set up integrals to find areas under a curve spanning an infinitely long interval. Given a limit, students will determine if L'Hopital's rule applies. If not, they will manipulate the function manually until it is in an indeterminate form for which L'Hopital's rule applies. April: How are parametric equations useful in describing the motion of objects and in other problems? How does one deal with derivatives and integrals when relationships are defined parametrically? What are some of the scientific applications of conic sections? Conics sections locus & geometric definitions and equation derivations, reflective properties of conics, area and circumference of an ellipse, rotation of axes, classifying conics based on the discriminant, focus/directrix definitions of conics, applications (problem solving). Parametric equations, parametizing curves and removing parameters, cycloids, find slopes of curves defined parametrically, as well as finding arc lengths, volumes of revolutions, and surface areas. Polar coordinates, polar functions and graphs, intersections and areas between polar curves, symmetry, slopes & arc lengths of polar curves, finding tangents at the pole. AP prep

7 Students will state and locus definitions of the conics and also define them clearly in terms of planes intersecting a double-naped cone and in terms of a constant ratio of distances (between a point and a focus and a point and a line). Students will set up and solve integrals to compute areas, volumes, and surface areas involving curves defined parametrically and via polar coordinates. AP practice tests May/June: How can certain relationships be defined in terms of polar coordinates? How does one use polar coordinates to reduce the complexity of a problem? How does one deal with polar coordinates when doing derivatives and integrals? How are vectors helpful in problem solving, and what do they have to do with calculus? Polar equations for conic sections. Integration technique for integrands involving rational expression in sine and cosine. Intro to vector calculus. AP prep. Students will write polar coordinate equations for parabolas, ellipses, and hyperbolas. Students will solve integrals with and without a calculator involving rational trig integrands. Students will perform calculus-based operations to vector-valued functions. AP practice tests.