Mathematics Scope & Sequence Calculus AB

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1 Mathematics Scope & Sequence Calculus AB Revised: March 2015 First Six Weeks (29 ) Limits and Continuity Limits of (including onesided limits) An intuitive understanding of the limiting process Calculating limits using algebra Estimating limits from graphs or tables of data Asymptotic and unbounded behavior Understanding asymptotes in terms of graphical behavior Describing asymptotic behavior in terms of limits involving infinity Comparing relative magnitudes of and their rates of change (for example, contrasting exponential growth, polynomial growth, and logarithmic growth) Continuity as a property of An intuitive understanding of continuity. (The function values can be made as close as desired by taking sufficiently close values of the domain.) Understanding continuity in terms of limits Geometric understanding of graphs of continuous (Intermediate Value Theorem). Evaluate limits using correct limit notation Evaluate limits using direct substitution, factoring, conjugate, LCD Know special limits of sine and cosine such as sin x x limx 0 = limx 0 = 1 x sin x, 1 cos x cos x 1 limx 0 = limx 0 = 0 x x Squeeze theorem Explore limits using technology Estimating limits from tables of data Explore limits at infinity algebraically, graphically, or from a table Evaluate limits as you approach both positive and negative infinity for horizontal asymptotes Evaluate limits include limits that do not exist Explore continuity graphically Prove a function is continuous using the three components of continuity to (limit must exist, function must exist and limit = function) Define type of discontinuity ( removable, non-removable) Demonstrate geometric understanding of graphs of continuous with the Intermediate Value Theorem (reminder is works for continuous only) 13 Textbook Sections: 2.2,2.3,2.4,2.5,4.5 First six weeks curriculum continued on next page.

2 Derivative, Tangent Line, Differentiability Concept of the derivative Derivative presented graphically, numerically, and analytically Derivative defined as the limit of the difference quotient Derivative interpreted as an instantaneous rate of change 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 Represent derivatives graphically, numerically, and analytically Explore local linearity graphically Compare secant to tangent lines Define the derivative as the limit of the difference quotient; also define the derivative using the alternate form of the derivative Investigate slope of a curve at a point. Emphasize examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents Describe three situations when a function is not differentiable (Sharp points, vertical tangent line, discontinuities) Interpret derivatives as an instantaneous rate of change Describe the relationship between differentiability and continuity Explore a tangent line to a curve at a point and local linear approximation (differentials) Define the instantaneous rate of change as the limit of average rate of change Approximate rate of change from graphs and tables of values 10 Textbook Sections: 3.1 Derivative Rules (Part 1) Computation of derivatives Knowledge of derivatives of basic, including power, exponential, logarithmic, trigonometric, and inverse trigonometric Derivative rules for sums, products, and quotients of Apply rules to evaluate derivatives of basic, including power, exponential, trigonometric. Apply rules to evaluate derivatives including sums, products, and quotients of Apply derivative rules to evaluate derivatives from numerical examples using tables 6 Textbook Sections: 3.2, 3.3

3 Second Six Weeks (24 ) Derivative Rules (Part 2) Computation of derivatives Chain rule and implicit differentiation Knowledge of derivatives of basic, including power, exponential, logarithmic, trigonometric, and inverse trigonometric Apply rules to evaluate derivatives of ln Apply chain rule to evaluate derivatives including numerical examples from tables Textbook Sections: 3.4 Implicit Differentiation Computation of derivatives Knowledge of derivatives of basic, including power, exponential, logarithmic, trigonometric, and inverse trigonometric Chain rule and implicit differentiation Use of implicit differentiation to find the derivative of an inverse function Differentiate implicitly Differentiate logarithmically Inverse Functions and Inverse Trig Functions Textbook Sections: 3.5 Computation of derivatives Knowledge of derivatives of basic, including power, exponential, logarithmic, trigonometric, and inverse trigonometric Use of implicit differentiation to find the derivative of an inverse function Textbook Sections: 3.6 Apply rules to evaluate derivatives of inverse trigonometric Evaluate the derivative of an inverse function 6

4 Related Rates Modeling rates of change, including related rates problems Textbook Sections: 3. Third Six Weeks (25 ) Model rates of change, including related rates problems First and Second Derivative Tests Derivative as a function Corresponding characteristics of graphs of f and f Relationship between the increasing and decreasing behavior of f and the sign of f 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 f, f, and f Relationship between the concavity of f and the sign of f Points of inflection as places where concavity changes Determine absolute max and min of a function by the Candidates test Extreme Value Theorem Investigate corresponding characteristics of graphs of f and f Find local extrema using the first derivative line test Create sign chart for organization of thoughts, must include sentence for proper justification. Investigate the relationship between the increasing and decreasing behavior of f and the sign of f Describe monotonic behavior Mean Value Theorem Write equations involving derivatives. Translate verbal descriptions into equations involving derivatives and vice versa Investigate corresponding characteristics of the graphs of f, f, and f Investigate the relationship between the concavity of f and the sign of f Define points of inflection as places where concavity changes Identify local extrema using the second derivative test 10 Textbook Sections: 4.1, 4.2, 4.3, 4.4 L Hopitals and Straight Line Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration Interpret the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration L Hopital s Rule 5 Textbook Sections: 8., 3.2

5 Fourth Six Weeks (32 ) Optimization Optimization, both absolute (global) and relative (local) extrema Model applications involving optimization 6 Textbook Sections: : 4. Anti-Derivatives and Riemann Sums Interpretations and properties of definite integrals Definite integral as a limit of Riemann sums Basic Properties of definite integrals (examples include additivity and linearity) Techniques of antidifferentiation Antiderivatives following directly from derivatives of basic Applications of antidifferentiation Finding specific antiderivatives using initial conditions, including applications to motion along a line Connect integration to graph and area under the curve. Approximate definite integrals of represented algebraically, graphically, and by tables of values using Riemann sums (using left, right, and midpoint evaluation points) and trapezoidal sums Define the definite integral as a limit of Riemann sums Apply basic properties of definite integrals (examples include additivity and linearity) Recognize antiderivatives follow directly from derivatives of basic (definite and indefinite integration) Apply basic rules to evaluate integrals including x power rule, e, and basic trig Find specific antiderivatives using initial conditions, including applications to motion along a line 10 Textbook Sections: 5.1, 5.2, 5.3 Fourth six weeks curriculum continued on next page.

6 Fundamental Theorem of Calculus Interpretations and properties of definite integrals Definite integral as the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval: bb ff (xx)dddd = ff(bb) ff(aa) aa 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 so defined. Second Fundamental Theorem Applications of antidifferentiation Definite definite integral as the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval: bb ff (xx)dddd = ff(bb) ff(aa) aa Evaluate definite integrals using the Fundamental Theorem Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of so defined. Second Fundamental Theorem of Calculus (derivative of an integral) Accumulated change from a rate of change. Integral defined Find specific antiderivatives using initial conditions, including applications to motion along a line. Calculate displacement and total distance traveled by a particle along a line Finding specific antiderivatives using initial conditions, including applications to motion along a line Textbook Sections: 5.4 Advanced Methods of Integration Techniques of antidifferentiation Antiderivatives following directly from derivatives of basic Antiderivatives by substitution of variables (including change of limits for definite integrals) Textbook Sections: 5.5, 5.6, 5. Apply basic rules to evaluate integrals including integration of logs, remaining trig, and inverse trig. Integrate by substitution of variables (including change of limits for definite integrals) 9

7 Slope Fields and Separation of Variables Applications of antidifferentiation Solving separable differential equations and using them in modeling (including the study of the equation yy = kkkk and exponential growth) Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations Textbook Sections: 6.1, 6.2, 6.3 Fifth Six Weeks (34 ) Solve separable differential equations and use them in modeling (include the study of the equation yy = kkkk and exponential growth) Interpret differential equations geometrically via slope fields and indentify the relationship between slope fields and solution curves for differential equations 8 Area and Volume Applications of integrals Area between two curves Volumes of revolution Find area between two curves Find volumes of revolution using disk/washer method Textbook Sections:.1,.2 Volume by Cross Sections Applications of integrals Volumes of a solid with a known cross section Find volumes of a solid with a known cross section 4 Textbook Sections:.2 AP Review AP Review AP Review 15 Textbook Sections: Released AP Questions

8 Sixth Six Weeks (33 ) AP Review AP Review AP Review 14 Textbook Sections: Released AP Questions After the AP Test Topics Shell Method Arc Length Integration by Parts Advanced Trig Integration Trig Substitution Partial Fractions Newton s Method Teachers may choose topics based on time in class 10 Textbook Sections:.3,.4, 8.2, 8.3, 8.4, 8.5, 3.8

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