Greenwich Public Schools Mathematics Curriculum Objectives. Calculus
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1 Mathematics Curriculum Objectives Calculus June 30, 2006
2 NUMERICAL AND PROPORTIONAL REASONING Quantitative relationships can be expressed numerically in multiple ways in order to make connections and simplify calculations using a variety of strategies, tools and technology. The concept of a limit is one of the foundations of calculus. f x as x approaches a given value or The limit of a function is the value approached by ( ) infinity. The derivative is the instantaneous rate of change at a given point. The integral is a function that can be used to determine the summation of an infinite set. Differentiation and definite integration are inverse operations. Essential Questions: How does the derivative represent an instantaneous rate of change? How does the integral represent the summation of an infinite set? How do you determine that a function is continuous and/or differentiable? Is there a way to visualize what a derivative is? Cluster: Functions, Graphs, and Limits Cal.1. Cal.2. Compare characteristics of major classes of functions and their rates of change (polynomial versus exponential versus logarithmic, etc.). Understand graphs of continuous functions using the Intermediate Value Theorem and Extreme Value Theorem. Cal.3. Cal.4. Cal.5. Cal.6. Understand the derivative when presented graphically, numerically, and analytically. Define the derivative as the limit of the difference quotient. Understand the relationship between differentiability and continuity. Understand the relationship between the increasing and decreasing behavior of f and the sign of f ' Cluster: Integrals Cal.7. Cal.8. Use the Fundamental Theorem of Calculus to find the derivative of a function defined by an integral. State the basic properties of definite integrals. 1
3 WORKING WITH DATA: PROBABILITY AND STATISTICS Data can be analyzed to make informed decisions using a variety of strategies, tools and technology. The slope of a line in algebra is the average rate of change while the slope of the tangent to a curve at a point in calculus is the instantaneous rate of change (the derivative of a functions). Essential Question: What does the graph of a function tell about the equation? How can calculus be used to solve problems in business and economics? How are derivatives used in optimization problems? Cluster: Functions, Graphs, and Limits Cal.9. Estimate limits from graphs or tables of data. Cal.10. Estimate function values using linear approximation. Cal.11. Approximate the rate of change from graphs and tables of values. 2
4 GEOMETRY AND MEASUREMENT Shapes and structures can be analyzed, visualized, measured and transformed using a variety of strategies, tools, and technology. The derivative of a function can be interpreted as an instantaneous rate of change. The definite integral can be used to find exact area, volume, or length by using the limit of Riemann sums. There is a defined relationship between the integral of function f, the function f, and the first and second derivatives of function f. Essential Question: How does the graph of a function relate to its equation? How are the following defined? (the area bounded by two curves, the volume generated by rotating a plane area, the length of a plane curve, the area of a surface revolution) What methods involving integrals can be used to find the volume of a solid? Cal.12. Understand the relationship between the concavity of f and the sign of f ''. Cal.13. Interpret the derivative as an instantaneous rate of change. Cal.14. Relate the graph of f ' to the graph of f. Cal.15. Apply the Mean Value Theorem and state its geometric consequences. Cal.16. State the geometric interpretation of differential equations via slope fields and the relationship between slope fields and solutions curves for differential equations. Cluster: Integrals Cal.17. Define the definite integral as a limit of Riemann sums. Cal.18. Apply the methods of integration as a rate of change to give accumulated change, finding the area of a region, the volume of a solid with known cross section, the average value of a function, and the distance traveled by a particle along a line. Cal.19. Use Riemann sums and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values. 3
5 ALGEBRAIC REASONING: PATTERNS AND FUNCTIONS Patterns and functional relationships can be represented and analyzed using a variety of strategies, tools, and technology. Derivatives can be used to solve a variety of problems involving instantaneous rate of change. Integrals can be used to solve a variety of problems related to area, velocity, acceleration, volume, area of a surface of revolution, length of a curve, and work. Essential Question: How can the concept of limits be applied in mathematics? How is the concept of a limit connected to a derivative and to an integral? How do the graphs of the first and second derivatives relate to the function graph? How is the rate of change reflected in its table and graph? Cluster: Functions, Graphs, and Limits Cal.20. Calculate limits using algebra. Cal.21. Understand asymptotes in terms of graphical behavior. Cal.22. Describe asymptotic behavior in terms of limits involving infinity. Cal.23. Understand continuity in terms of limits. Cal.24. Relate the graph of f '' to the graphs of f ' and f. Cal.25. Calculate the slope of a curve at a point, determining points of non-differentiability, corners, cusps, and vertical tangents). Cal.26. Find the tangent line to a curve at a point. Cal.27. Calculate the instantaneous rate of change as the limit of average rate of change. Cal.28. Calculate points of inflection. Cal.29. Apply mathematical optimization in solving problems, finding both absolute (global) and relative (local) extrema. Cal.30. Model rates of change, including related rates problems. Cal.31. Find derivatives using implicit differentiation. Cal.32. Analyze rates of change of inverse functions. Cal.33. Interpret the derivative as a rate of change in varied applied contexts, including velocity speed, and acceleration. Cal.34. Find the derivative of basic functions including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions Cal.35. Find the derivatives of sums, products, and quotients of functions. 4
6 Cal.36. Apply the Chain rule to differentiate a composite function. Cluster: Integrals Cal.37. State the definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval. Cal.38. Interpret the integral of a rate of change as a total change: f '( x) dx= f ( b) f ( a) Cal.39. Use the Fundamental Theorem of Calculus to evaluate definite integrals. Cal.40. Use the Fundamental Theorem of Calculus to represent a general antiderivative, and the analytical and graphical analysis of functions so defined. Cal.41. Find antiderivatives by substitution of variables. Cal.42. Find specific antiderivatives using initial conditions, including application to motion along a line. b a 5
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