How do limits demonstrate the dynamic nature of Calculus? How will the three processes confirm the uniqueness of a limit?

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1 WDHS Curriculum Map: created by Andrea Kappre Course: Honors Calculus March 2012 Time Interval/ Content Standards/ Strands Essential Questions Skills Assessment Unit 1: Limits Sections F-IF.7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases F-IF.9. Compare properties of two functions each represented in a different way How do limits demonstrate the dynamic nature of Calculus? How will the three processes confirm the uniqueness of a limit? Relevant vocabulary, notation Finding limits graphically, numerically, and analytically. Continuity of a function and one sided limits. Infinite limits and their properties. notation Evaluate limits graphically, numerically and analytically, including one sided limits. Use the definition of continuity. Identify removable and nonremovable types of discontinuities. Determine infinite limits Use the properties of infinite limits to find vertical asymptotes. Apply the intermediate value theorem.

2 Unit 2: Differentiation sections: F-IF.4 For a function that models a relationship between two quantities interpret key features of graphs and tables in terms of quantities, and sketch graphs showing key features given verbal description of relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. What does it mean to say that a line is tangent to a curve at a point? How are the properties of differentiation useful in mathematics? Relevant vocabulary, notation The definition of tangent line with slope m. Relationship between differentiability and continuity. Basic differentiation rules and rates of change. Derivative of Trigonometric functions. Higher order derivatives. The Chain Rule. Implicit Differentiation. (Orbits) F-IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. notation Find the equation of a tangent line. Investigate the relationship between differentiability and continuity. Use differentiation rules to determine derivatives. Unit 3: Application of Differentiation sections: F-IF.4 For a function that models a used to describe the motion of an object? used to find the rates of change of two or more related variables that are Relevant vocabulary, notation Position, velocity and acceleration. Rates of change Related rates

3 relationship between two quantities interpret key features of graphs and tables in terms of quantities, and sketch graphs showing key features given verbal description of relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. F-IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. changing with respect to time? used to sketch a complete and accurate graph on a given interval? used to determine maximum and minimum values in real world applications? Extrema Applications of first and second derivative test. Curve Sketching Optimization. notation Solve applications of velocity and acceleration. Solve related rates problems. Find and classify extrema Sketch curves using derivatives. Solve optimization problems using derivatives. Unit 4: Integration sections: F-IF.8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. What does is mean to find the area of a plane region that is bounded by the graphs of functions? How are the properties of integration useful in mathematics? Relevant vocabulary, notation, Anti-derivatives and Indefinite integration. The definite integral The Fundamental Theorem of Calculus. Integration by U-substitution. notation Evaluate indefinite integrals Evaluate definite integrals Find the area under a curve

4 Apply the Fundamental Theorem of Calculus Recognize patterns that involve u-substitution. Unit 5: Applications of Integration. sections: F-IF.8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. How can integrals be used to find the area between two curves? How can integrals be used to find the volume of a solid of revolution? Relevant vocabulary, notation, Area of a region between two curves. Volume of a region rotated around an axis. notation Use integration to determine the area between two curves. Use the washer, disc, and shell methods to find volumes. (Perfect Doughnut) Unit 6: Inverses sections: and 8.2 F-BF.5 - Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. F-TF.7 - Use Inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context What is a logarithm and how can a natural log be defined in terms of an integral? How does one deal with exponential and logarithmic functions in derivatives and integrals? What role do inverse Relevant vocabulary, notation, Integral definition of the natural logarithm Derivations of log properties, inverse properties The calculus definition of the number e, logs and exponentials of other bases.

5 trigonometric functions play in calculus? Integration techniques of exponentials, and natural logarithms. How to use inverse trig functions to integrate. notation Derive various properties of exponential and logarithmic functions. Integrate trigonometric, polynomial, exponential, and logarithmic functions. Identify integrals that involve inverse trig functions, make appropriate substitutions and integrate.