2011-12+AP+Calculus



=**__Course Outline/Syllabus__**=
 * Mount Union Area School District Math Department **
 * //Advanced Calculus AB //**


 * ** Review (Approximately 2 Weeks) ** ||


 * < Graphs and models ||< * Sketch the graph of an equation by hand (basic parent graphs and basic transformations).
 * Use graphing calculator to graph functions. (Skill Sheet 1)
 * Find the intercepts of a graph algebraically.
 * Use graphing calculator to find roots and y-intercepts of functions. (Skill Sheet 2)
 * Test a graph for symmetry with respect to an axis and the origin.
 * Determine whether a function is even, odd, or neither.
 * Find the points of intersection of two graphs algebraically.
 * Use graphing calculators to find points of intersection of two functions. (Skill Sheet 3)
 * Interpret mathematical models for real-life data. ||
 * < Functions and their graphs ||< * Use function notation to represent and evaluate a function.
 * Find the domain and range of a function.
 * Use appropriate viewing windows and scales to view a function on the graphing calculator.
 * Sketch, compare, and contrast the graphs of functions (linear, quadratic, cubic, power, exponential, logarithmic, trigonometric).
 * Identify different types of transformations of functions.
 * Classify functions and recognize combinations of functions.
 * Use a graphing calculator to graph composite functions. (Skill Sheet 4) ||
 * < Fitting models to data ||< * Fit a linear model to a real-life data set.
 * Fit a quadratic model to a real-life data set.
 * Fit a trigonometric model to a real-life data set. ||


 * ** Unit I: Limits and Their Properties (Approximately 2 Weeks) ** ||


 * < Finding limits graphically and numerically ||< * Estimate a limit using a numerical or graphical approach.
 * Use the TRACE and TABLE features on the graphing calculator to estimate limits. (Skill Sheet 5)
 * Learn different ways that a limit can fail to exist.
 * Study and use a formal definition of a limit. ||
 * < Evaluating limits analytically ||< * Evaluate a limit using properties of limits.
 * Develop and use a strategy for finding limits.
 * Evaluate a limit using cancellation and rationalization techniques.
 * Understand that the graphing calculator may not distinguish clearly between rational algebraic functions and their algebraic equivalents.
 * Evaluate a limit using the Squeeze Theorem. ||
 * < Continuity and one-sided limits ||< * Determine continuity at a point and continuity on an open interval.
 * Determine one-sided limits and continuity on a closed interval.
 * Use properties of continuity.
 * Use the TRACE and TABLE features on the graphing calculator to determine continuity. (Skill Sheet 5)
 * Understand and use the Intermediate Value Theorem. ||
 * < Infinite limits and vertical asymptotes ||< * Determine infinite limits from the left and from the right.
 * Find and sketch the vertical asymptotes of the graph of a function. ||


 * ** Unit II: Differentiation (Approximately 5 Weeks) ** ||


 * < Tangents lines and the derivative ||< * Find the slope of the tangent line to a curve at a point.
 * Use the graphing calculator to plot tangent lines to a curve at a point. (Skill Sheet 7)
 * Use the nDeriv command on the TI83+ to compute the first derivative of a function at a given value.
 * Use the limit definition to find the derivative of a function.
 * Understand and explain the relationship between differentiability and continuity.
 * Represent derivatives graphically, numerically, and algebraically.
 * Describe the average rate of change as the slope of the secant line connecting two points on the graph.
 * Describe the difference quotient as the average rate of change.
 * Describe the derivative as the limit of the difference quotient.
 * Use a graphing calculator to graph the difference quotient, graphically find the limit of that difference quotient, and identify that limit as the derivative.
 * Describe the derivative as the slope of the tangent line to the graph at a point.
 * Describe the derivative as an instantaneous rate of change.
 * Estimate derivatives from graphs.
 * Estimate derivatives from tables of values.
 * Graph derivatives using the nDeriv command on the TI83+.
 * Use the graphing calculator to interpret the graph of a first derivative. (Skill Sheet 9) ||
 * < Differentiation rules and rates of change ||< * Find the derivative of a function using the Constant Rule.
 * Find the derivative of a function using the Power Rule.
 * Find the derivative of a function using the Constant Multiple Rule.
 * Find the derivative of a function using the Sum and Difference Rules.
 * Find the derivative of the sine function and cosine function.
 * Use derivatives to find rates of change. ||
 * < Position, velocity and speed, and acceleration ||< * Interpret the derivative of displacement as velocity.
 * Interpret the derivative of velocity as acceleration.
 * Use the graphing calculator to compare position, velocity, and acceleration graphs. ||


 * < Product and Quotient rules and higher order derivatives ||< * Find the derivative of a function using the Product Rule.
 * Find the derivative of a function using the Quotient Rule.
 * Find the derivative of the tangent, cosecant, secant, and cotangent functions.
 * Find a higher-order derivative of a function.
 * Use the graphing calculator to interpret the graph of a second derivative. (Skill Sheet 10)
 * Distinguish among graphs of functions and their first and second derivatives.
 * Describe corresponding characteristics of graphs of f, f’, and f’’. ||
 * < The Chain Rule ||< * Find the derivative of a composite function using the Chain Rule.
 * Find the derivative of a function using the General Power Rule.
 * Simplify the derivative of a function using algebra.
 * Find the derivative of a trigonometric function using the Chain Rule. ||
 * < Implicit Differentiation ||< * Distinguish between functions written in implicit form and explicit form.
 * Use implicit differentiation to find the derivative of a function. ||
 * < Related Rates ||< * Find a related rate.
 * Use related rates to solve real-life problems. ||


 * ** Unit III: Applications of Derivatives (Approximately 4 Weeks) ** ||


 * < Extreme Value Theorem ||< * Understand the definition of extrema of a function on an interval.
 * Understand the definition of relative extrema of a function on an open interval.
 * Find extrema on a closed interval.
 * Use the graphing calculator to find local maxima and minima. (Skill Sheet 8)
 * Understand and use the Extreme Value Theorem. ||
 * < Rolle’s Theorem/Mean Value Theorem ||< * Understand and use Rolle’s Theorem.
 * Understand and use the Mean Value Theorem. ||
 * < Increasing and decreasing functions ||< * Determine intervals on which a function is increasing or decreasing.
 * Apply the First Derivative Test to find relative extrema of a function. ||
 * < Concavity and second derivatives ||< * Determine intervals on which a function is concave upward or concave downward.
 * Find any points of inflection of the graphs of a function.
 * Apply the Second Derivative Test to find relative extrema of a function. ||
 * < Limits at infinity and horizontal asymptotes ||< * Determine (finite) limits at infinity.
 * Determine the horizontal asymptotes, if any, of the graph of a function.
 * Determine infinite limits at infinity. //Describe in terms of asymptotic behavior.// ||
 * < General curve sketching principles ||< * Analyze and sketch the graph of a function.
 * //Given the graph of f, sketch f’.//
 * //Given the graph of f’, sketch f.// ||


 * ** Unit IV: Integration (Approximately 5 Weeks) ** ||


 * < Introduction to differential equations ||< * Write the general solution of a differential equation.
 * Use the indefinite integral notation for antiderivatives.
 * Use basic integration rules to find antiderivatives.
 * Find a particular solution of a differential equation. ||
 * < Riemann sums and the definite integral ||< * Understand the definition of a Riemann sum (//definite integral = limit of Riemann sum//).
 * Use Riemann Sums (using upper, lower, and midpoint evaluation points) to approximate definite integrals.
 * Use a graphing calculator program for approximating the area under a curve using rectangles. (Skill Sheets 11 and 12)
 * Evaluate a definite integral using limits.
 * Evaluate a definite integral using properties of definite integrals. ||
 * < The Fundamental Theorem of Calculus ||< * Evaluate a definite integral using the Fundamental Theorem of Calculus.
 * Use a graphing calculator to shade a definite integral. (Skill Sheet 17)
 * Use a graphing calculator to numerically compute a definite integral. (Skill Sheet 18)
 * Understand and use the Mean Value Theorem for Integrals.
 * Find the average value of a function over a closed interval.
 * Understand and use the Second Fundamental Theorem of Calculus. ||
 * < Differential equations and integration by substitution ||< * Use pattern recognition to evaluate an indefinite integral.
 * Use a change of variables to evaluate an indefinite integral.
 * Use the General Power Rule for Integration to evaluate an indefinite integral.
 * Use a change of variables to evaluate a definite integral.
 * Evaluate a definite integral involving an even or odd function. ||
 * < Numerical Integration ||< * Approximate a definite integral using the trapezoidal rule (functions represented algebraically, graphically, and by tables of values).
 * Use a graphing calculator program for approximating the area under a curve using trapezoids. (Skill Sheets 13 and 14)
 * Approximate a definite integral using Simpson’s Rule.
 * Use a graphing calculator program for approximating the area under a curve using Simpson’s method. (Skill Sheets 15 and 16)
 * Analyze the approximate error in the Trapezoidal Rule and in Simpson’s Rule. ||


 * ** Unit V: Differential Equations (Approximately 5 Weeks) ** ||


 * < Definition of Logarithm as an Integral ||< * Develop and use properties of the natural logarithmic function.
 * Understand the definition of the number e.
 * Find derivatives of functions involving the natural logarithmic function. ||
 * < Slope Fields ||< * Use a slope field to sketch solutions of a differential equation.
 * Use Euler’s Method to approximate a solution of a differential equation.
 * Solve a first-order linear differential equation. ||
 * < Differential equations and logarithms ||< * Use the Log Rule for Integration to integrate a rational function.
 * Integrate trigonometric functions. ||
 * < Inverse functions and their derivatives ||< * Verify that one function is the inverse function of another function.
 * Determine whether a function has an inverse function.
 * Find the derivative of an inverse function. ||
 * < Exponential function is inverse of logarithmic function ||< * Develop properties of the natural exponential function.
 * Differentiate natural exponential functions.
 * Integrate natural exponential functions. ||
 * < Differential equations and the exponential function ||< * Develop properties of the natural exponential function.
 * Differentiate natural exponential functions.
 * Integrate natural exponential functions. ||
 * < Logistic differential equation ||< * Define exponential functions that have bases other than //e//.
 * Differentiate and integrate exponential functions that have bases other than //e//.
 * Use exponential functions to model compound interest and exponential growth. ||
 * < Growth and decay models ||< * Use separation of variables to solve a simple differential equation.
 * Use exponential functions to model growth and decay in applied problems. ||
 * < Models using separable differential equations ||< * Use initial conditions to find a particular solution of a differential equation.
 * Recognize and solve differential equations that can be solved by separation of variables.
 * Recognize and solve homogeneous differential equations.
 * Use a differential equation to model and solve an applied problem. ||
 * < Inverse trigonometric functions ||< * Develop properties of the six inverse trigonometric functions.
 * Differentiate an inverse trigonometric function.
 * Review the basic differentiation formulas for elementary functions. ||


 * ** Unit VI: Applications of Integration (Approximately 4 Weeks) ** ||


 * < Area between two curves ||< * Find the area of a region between two curves using integration.
 * Find the area of a region between intersecting curves using integration.
 * Describe integration as an accumulation process. ||
 * < Volumes by disks and known cross sections ||< * Find the volume of a solid of revolution using the disk method.
 * Find the volume of a solid of revolution using the washer method.
 * Find the volume of a solid with known cross sections. ||
 * < Volumes by the shell method ||< * Find the volume of a solid of revolution using the shell method.
 * Compare the uses of the disk method and the shell method. ||


 * ** Unit VII: Advanced Applications and Problem Solving (Approximately 5 Weeks) ** ||
 * Topics selected from practice AP Exams. ||

These topics will be chosen by the students according to their post-secondary school plans. ||
 * ** Unit VIII: Advanced Topics (Approximately 4 Weeks) ** ||
 * Students will spend the remaining time after the AP Exam studying advanced topics.


 * <  ||< **// Unit //** ||< **// Number of Objectives //** ||< **// Weeks //** ||
 * < MP 1 ||< P Review ||< 19 ||< 2 Weeks ||
 * ^  ||< I. Limits and Their Properties ||< 16 ||< 2 Weeks ||
 * ^  ||< II. Differentiation ||< 40 ||< 5 Weeks ||
 * < MP 2 ||< III. Applications of Derivatives ||< 18 ||< 4 Weeks ||
 * ^  ||< IV. Integration ||< 25 ||< 5 Weeks ||
 * < MP 3 ||< V. Differential Equations ||< 29 ||< 5 Weeks ||
 * ^  ||< VI. Applications of Integration ||< 8 ||< 4 Weeks ||
 * < MP 4 ||< VII. Advanced Applications and Problem Solving ||< TBA ||< 5 Weeks ||
 * ^  ||< VIII. Advanced Topics ||< TBA ||< 4 Weeks ||