Department/Grade:  Math/11-12            Course:  AP Calculus                Revision Date:  August, 2007

                                               

Required/Elective:  Elective                  Course Length:  1 year             Prerequisite:  Precalculus or teacher

                                                                                                                                  recommendation

 

Course Description:  This course consists of a full high school year of work comparable to calculus courses in colleges and universities. Topics of study include differential and integral calculus, and discussions and exercises that emphasize a multi-representational approach in which concepts, results, and problems are expressed graphically, numerically, analytically and verbally. Students will be required to take the AP exam and are expected to seek college credit at higher institutions.

 

COURSE OBJECTIVES

CONTENT OUTLINE

Graphs and models

·          Sketch the graph of an equation by hand (basic parent graphs and basic transformations).

·          Use graphing calculator to graph functions.

·          Find the intercepts of a graph algebraically.

·          Use graphing calculator to find roots and y-intercepts of functions.

·          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.

·          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.

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.

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.

·          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.

·          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.

Tangent 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.

·          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.

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.

·          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.

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.

·          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.

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.

·          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.

·          Use a graphing calculator to numerically compute a definite integral. 

·          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.

·          Approximate a definite integral using Simpson’s Rule.

·          Use a graphing calculator program for approximating the area under a curve using Simpson’s method.

·          Analyze the approximate error in the Trapezoidal Rule and in Simpson’s Rule.

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.

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.

RESOURCE MATERIALS

ASSESSMENT OPTIONS

·          Major Text – Calculus of a Single Variable (Larson, Hostetler, Edwards)

·          www.collegeboard.com

 

·          Unit Tests

·          Cumulative Tests

·          Practice AP Exams

·          Final Exam

·          Open-Ended Assessments

·          Quizzes

·          Projects

·          Oral Presentations

·          Observation

·          Evaluate written work

·          Performance assessments

·          Problem solving journal/activity

·          Evaluate oral response

·          Anecdotal records

·          Performance tasks