AP Calculus BC Syllabus

Transcription

1 AP Calculus BC Syllabus Course Overview and Philosophy This course is designed to be the equivalent of a college-level course in single variable calculus. The primary textbook is Calculus, 7 th edition, by Ron Larson, Robert P. Hostetler, and Bruce H. Edwards; copyright 2002 by Houghton Mifflin Company. The main focus of this course is to develop the students understanding of the concepts of calculus and to provide experience with its methods and applications. The course emphasizes a multirepresentational approach to calculus, with concepts, results, and problems being expressed graphically, numerically, analytically, and verbally. The connections among these representations also are important. Graphing Calculator Technology An approved graphing calculator is required for this course. I recommend either the TI- 83, TI-83plus, or TI-89. Teacher and students will use the calculator as a tool to illustrate ideas and topics. I stress the four required functionalities of graphing technology: 1) Finding a root. 2) Sketching a function in a specified window. 3) Approximating the derivative at a point using numerical methods. 4) Approximating the value of a definite integral using numerical methods. Technology will be used regularly by students and teacher to reinforce the relationships among the multiple representations of functions, to confirm written work, to implement experimentation, and to assist in interpreting results. Multiple Representations/Communication Teacher and students will work with functions representing them graphically, numerically, analytically, and verbally and make connections between these representations. v(t) (1,1) t (2,-1) For example, in the function above if v(t) represents the velocity of an object in miles per

2 hour at any time t, and t represents the number of hours, then the time at which the object returns to its original position for 0 t 3 can be represented geometrically as the time when the area of the triangle in the first quadrant is equal to the area of the trapezoid in the fourth quadrant or as the value of x where when t 2. 5 hours. 0 x v( t) dt 0. In either case, the answer is Students will be encouraged to work together and explain solutions and variations in problem solving techniques with their classmates using proper vocabulary and terms. Often, students will work together in groups and present their conclusions in a single paper that is graded based upon the correctness of the mathematics and the quality of the presentation. Group Work and Experimentation At times students will work together in labs designed for them to learn through collaboration and discovery. Some labs will require the use of technology. *In one lab, students will graph y sin x in a standard trigonometric viewing window. Estimate the slope of the tangent line at various x-values and plot the slope values as a function of x on the overhead screen. (The slope values are clearly zero at the turning points and can be estimated to be +1 or -1 at the x-intercepts. A few more estimates will enable students to guess the curve.) Students should see that the slope curve follows the path of the cosine function. To test this conjecture, graph the numerical derivative of the sine in the same window. Then graph the cosine function and note that the two graphs are superimposed. Tracing gives the same values on both curves. From this point it is d easy to proceed to an analytic proof of (sin x) cosx. dx 2 *In another lab, students will calculate the approximate area under the curve f ( x) x in the first quadrant on the interval [0, 10] using both inscribed and circumscribed rectangles. They will start by using two rectangles, then five, then 10, and so on. They will keep a record of the areas of both the lower and upper sums. Students should agree that as the number of rectangles increases, the area obtained gets closer to the area under the curve. Students should also agree that the exact area under the curve should lie somewhere between the upper and lower sum for any given number of rectangles. Through careful investigation and observation, students eventually use sigma notation to 1000 set up and solve for the exact area under the curve,, by taking the limit as the 3 number of rectangles approaches infinity. Answers will be checked by using the graphing calculator to obtain the approximate area This will eventually lead into the Fundamental Theorem of Calculus, where students can verify that the area is x 1000 x dx

3 Course Sequence (The timeline for this course will depend upon individual student classes. However, it is tentatively planned that we will cover all chapters leading up through chapter 6 by the end of the first semester. This will leave chapters 7 through 12 for the second semester. ) Chapter P (Preparation for Calculus) P.1 Graphs and Models P.2 Linear Models and Rates of Change P.3 Functions and Their Graphs P.4 Fitting Models to Data Chapter 1 (Limits and Their Properties) 1.1 A Preview of Calculus -Understand what calculus is and how it compares to precalculus. -Understand that the tangent line problem is basic to calculus. -Understand that the area problem is also basic to calculus. 1.2 Finding Limits Graphically and Numerically -Estimate a limit using a numerical or graphical approach. -Learn different ways that a limit can fail to exist. -Study and use a formal definition of a limit. 1.3 Evaluation Limits Analytically -Evaluate a limit using properties of limits. -Develop and use a strategy for finding limits. -Evaluate a limit using dividing out and rationalizing techniques. -Evaluate a limit using the Squeeze Theorem. 1.4 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. -Understand and sue the Intermediate Value Theorem. 1.5 Infinite Limits -Determine infinite limits from the left and from the right. -Find and sketch the vertical asymptotes of the graph of a function. Chapter 2 (Differentiation) 2.1 The Derivative and the Tangent Line Problem -Find the slope of the tangent line to a curve at a point. -Use the limit definition to find the derivative of a function. -Understand the relationship between differentiability and continuity. 2.2 Basic 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 of the cosine function. -Use derivatives to find rates of change. 2.3 The 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.

4 -Find the derivative of a trigonometric function. -Find a higher-order derivative of a function. 2.4 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. 2.5 Implicit Differentiation -Distinguish between functions written in implicit form and explicit form. -Use implicit differentiation to find the derivative of a function. 2.6 Related Rates -Find a related rate. -Use related rates to solve real-life problems. Chapter 3 (Applications of Differentiation) 3.1 Extrema on an Interval -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. 3.2 Rolle s Theorem and the Mean Value Theorem -Understand and use Rolle s Theorem. -Understand and use the Mean Value Theorem. 3.3 Increasing and Decreasing Functions and the First Derivative Test -Determine intervals on which a function is increasing or decreasing. -Apply the First Derivative Test to find relative extrema of a function. 3.4 Concavity and the Second Derivative Test -Determine intervals on which a function is concave upward or concave downward. -Find any points of inflection of the graph of a function. -Apply the Second Derivative Test to find relative extrema of a function. 3.5 Limits and Infinity -Determine (finite) limits at infinity. -Determine the horizontal asymptotes, if any, of the graph of a function. -Determine infinite limits at infinity. 3.6 A Summary of Curve Sketching -Analyze and sketch the graph of a function. 3.7 Optimization Problems -Solve applied minimum and maximum problems. 3.8 Newton s Method -Approximate a zero of a function using Newton s Method. 3.9 Differentials -Understand the concept of a tangent line approximation. -Compare the value of the differential, dy, with the actual change in y, y. -Estimate a propagated error using a differential. -Find the differential of a function using differentiation formulas. Chapter 4 (Integration)

5 4.1 Antiderivatives and Indefinite Integration -Write the general solution of a differential equation. -Use indefinite integral notation for antiderivatives. -Use basic integration rules to find antiderivatives. -Find a particular solution of a differential equation. 4.2 Area -Use sigma notation to write and evaluate a sum. -Understand the concept of area. -Approximate the area of a plane region. -Find the area of a plane region using limits. 4.3 Riemann Sums and Definite Integrals -Understand the definition of a Riemann sum. -Evaluate a definite integral using limits. -Evaluate a definite integral using properties of definite integrals. 4.4 The Fundamental Theorem of Calculus -Evaluate a definite integral using the Fundamental Theorem of Calculus. -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. 4.5 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. 4.6 Numerical Integration -Approximate a definite integral using the Trapezoidal Rule. -Approximate a definite integral using Simpson s Rule. -Analyze the approximate error in the Trapezoidal Rule and in Simpson s Rule. Chapter 5 (Logarithmic, Exponential, and Other Transcendental Functions) 5.1 The Natural Logarithmic Function: Differentiation -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. 5.2 The Natural Logarithmic Function: Integration -Use the Log Rule for Integration to integrate a rational function. -Integrate trigonometric functions. 5.3 Inverse Functions -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. 5.4 Exponential Functions: Differentiation and Integration -Develop properties of the natural exponential function. -Differentiate natural exponential functions.

6 -Integrate natural exponential functions. 5.5 Bases other than e and Applications -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. 5.6 Differential Equations: Growth and Decay -Use separation of variables to solve a simple differential equation. -Use exponential functions to model growth and decay in applied problems. 5.7 Differential Equations: Separation of Variables -Use initial conditions to find particular solutions of differential equations. -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. 5.8 Inverse Trigonometric Functions: Differentiation -Develop properties of the six inverse trigonometric functions. -Differentiate an inverse trigonometric function. -Review the basic differentiation formulas for elementary functions. 5.9 Inverse Trigonometric Functions: Integration -Integrate functions whose antiderivatives involve inverse trigonometric functions. -Use the method of completing the square to integrate a function. -Review the basic integration formulas involving elementary functions Hyperbolic Functions -Develop properties of hyperbolic functions. -Differentiate and integrate hyperbolic functions. -Develop properties of inverse hyperbolic functions. -Differentiate and integrate functions involving inverse hyperbolic functions. Chapter 6 (Applications of Integration) 6.1 Area of a Region 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. 6.2 Volume: The Disk Method -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. 6.3 Volume: 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. 6.4 Arc Length and Surfaces of Revolution -Find the arc length of a smooth curve.

7 -Find the area of a surface of revolution. 6.5 Work -Find the work done by a constant force. -Find the work done by a variable force. 6.6 Moments, Centers of Mass, and Centroids -Understand the definition of mass. -Find the center of mass in a one-dimensional system. -Find the center of mass in a two-dimensional system. -Find the center of mass of a planar lamina. -Use the Theorem of Pappus to find the volume of a solid of revolution. 6.7 Fluid Pressure and Fluid Force -Find fluid pressure and fluid force. Chapter 7 (Integration Techniques, L Hopital s Rule, and Improper Integrals) 7.1 Basic Integration Rules -Review procedures for fitting an integrand to one of the basic integration rules. 7.2 Integration by Parts -Find an antiderivative using integration by parts. -Use a tabular method to perform integration by parts. 7.3 Trigonometric Integrals -Solve trigonometric integrals involving powers of sine and cosine. -Solve trigonometric integrals involving powers of secant and tangent. -Solve trigonometric integrals involving sine-cosine products with different angles. 7.4 Trigonometric Substitution -Use trigonometric substitution to solve an integral. -Use integrals to model and solve real-life applications. 7.5 Partial Fractions -Understand the concept of a partial fraction decomposition. -Use partial fraction decomposition with linear factors to integrate rational functions. -Use partial fraction decomposition with quadratic factors to integrate rational functions. 7.6 Integration by Tables and Other Integration Techniques -Evaluate an indefinite integral using a table of integrals. -Evaluate an indefinite integral using reduction formulas. -Evaluate an indefinite integral involving rational functions of sine and cosine. 7.7 Indeterminate Forms and L Hopital s Rule -Recognize limits that produce indeterminate forms. -Apply L Hopital s Rule to evaluate a limit. 7.8 Improper Integrals -Evaluate an improper integral that has an infinite limit of integration. -Evaluate an improper integral that has an infinite discontinuity. Chapter 8 (Differential Equations)

8 8.1 Slope Fields and Euler s Method -Use initial conditions to find particular solutions of differential equations. -Use slope fields to approximate solutions of differential equations. -Use Euler s Method to approximate solutions of differential equations. 8.2 Differential Equations: Growth and Decay -Use separation of variables to solve a simple differential equation. -Use exponential functions to model growth and decay in applied problems. 8.3 Differential Equations: Separation of Variables -Recognize and solve differential equations that can be solved by separation of variables. -Recognize and solve homogeneous differential equations. -Use differential equations to model and solve applied problems. 8.4 The Logistic Equation -Solve and analyze logistic differential equations. -Use logistic differential equations to model and solve applied problems. Chapter 9 (Infinite Series) 9.1 Sequences -List the terms of a sequence. -Determine whether a sequence converges or diverges. -Write a formula for the nth terms of a sequence. -Use properties of monotonic sequences and bounded sequences. 9.2 Series and Convergence -Understand the definition of a convergent infinite series. -Use properties of infinite geometric series. -Use the nth-term Test for Divergence of an infinite series. 9.3 The Integral Test and p-series -Use the Integral Test to determine whether an infinite series converges or diverges. -Use properties of p-series and harmonic series. 9.4 Comparisons of Series -Use the Direct Comparison Test to determine whether a series converges or diverges. -Use the Limit Comparison Test to determine whether a series converges or diverges. 9.5 Alternating Series -Use the Alternating Series Test to determine whether an infinite series converges. -Use the Alternating Series Remainder to approximate the sum of an alternating series. -Classify a convergent series as absolutely or conditionally convergent. -Rearrange an infinite series to obtain a different sum. 9.6 The Ratio and Root Tests -Use the Ratio Test to determine whether a series converges or diverges. -Use the Root Test to determine whether a series converges or diverges. -Review the tests for convergence and divergence of an infinite series.

9 9.7 Taylor Polynomials and Approximations -Find polynomial approximations of elementary functions and compare them with the elementary functions. -Find Taylor and Maclaurin polynomial approximations of elementary functions. -Use the remainder of a Taylor polynomial. 9.8 Power Series -Understand the definition of a power series. -Find the radius and interval of convergence of a power series. -Determine the endpoint convergence of a power series. -Differentiate and integrate a power series. 9.9 Representation of Functions by Power Series -Find a geometric power series that represents a function. -Construct a power series using series operations Taylor and Maclaurin Series -Find a Taylor or Maclaurin series for a function. -Find a binomial series. -Use a basic list of Taylor series to find other Taylor series. Chapter 10 (Conics, Parametric Equations, and Polar Coordinates) 10.1 Conics and Calculus -Understand the definition of a conic section. -Analyze and write equations of parabolas using properties of parabolas. -Analyze and write equations of ellipses using properties of ellipses. -Analyze and write equations of hyperbolas using properties of hyperbolas Plane Curves and Parametric Equations -Sketch the graph of a curve given by a set of parametric equations. -Eliminate the parameter in a set of parametric equations. -Find a set of parametric equations to represent a curve. -Understand two classic calculus problems, the tautochrone and brachistochrone problems Parametric Equations and Calculus -Find the slope of a tangent line to a curve given by a set of parametric equations. -Find the arc length of a curve given by a set of parametric equations. -Find the area of a surface of revolution (parametric form) Polar Coordinates and Polar Graphs -Understand the polar coordinate system. -Rewrite rectangular coordinates and equations in polar form and vice versa. -Sketch the graph of an equation given in polar form. -Find the slope of a tangent line to a polar graph. -Identify several types of special polar graphs Area and Arc Length in Polar Coordinates -Find the area of a region bounded by a polar graph. -Find the points of intersection of two polar graphs.

10 -Find the arc length of a polar graph. -Find the area of a surface of revolution (polar form) Polar Equations of Conics and Kepler s Laws -Analyze and write polar equations of conics. -Understand and use Kepler s Laws of planetary motion. Chapter 11 (Vectors and the Geometry of Space) 11.1 Vectors in the Plane -Write the component form of a vector. -Perform vector operations and interpret the results geometrically. -Write a vector as a linear combination of standard unit vectors. -Use vectors to solve problems involving force or velocity. Chapter 12 (Vector Valued Functions) 12.1 Vector Valued Functions -Analyze and sketch a space curve given by a vector-valued function. -Extend the concepts of limits and continuity to vector-valued functions Differentiation and integration of Vector-Valued Functions -Differentiate a vector-valued function. -Integrate a vector-valued function Velocity and Acceleration -Describe the velocity and acceleration associated with a vector-valued function. -Use a vector-valued function to analyze projectile motion Tangent Vectors and Normal Vectors -Find a unit tangent vector at a point on a space curve. -Find the tangential and normal components of acceleration Arc Length and Curvature -Find the arc length of a space curve. -Use the arc length parameter to describe a plane curve or space curve. -Find the curvature of a curve at a point of the curve. -Use a vector-valued function to find frictional force. Web Resources AP Central (apcentral.collegeboard.com)