AP Calculus B C Syllabus

Transcription

1 AP Calculus B C Syllabus Course Textbook Finney, Ross L., et al. Calculus: Graphical, Numerical, Algebraic. Boston: Addison Wesley, Additional Texts & Resources Best, George, Stephen Carter, and Douglas Crabtree. Concepts and Calculators in Calculus. 2 nd edition, Andover, Mass.: Venture Publishing. Foerster, Paul A. Calculus-Concepts and Applications. Emeryville, Calif.: Key Curriculum Press. Stewart, James. Calculus: Concepts and Contexts. 3 rd ed. Pacific Grove, Calif.: Brooks/Cole Publishing Co. D & S Marketing Systems Multiple-Choice and Free-Response Questions in Preparation for the AP Calculus (AB) Examination Multiple-Choice and Free-Response Questions in Preparation for the AP Calculus (BC) Examination Edwards, Bruce H., Ron Larson, and Robert P. Hostetler. Preparing for the AP Calculus AB and Calculus BC Examinations. 7 th and 8 th ed. Boston: Houghton Mifflin. Venture Publications Preparing for the Calculus AB Exam Preparing for the Calculus BC Exam Mathematics Criterion Center Calculus AB Student Workbook General Course Information Methods of instruction include, but are not limited to: lecture, discovery, group work, and demonstration. TI-83, 84 and 89 graphing calculator models are used to enhance learning. Calculator skills as required by the College Board Course Description are also taught. Throughout the course, students are taught and tested on their ability to communicate mathematically by explaining their results in sentences. They also present problems on the board to their peers and explain them. Graphing Calculators

2 Graphing calculators are used to enhance and compliment daily learning. For instance, we find the equation of a tangent line at a point, and then graph both the line and the curve and zo om in at the point of tangency to see the local linearity. After analytically finding where a function is increasing and decreasing, is concave up and concave down, and its extreme values, students are excited to verify their results with the graphing calculator. The calculator allows students to visualize their abstract solutions and understand exactly what they have found. We also use the table function to find limits by changing the x-values for a function and seeing the resulting y-value. This allows students to see limits as x approaches infinity, and also to see what happens as x approaches a vertical asymptote. As a class we can then discuss the implications of this information. We also spend time discussing what good calculator use is. The students are not only capable of finding roots, graphing equations, and finding numerical derivative and integrals, but also know when it is appropriate to use technology. The addition of technology to the curriculum has allowed us to examine more re al-life problems without getting lost in the grind of numerical calculation. Major Assignments & Assessments Each topic is assessed both mid-unit and at the conclusion of the chapter. Each assessment has a calculator and non-calculator section. Since this is the way the AP Exam is divided, it is important for the students to see problems both ways. Additionally, spiraling of past topics is included on assessments. Rule of Four The course provides students with the opportunity to work with functions represented in a variety of ways - graphically, numerically, analytically, and verbally and emphasizes the connections among these representations. Homework problems are chosen from the book to make sure a wide variety of problems are used. In addition, the students are introduced to previously released AP free response and multiple-choice problems relating to the current topic. On all homework, quizzes and tests, students are required to give complete explanations to all problems. This includes showing all mathematical steps for analytical problems and detailed verbal descriptions along with any appropriate mathematical notation for problems involving graphs or tables. A lot of time is spent explaining to the students exactly how they will be graded on the AP Exam and what it means to j ustify your answer.

3 The students are encouraged to work cooperatively both in and out of the classroom. They are frequently asked to work in groups to go over their homework and make test corrections. This gives them additional opportunity to verbalize their understanding of the problems by explaining the solutions to their peers. They are also asked to present free response problems to the class. This goes a long way in helping them internalize the concepts. AP Review At least six (6) full weeks are set-aside at the end of the year to tie all of the individual concepts together and make connections between them. During the review period, the students work several AP Free-Response & Multiple-Choice Tests from various publishers. The students are then asked to make corrections in order to see what concepts in which they need to improve. The free response questions are always graded according to AP standards so that the students can see if their justifications are in line with the AP requirements. Course Overview CH. 1 - PREREQUISITES FOR CALCULUS Review functions, symmetry, domain & range and trigonometry. Graphing equations, finding zeros of a function. CH. 2 - LIMITS AND CONTINUITY (11 DAYS) Graphical understanding of limits. Evaluating limits algebraically. Applying the properties of limits. Using the sandwich/squeeze theorem when necessary. Find and verify end-behavior models. Calculate limits as x Identify and define vertical and horizontal asymptotes using limits. Identify the intervals upon which a given function is continuous using the definition of continuity. Determine whether a function is continuous at an endpoint using the Extreme Value Theorem. Remove a removable discontinuity by extending or modifying a function. Apply the Intermediate Value Theorem and Extreme Value Theorem. Apply the properties of algebraic combinations and composites of continuous functions. Apply directly the definition of the slope of a curve in order to calculate slope at a point.

4 Find the equations of the tangent line and normal line to a curve at a given point. Find the average rate of change of a function. CH. 3 DERIVATIVES (17 DAYS) Finding limits using tables. Also used to demonstrate the sandwich/squeeze theorem and end behavior models. Calculate slopes and derivatives using the definition of the derivative. Graph f from the graph of f, graph f from the graph of f, and graph the derivative of a function given numerically with data. Find whether a function is not differentiable and distinguish between the corners, cusps, discontinuities, and vertical tangents. Approximate derivatives numerically and graphically. Use the rules of differentiation to calculate derivatives, including second and higher order derivatives. Use derivatives to analyze straight line motion and solve other problems involving instantaneous rates of change. Find derivatives of functions involving the 6 trigonometric functions. Differentiate composite functions using the Chain Rule. Find slopes of parameterized curves. Find derivatives using implicit differentiation. Find derivatives using the Power Rule of rational powers of x. Calculate derivatives of functions involving inverse functions. Calculate derivatives of exponential and logarithmic functions. Find the numerical value of a derivative at a point. CH. 4 - APPLICATIONS OF DERIVATIVE (17 DAYS) Determine the local or global extreme values of a function. Apply the Mean Value Theorem. Find the intervals on which a function is increasing, decreasing or monotonic. Use the First and Second Derivative Tests to determine the local extreme values of a function. Determine the concavity of a function and locate the points of inflection by analyzing the second derivative. Graph f using information about f and f. Solve application problems involving finding minimum or maximum values of functions.

5 Find linearization. Estimate the change in a function using differentials. Solve related rate problems. Used to illustrate linearization. CH. 5 - THE DEFINITE INTEGRAL (15 DAYS) Construct and interpret slope fields as visualizations of differential equations. Use basic properties to find antiderivatives, including problems with initial conditions. Approximate the area under the graph of a nonnegative continuous function by using rectangle and trapezoid approximation methods. Interpret the area under a graph as a net accumulation of a rate of change. Express the area under a curve as a definite integral and as a limit of Riemann sums. Compute the area under a curve using a numerical integration procedure. Apply the Fundamental Theorem of Calculus. Demonstrate the relationship between the derivative and the definite integral as expressed in both parts of the Fundamental Theorem of Calculus. Apply rules for definite integrals and find the average value of a function over a closed interval. Find the value of a definite integral. CH. 6 - DIFFERENTIAL EQ. & MATHEMATICAL MODELING (14 DAYS) Compute indefinite and definite integrals by the method of substitution. Solve a differential equation of the form dy/dx = f(x), in which the variables are separable. Use integration by parts to evaluate indefinite and definite integrals. Use tabular integration or the method of solving for the unknown integral in order to evaluate integrals that require repeated use of integration by parts. Solve problems involving exponential growth and decay in a variety of applications. Solve problems involving exponential or logistic population growth.

6 CH. 7 - APPLICATIONS OF DEFINITE INTEGRALS (12 DAYS) Solve problems in which a rate is integrated to find the net change over time in a variety of applications. Use integration to calculate areas of regions in a plane. Use integration to calculate volumes of solids by slicing (cross-sections). Use integration to calculate volumes of solids using washers and shells. Use integration to calculate surface areas of solids of revolution. Use integration to calculate lengths of curves in a plane. CH. 8 - L HOPITAL S RULE, IMPROPER INTEGRALS, & PARTIAL FRAC. (7 DAYS) Solve a limit problem given in indeterminate form using L Hopital s rule. Determine which of two functions grows faster. Integrate an improper integral. Determine if an improper integral converges or diverges using L Hopital s as necessary. Use direct comparison test or the limit comparison test to see if an integral converges. Solve integration problems using the technique of partial fractions. CH. 9 - INFINITE SERIES (21 DAYS) Define series as a sequence of partial sums. Identify and sum geometric series. Representing functions with a series, including power series. Constructing Maclaurin & Taylor series. Manipulating Taylor series. Estimating the error of a series using Lagrange Error Bound for Taylor polynomials. Testing for convergence of a series using: Nth-term test, Direct Comparison Test, Ratio Test, Integral Test, P-series, Limit-Comparison, and Alternating Series Test, employing L Hopital s as necessary. Finding the error of an Alternating Series given a specific number of terms. Find the interval of convergence of a series, including power series. Test for convergence at endpoints. Identify harmonic series.

7 Determine whether a series is absolutely or conditionally convergent. Illustrate the Taylor approximation of a series to a function. CH PARAMETRIC, VECTOR, AND POLAR FUNCTIONS (12 DAYS) Find the first & second derivatives of parametric functions. Find the arc length of a parameterized curve. Find the surface area of a parameterized curve. Determine the velocity, speed, acceleration & direction of a vector. Integrate a vector. Differentiate a vector. Graph polar functions. Find the derivative of a polar curve. Find the area between a polar curve and the origin and the area between two polar curves. Find the length of a polar curve. Find the surface area of a solid in polar coordinates.