ࡱ> _ ;bjbj tbb3 $666PTl6JZFFFFFYYYYYYY$,\^Y"YFFmZ333XFFY3Y3325F L;:46YZ0JZ4_u_l5_5#0"3'CYY3JZ_ : PLEASANT VALLEY SCHOOL DISTRICT PLANNED COURSE CURRICULUM GUIDE CALCULUS (AB/BC) - AP Grade 12 I. COURSE DESCRIPTION AND INTENT: AP Calculus is designed for mathematically well-prepared students as a formal introduction to calculus. Topics include differentiation and integration of transcendental and non-transcendental functions as well as their applications. The students who successfully complete this course will have a thorough knowledge of first-semester and second-semester college level calculus and will be prepared for the Advanced Placement Examination in Calculus. The AP Calculus examination is not a course requirement.  INSTRUCTIONAL TIME: Class Periods: 6 per 6-day cycle Length of Class Periods (minutes): 56 Length of Course: One (1) Year Unit of Credit: 1.00 Updated: May 2012 COURSE: Calculus (AB/BC) APGRADE(S): 12STRAND: 2.2 (Part 1)TIME FRAME: One (1) Year PA COMMON CORE STANDARDSCC.2.2.HS.C.2  ASSESSMENT ANCHORS  RESOURCES Calculus of a Single Variable - Swokowski Previous AP exam problems College textbooks Graphing calculators Access computer software package TestGen Worksheets CBL (Calculator Based Lab)  OBJECTIVES The learner will identify the concept of a limit and apply limit theorems to solve problems.  ESSENTIAL CONTENT Find the value of a limit by examining the behavior of algebraic functions at various values especially values at which they are undefined. Find limits graphically, numerically, and algebraically. Define a limit mathematically and use the definition to demonstrate why limits exist or fail to exist. Explore properties of limits. Use limits involving infinity to connect the concept of horizontal and/or vertical asymptotes of rational functions. Apply the properties of limits and limit theorems to solve problems. Define the concept of a one-sided limit. Define continuity. Apply the definition of continuity to determine the continuity of a function at a given value and/or on an interval. Discuss the Intermediate Value Theorem and use it to solve problems. Use LHopitals Rule to evaluate limits. Use limits to determine the relative rates of growth of various functions. INSTRUCTIONAL STRATEGIES Use graphing calculators to enhance the visualization of a limit. Use graphing calculator to calculate limits that cannot be solved using limit theorems. Math journal entry describing the definition of limit. Warm-up problems from previous AP exams.  ASSESSMENTS  Teacher designed tests and quizzes Worksheets designed to demonstrate knowledge of the concepts taught Portfolio assessment Written or oral presentation of projects and/or homework Homework assessment Cooperative group assessments/competitions  CORRECTIVES/EXTENSIONS Correctives: Math tutoring lab. After school teacher help. Computer generated worksheets. Extensions: Bonus problems. Student generated proofs. Student generated problems applying limits to real-world situations. Assign student projects to be presented to class.  COURSE: Calculus (AB/BC) APGRADE(S): 12STRAND: 2.2 (Part 2)TIME FRAME: One (1) Year PA COMMON CORE STANDARDSCC.2.1.HS.F.4 CC.2.2.HS.C.2 CC.2.2.HS.C.6 CC.2.2.HS.C.8 CC.2.2.HS.D.9 CC.2.2.HS.D.10 ASSESSMENT ANCHORS  RESOURCES Calculus of a Single Variable - Swokowski Previous AP exam problems College textbooks Graphing calculators Access computer software package TestGen Worksheets CBL (Calculator Based Lab)  OBJECTIVES The learner will define the derivative, calculate derivatives, and apply the derivative to solve a variety of computational problems and application problems.  ESSENTIAL CONTENT Define the derivative in terms of the limit of a difference quotient. Discuss the importance of the derivative as a measurement of rate of change. Explore values at which a derivative does not exist. Apply the derivative to instantaneous velocity by exploring average velocity on very small time intervals. Connect the idea of one-sided limits to the definition of right handed and left handed derivatives at a value. Discover the connection between differentiability and continuity. Apply the power rule, product rule, quotient rule, and chain rule. Apply the above to functions which require more than one of these rules. Define the differential. Apply differentials to application problems. Define absolute error, percentage error, and relative error. Apply the rules of differentiation to algebraic functions. Apply the theorems for differentiation of natural exponential functions and natural logarithmic functions. Extend the theorems for differentiation of natural exponential functions and natural logarithmic functions to common exponential and logarithmic functions. Define higher order derivatives. Calculate first, second, and third derivatives of functions where they exist. Establish the intervals on which a function is increasing or decreasing. Utilize the concepts of increasing and decreasing to establish relative maximum and minimum points of a function. Prove Rolle's Theorem and the Mean Value Theorem. Use higher order derivatives to determine the concavity of the graph of a function. Find the points of inflection, if they exist, of the graph of a function. Develop a strategy for applying the idea of extreme values to practical applied max/min problems. Find the velocity and acceleration of a particle moving along a line. Apply the rules of differentiation to physical problems involving related rates. Introduce the concept of an antiderivative. Use vectors to find the velocity and/or the acceleration of an object. Find the slope of a curve defined parametrically. Find the slope of a curve defined using polar coordinates.  INSTRUCTIONAL STRATEGIES Use graphing calculator to have students discover the derivatives of trig functions and exponential functions. Use graphing calculator to enhance understanding of relative max and min values. Use calculus "match game" to have students identify a graph based on its derivative and vice-versa. Guided discovery. Cooperative learning activities. Warm-up problems from previous AP exams  ASSESSMENTS  Teacher designed tests and quizzes Worksheets designed to demonstrate knowledge of the concepts taught Portfolio assessment Written or oral presentation of projects and/or homework Homework assessment Cooperative group assessments/competitions CORRECTIVES/EXTENSIONS Correctives: Math tutoring lab. After school teacher help. Computer generated worksheets. Extensions: Bonus problems. Student generated proofs. Student generated problems applying the derivative to real-world situations. Assign student projects to be presented to class.  COURSE: Calculus (AB/BC) APGRADE(S): 12STRAND: 2.2 (Part 3)TIME FRAME: One (1) Year PA COMMON CORE STANDARDSCC.2.1.HS.F.4 CC.2.2.HS.C.2 CC.2.2.HS.C.6 CC.2.2.HS.D.6 CC.2.2.HS.D.9 CC.2.3.HS.A.14  ASSESSMENT ANCHORS  RESOURCES Calculus of a Single Variable - Swokowski Previous AP exam problems College textbooks Graphing calculators Access computer software package TestGen Worksheets CBL (Calculator Based Lab)  OBJECTIVES The learner will define the definite integral, calculate definite and indefinite integrals, and apply the definite integral to solve a variety of computational problems and application problems.  ESSENTIAL CONTENT Express a series using summation notation. Use the concept of a limit along with simple geometry to calculate the area under a curve. Define the definite integral as a limit of a Riemann Sum. Use the concept of signed area to relate the area under a curve to the value of a definite integral. Make the connection between the properties of a definite integral and the properties of a limit. Prove the Mean Value Theorem for definite integrals. Discover/Prove the Fundamental Theorem of Calculus. Apply the theorems for integration of natural exponential functions and natural logarithmic functions. Extend the theorems for integration of natural exponential functions and natural logarithmic functions to common exponential and logarithmic functions. Define indefinite integrals. Use the power rule for indefinite integration. Prove the change in variable theorem and use substitution to evaluate integrals. Define the Trapezoidal Rule and the Error Estimate for same. Define Simpson's rule and Error Estimate for same. Solve separable differential equations. Apply differential equations to problems involving growth and decay. Use the Trapezoidal Rule and Simpson's rule to approximate the definite integrals for stated values. Apply the idea of integrals to find the area between two curves. Find the volume of a solid of revolution by washers and discs, cylindrical shells, and by slicing. Define Hooke's Law. Use Hooke's Law to solve work problems. Apply the definite integral to solve force problems. Apply the definite integral to find the length of irregular arcs. Use the definite integral as an accumulation function. Approximate function values using Eulers Method. Evaluate integrals using integration by parts. Evaluate integrals using integration by partial fractions. Identify and evaluate improper integrals. Use logistic growth models to solve application problems. Find the length of a curve defined parametrically. Use the integral along with vectors to find the distance travelled and/or the displacement of an object. Find areas bounded by polar curves.  INSTRUCTIONAL STRATEGIES Use graphing calculator to enhance student visualization of calculating area under a curve by use of inscribed and circumscribed polygons. Use graphing calculator programs to do various techniques of numeric integration. Guided discovery. Cooperative learning activities. Warm-up problems from previous AP exams.  ASSESSMENTS  Teacher designed tests and quizzes Worksheets designed to demonstrate knowledge of the concepts taught Portfolio assessment Written or oral presentation of projects and/or homework Homework assessment Cooperative group assessments/competitions  CORRECTIVES/EXTENSIONSCorrectives: Math tutoring lab. After school teacher help. Computer generated worksheets. Extensions: Bonus problems. Integration by Trig Substitutions (alternate topic) Student generated proofs. Student generated problems applying the definite integral to real-world situations. Assign student projects to be presented to class.  COURSE: Calculus (BC) APGRADE(S): 12STRAND: 2.2 (Part 4)TIME FRAME: One (1) Year PA ACADEMIC STANDARDSCC.2.2.HS.C.2 CC.2.2.HS.C.3  ASSESSMENT ANCHORS  RESOURCES Calculus of a Single Variable - Swokowski Previous AP exam problems College textbooks Graphing calculators Access computer software package TestGen Worksheets CBL (Calculator Based Lab)  OBJECTIVES The learner will define infinite series, use various methods to determine the convergence of these series, and construct polynomials to converge to various rational, trigonometric, or transcendental functions.  ESSENTIAL CONTENT Identify geometric series and power series. Establish the convergence or divergence of geometric series. Extend the convergence of geometric series to the idea of a power series and the function it converges to on its interval of convergence. Represent various functions using infinite series. Create convergent series using integration & differentiation of known convergent series. Construct a Taylor Polynomial for various functions including sine, cosine, exponential functions, and logarithmic functions. Use common Maclaurin Series to generate other more complicated Maclaurin Series. Apply the Remainder Estimation Theorem Discover and prove the divergence of the harmonic series. Determine if a series is convergent or divergent using a variety of tests including ratio test, direct comparison test, limit comparison test, integral test, p-series test, alternating series test, nth term test, nth root test, and others. Find radius of convergence for a series Discover the difference between absolute and conditional convergence. Test for convergence of a series at the endpoints of the interval of convergence.  INSTRUCTIONAL STRATEGIES Use graphing calculators to enhance student comprehension of Taylor Polynomials. Guided discovery. Cooperative learning activities to enhance curriculum. Warm-up problems from previous AP exams.  ASSESSMENTS  Teacher designed tests and quizzes Worksheets designed to demonstrate knowledge of the concepts taught Portfolio assessment Written or oral presentation of projects and/or homework Homework assessment Cooperative group assessments/competitions  CORRECTIVES/EXTENSIONS Correctives: Math tutoring lab. After school teacher help. Computer generated worksheets. Extensions: Bonus problems. Student generated proofs. Student generated problems applying the infinite series and/or Taylor Polynomials to real-world situations especially in the realm of computer science. 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