# PLEASANT VALLEY SCHOOL DISTRICT

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.

II. 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) AP |GRADE(S): 12 |

|STRAND: 2.2 (Part 1) |TIME FRAME: One (1) Year |

|PA COMMON CORE STANDARDS |

|CC.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 L’Hopital’s 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) AP |GRADE(S): 12 |

|STRAND: 2.2 (Part 2) |TIME FRAME: One (1) Year |

|PA COMMON CORE STANDARDS |

|CC.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) AP |GRADE(S): 12 |

|STRAND: 2.2 (Part 3) |TIME FRAME: One (1) Year |

|PA COMMON CORE STANDARDS |

|CC.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 Euler’s 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/EXTENSIONS |

|Correctives: |

|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) AP |GRADE(S): 12 |

|STRAND: 2.2 (Part 4) |TIME FRAME: One (1) Year |

|PA ACADEMIC STANDARDS |

|CC.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. |

|Assign student projects to be presented to class. |

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