AP Calculus (AB) Grades Recommended: 11, 12 Course Code: MA 590, Ma 591 Prerequisites: MA321, MA521 Length of course: one year Description The student will study all of the topics required for the Advanced Placement Calculus Exam (AB). Applications include approximations by differentials, work, maximum, minimum and solids of revolution. The fundamental ideas of calculus: limits, continuity, sequences and series, curves in the plane, derivatives and integrals and their applications are studied. A Graphing Calculator is required for this course. Students taking this course are expected to take the AP exam. Objectives Students should be able to work with functions represented in a variety of ways: graphical, numerical, analytical, or verbal. They should understand the connections among these representations. Students should understand the meaning of the derivative in terms of a rate of change and local linear approximation and should be able to use derivatives to solve a variety of problems. Students should understand the meaning of the definite integral both as a limit of Riemann sums and as the net accumulation of change and should be able to use integrals to solve a variety of problems. Students should understand the relationship between the derivative and the definite integral as expressed in both parts of the Fundamental Theorem of Calculus. Students should be able to communicate mathematics both orally and in well-written sentences and should be able to explain solutions to problems. Students should be able to model a written description of a physical situation with a function, a differential equation, or an integral. Students should be able to use technology to help solve problems, experiment, interpret results, and verify conclusions. Students should be able to determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement. Students should develop an appreciation of calculus as a coherent body of knowledge and as a human accomplishment. Technology and Other Resources Calculus textbook (James Stewart) Student Solutions Manual Graphing Calculator Grading System 50% Tests and Projects, 50% Homework and Class Participation
Syllabus AP Calculus (AB) Unit 1: Functions Graphs and Limits +( 1 st 9 weeks) All students will be asked to analyze graphs, determine limits of functions, understand asymptotic and unbounded behavior an continuity. Main Topics A. Functions and Models (Chapter 1*) B. Limits and Rates of Change. (Chapter 2*). C. Derivatives (Chapter 3) 1. Derivatives (3.1) 2. The Derivative as a Function (3.2) (* indicates entire chapter) Learning Outcomes (objectives from College Board): ASW Analysis of graphs: Recognize the interplay between geometric and analytic information Use calculus to predict and to explain the observed local and global behavior of a function Limits of functions: Develop an intuitive understanding of the limiting process Calculate limits using algebra Estimate limits from graphs or tables of data Continuity as a property of functions: Develop an intuitive understanding of continuity. (The function values can be made as close as desired by taking sufficiently close values of the domain.) Understand continuity in terms of limits. Geometric understanding of graphs of continuous functions (Intermediate Value Theorem) Concept of the derivative: Present the derivative graphically, numerically, and analytically Interpret the derivative as an instantaneous rate of change Understand the relationship between differentiability and continuity. Derivative at a point: Identify the slope of a curve at a point. (Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents.) Determine the tangent line to a curve at a point and local linear approzimation Derivative as a function: Demonstrate an understanding of corresponding characteristics of the graphs of f, f, and f
Products/Projects The students will research the history of calculus, choose one topic and present their material to their classmates and parents as a paper, role play and interview, poster, diorama, etc. Major Assessments The project described above ( or one comparable chosen by the teacher) and two section tests.
Syllabus: AP Calculus (AB) Unit 2: Derivatives and their Applications(2 nd -9 weeks) All students will demonstrate knowledge of!) concept of the derivative, 2) the derivative at a point, 3) the derivative as a function, 4) second derivatives, 5) the applications of derivatives, 6) antidrivatives. Main Topics D. Derivatives (Chapter 3) 1. Differentiation Formulas (3.3) 2. Rates of Change in the Natural and Social Sciences (3.4) 3. Derivatives of Trigonomic Functions (3.5) 4. The Chain Rule (3.6) 5. Implicit Differentation (3.7) 6. Higher Derivatives (3.8) 7. Related Rates (3.9) 8. Linear Approximations and Differentials (3.10) E. Applications of Differentiation (Chapter 4) 1. Maximum and Minimum Values (4.1) 2. The Mean Value Theorem (4.2) 3. How Derivatives Affect the Shape of a Graph (4.3) 4. Limits at Infinity; Horizontal Asymptotes (4.4) 5. Summary of Curve Sketching (4.5) Learning Outcomes (objectives from College Board)): ASW Asymptotic and unbounded behavior: Understand asymptotes in terms of graphical behavior Describe asymptotic behavior in terms of limits involving infinity Continuity as a property of functions: Develop a geometric understanding of graphs of continuous functions (Extreme Value Theorem) Derivative at a point: Develop an understanding of instantaneous rate of change as the limit of average rate of change. Find approximate rate of change from graphs and tables of values Derivative as a function: Understand the relationship between the increasing and decreasing behavior of f and the sign of f Translate verbal descriptions into equations involving derivatives and vice versa. Second derivatives: Demonstrate an understanding of corresponding characteristics of the graphs of f, f, and f Determine the relationship between concavity of f and the sign of f Recognize the points of inflection as places where concavity changes Applications of derivatives: Analyze curves, including the notions of monotonicity and concavity
Investigate optimization, both absolute (global) and relative (local) extrema. Model rates of change, including related rates problems Use implicit differentiation to find the derivative of an inverse function. Interpret the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration Computation of derivatives: Demonstrate knowledge of derivatives of basics trigonometric functions Apply the basic rules for the derivative of sums, products, and quotients of functions Use the Chain Rule and implicit differentiation Products/Projects Using the knowledge the students have acquired they will build a solution to a unique calculus problem, display and explain this solution to the class and parents. Major Assessments The project described above and two section tests.
Syllabus AP Calculus (AB) Unit 3: More Applications of Differentiation, Integrals and their Applications (3 rd 9-weeks) All students will interpret integrals, identify the properties of the definite integral, and use appropriate integrals to model physical, biological, or economic situations. All students will understand and use the Fundamental Theorem of Calculus, the use of Riemann sums to approximate definite integrals of functions represented in various formats. Main Topics F. Applications of Differentiation (Chapter 4) 1. Graphing with Calculus and Calculators (4.6) 2. Optimization Problems (4.7) 3. Applications to Economics (4.8) 4. Antiderivatives (4.10) G. Integrals (Chapter 5*) H. Applications of Integration 1. Areas between Curves (6.1) 2. Volumes (6.2) (* indicates entire chapter) Learning Outcomes (objectives from College Board): ASW Techniques of antidifferentiation: Antidrivatives following directly from derivatives of basic functions Antiderivatives by substitution of variables (including change of limits for definite integrals) Applications of antiderentaition: Find specific antiderivatives using initial conditions, including applications to motion along a line Solving separable differential equations and using them in modeling (in particular, studying the equation y = ky and exponential growth) Interpretations an properties of definite integrals Definite integral as a limit of Riemann sums Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval: b _ ƒ(x)dx = ƒ(b) - ƒ(a) a Basic properties of definite integrals (examples include additivity and linearity) Fundamental Theorem of Calculus Use of the Fundamental Theorem to evaluate definite integrals Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined Applications of integrals
Appropriate integrals are used in a variety of applications to model physical, biological, or economic situations. Although only a sampling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems. Whatever applications are chosen, the emphasis is on using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include using the integral of a rate of change to give accumulated change, finding the area of a region (including a region bounded by polar curves), the volume of a solid with known cross sections, the average value of a function, the distance traveled by a particle along a line, and the length of a curve. Numerical approximations to definite integrals Use of Riemann sums (using left, right, and midpoint evaluation points) and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values Products/Projects Build a model of the volume under a curve, display the model with correct explanation of the calculations of the volume. Major Assessments The project described above and three section tests
Syllabus AP Calculus (AB) Unit 4: Applications of Integration and Differential Equations (4 rd 9-weeks) All students will investigate and apply the definite integral by using it to compute areas between curves, volumes of solids, and the work done by varying force. Students will also use it to analyze a differential equation that arises in the process of modeling phenomenon s of physical science or social science. Main Topics L. Applications of Integration (6.3, 6.4, 6.5) 1. Volumes by Cylindrical Shells (6.3) 2. Work (6.4) 3. Average Value of a Function M. Differential Equations (Chapter 10*) N. AP Test Practice (* indicates entire chapter) Learning Outcomes (state objectives by number): ASW Applications of integrals Appropriate integrals are used in a variety of applications to model physical, biological, or economic situations. Although only a sampling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems. Whatever applications are chosen, the emphasis is on using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include using the integral of a rate of change to give accumulated change, finding the area of a region (including a region bounded by polar curves), the volume of a solid with known cross sections, the average value of a function, the distance traveled by a particle along a line, and the length of a curve. Numerical approximations to definite integrals Use of Riemann sums (using left, right, and midpoint evaluation points) and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and value Applications of Derivatives Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations
Products/Projects AP Calculus Exam (AB). Major Assessments The project described above and two section tests.