Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
AP Calculus AB 2014 – 2015 Brief Description of Course Calculus AB is a course in single-variable calculus that includes techniques and applications of the derivative, techniques and applications of the definite integral, and the Fundamental Theorem of Calculus. Algebraic, numerical, and graphical representations are emphasized throughout the course. Expectations Students will be expected to work in collaborative small groups. Each student will be teacher; each will be learner as they help each other master the material. Emphasis will be placed on student’s communicating their mathematical thinking within their groups, to the class as a whole, and on assessments. Students will be expected to explain their processes and justify their results both orally and in written form, in clear, concise, well-crafted sentences. Objectives and Goals Students should understand the meaning of the derivative in terms of rate of change and local linear approximations. Students should be able to work with functions represented graphically, numerically, analytically, or verbally, and should understand the connections among these representations. Students should understand the meaning of the definite integral both as a limit of Riemann sums and as a net accumulation of a rate of change; and understand the relationship between the derivative and integral. Students should be able to model problem situations with functions, differential equations, or integrals, and communicate both orally and in written form. Students should be able to represent differential equations with slope fields, solve separable differential equations analytically, and solve differential equations using numerical techniques such as Euler’s method. Technology Our study of calculus will include the continual use of a variety of technologies; chief among them will be the graphing calculator. Urbandale High School requires students to have a graphing calculator, with a recommendation of the TI-84 Plus, to use daily to explore, discover and reinforce the concepts of calculus. Students may use the graphing calculators on most but not all assessments. Students will be able to use their calculator to: Graph a function within an arbitrary viewing window. Solve an equation graphically and numerically. Analyze and interpret their results. Compute the numerical derivative of a function. Compute definite integrals numerically. Conduct explorations. Examples: Students will use the graph and zoom feature to investigate lim x®0 sinq q . They will support their conclusion using the table feature. After finding the derivative of a function students will graph f(x) and f´(x). Students will then graph nDeriv(f(x)) to support their calculation of the derivative. Timeline This is a full year course (2 semesters or 36 weeks). I. II. III. IV. V. VI. VII. Prerequisites for Calculus (1 week) Completed as an independent study. Limits and Continuity (5 weeks) One formative, one summative assessment. Derivatives (6 weeks) Two formative, one summative assessment. Applications of Derivatives (6 weeks) Two formative, one summative assessment. The Definite Integral (6 weeks) Two formative, one summative assessment. Differential Equations and Mathematical Modeling (5 weeks) Two formative, one summative assessment. Applications of Definite Integrals (6 weeks) Two formative, one summative assessment. Major Topics I Prerequisites for Calculus 1.1 Lines Increments; Slope of a Line; Parallel and Perpendicular Lines; Equations of Lines; Applications 1.2 Functions and Graphs Functions; Domains and Range; Viewing and Interpreting Graphs; Even and Odd Functions; Symmetry; Functions Defined in Pieces; Absolute Value Functions; Composite Functions 1.3 Exponential Functions Exponential Growth and Decay; Applications; The Number e 1.4 Parametric Equations Relations; Circles, Ellipses, Lines and Other Curves 1.5 Functions and Logarithms One-to-One Functions; Inverses; Finding Inverses; Logarithmic Functions; Properties of Logarithms; Applications 1.6 Trigonometric Functions Radian Measure; Graph’s Periodicity; Even and Odd Trigonometric Functions; Transformations of Trigonometric Graphs; Inverses II Limits and Continuity 2.1 Rates of Change and Limit Average and Instantaneous Speed; Definition of Limit; Properties of Limits; Onesided and Two-sided Limits; Sandwich Theorem 2.2 Limits Involving Infinity Finite Limits as x±∞; Sandwich Theorem Revisited; Infinite Limits as xa; End Behavior Models 2.3 Continuity Continuity at a Point; Continuous Functions; Algebraic Combinations, Composites; Intermediate Value Theorem for Continuous Functions 2.4 Rates of Change and Tangent Lines Average Rates of Change; Tangent to a Curve; Slope of a Curve; Normal to a Curve; Speed Revisited III Derivatives 3.1 Derivative of a Function Definition; Notation; Relationship Between the Graphs of f and f ´; Graphing the Derivative from Data; One-sided Derivatives 3.2 Differentiability How f ´(a) Might Fail to Exist; Differentiability Implies Local Linearity; Derivatives on a Calculator; Differentiability Implies Continuity; Intermediate Value Theorem for Derivatives 3.3 Rules for Differentiation Positive Integer Powers; Multiples, Sums, and Differences; Products and Quotients; Negative Integer Powers of x; Second and Higher Order Derivatives 3.4 Velocity and Other Rates of Change Instantaneous Rates of Change; Motion along a Line; Sensitivity to Change; Derivatives in Economics 3.5 Derivatives of Trigonometric Functions Derivative of the Sine and Cosine Functions; Simple Harmonic Motion; Jerk; Derivatives of Other Basic Trigonometric Functions 3.6 Chain Rule Derivative of a Composite Function; Outside-Inside Rule; Repeated use of the Chain Rule; Slopes of Parameterized Curves; Power Chain Rule 3.7 Implicit Differentiation Implicitly Defined Functions; Tangents and Normal Lines; Derivatives of Higher Order; Rational Powers of Differentiable Functions 3.8 Derivatives of Inverse Trigonometric Functions Derivatives of Arcsine, Arccosine, Arctangent, Arc cotangent, Arc secant, and Arc cosecant functions 3.9 Derivatives of Exponential and Logarithmic Functions Derivative of ex; Derivative of ax; Derivative of ln x; Derivative of logax; Power Rule for Arbitrary Real Powers IV Applications of Derivatives 4.1 Extreme Values of Functions Absolute (Global) Extreme Values; Local (Relative) Extreme Values; Locating Extreme Values 4.2 Mean Value Theorem Mean Value Theorem; Physical Interpretation Increasing and Decreasing Functions; Other Consequences 4.3 Connecting f and f ´´with the graph of f First Derivative Test; Concavity; Points of Inflection; Second Derivative Test; Learning about Functions from Derivatives 4.4 Modeling and Optimization Examples from Mathematics, Business and Industry; Economics Modeling; Discrete Phenomena with Differentiable Functions 4.5 Linearization and Newton’s Method Linear Approximation; Newton’s Method; Differentials Estimating; Change with Differentials; Absolute, Relative and Percentage Change; Sensitivity to Change 4.6 Related Rates Related Rate Equations Solution Strategy; Simulating Related Motion V The Definite Integral 5.1 Estimating with Finite Sums Distance Traveled; Rectangular Approximation Method (RAM); Volume of a Sphere; Cardiac Output 5.2 Definite Integrals Riemann Sums; Terminology and Notation of Integration; Definite Integral and Area; Constant Functions; Integrals on a Calculator; Discontinuous Integrals Functions 5.3 Definite Integrals and Antiderivatives Properties of Definite Integrals; Average Value of a Function; Mean Value Theorem for Definite Integrals; Connecting Differential and Integral Calculus 5.4 Fundamental Theorem of Calculus Fundamental Theorem, Part I; Graphing the Function ò x a f (t)dt ; Fundamental Theorem, Part II; Area Connection; Analyzing Antiderivatives Graphically 5.5 Trapezoidal Rule and Simpson’s Rule Trapezoidal Approximations; Approximations by Simpson’s Rule; Error Analysis VI Differential Equations and Mathematical Modeling 6.1 Slope Fields and Euler’s Method Differential Equations; Slope Fields; Euler’s Method 6.2 Antidifferentiation by Substitution Indefinite Integrals; Leibniz Notation and Antiderivatives; Substitution in Indefinite Integrals; Substitution in Definite Integrals 6.3 Antidifferentiation by Parts Product Rule in Integral Form; Solving for the Unknown; Integral Tabular Integration; Inverse Trigonometric and Logarithmic Functions 6.4 Exponential Growth and Decay Separable Differential Equations; Law of Exponential Change; Continuously Compounded Interest; Radioactivity; Modeling Growth with Other Bases; Newton’s Law of Cooling 6.5 Logistic Growth How Populations Grow; Partial Fractions; The Logistic Differential Equation; Logistic Growth Models VII Applications of Definite Integrals 7.1 Integral As Net Change Linear Motion Revisited; General Strategy Consumption Over Time; Net Change from Data; Work 7.2 Areas in the Plane Area Between Curves; Area Enclosed by Intersecting Curves; Boundaries with Changing Functions; Integrating with Respect to y; Saving Time with Geometry Formulas 7.3 Volumes Volume As an Integral; Square Cross Sections; Circular Cross Sections; Cylindrical Shells; Other Cross Sections 7.4 Lengths of Curves A Sine Wave Length of Smooth Curve; Vertical Tangents, Corners and Cusps 7.5 Applications from Science and Statistics Work Revisited; Fluid Force and Fluid Pressure; Normal Probabilities Textbooks Title: Calculus: Graphical, Numerical, Algebraic, AP Edition Publisher: Addison Wesley Publishing Company Published Date: 2006-02 Author: Franklin Demana Second Author: Bert K. Waits Third Edition 2007 Other Course Materials Student Practice Workbook ISBN 0-13-201411-4 Barron’s, AP Calculus, 10th edition, Hockett, Bock Pearson Education, AP Calculus, Barton, Brunsting, Diehl, Hill, Tyler, Wilson