Calculus AB

Mrs. Reynolds and Mr. Marcy

2015-2016

Inspire School of Arts and Sciences

Course Description

This course is the equivalent of a first semester college Calculus course and is designed to prepare students to take the AB Calculus AP Test. Topics include functions, graphs, limits, derivatives, applications of derivatives, integrals, and applications of integrals. The course provides students with the opportunity to work with functions represented in a variety of ways – graphically, numerically, analytically, and verbally and emphasized the connections among these representations. The course teaches students how to communicate mathematics and explain solutions to problems both verbally and in written sentences. Graphing calculators are used in this course. The course teaches students how to use graphing calculators to help solve problems, experiment, interpret results, and support conclusions. The course requires a solid background in Algebra.

The emphasis is on understanding concepts, therefore topics will be presented geometrically, numerically, verbally, and algebraically. Part of the aim of this course is to train you to think logically. Learn to write the solutions of the exercises in a connected, step by step fashion with explanatory sentences – not just a string of disconnected equations or formulas. (Stewart)

Course Text

Stewart, James. Single Variable Calculus with Vector Functions Early Transcendentals. 7th Edition, AP Edition. 2012 Brooks/Cole, Cengage Learning.

In Calculus, reading the text is an essential part of the learning curve and will, in the end, save you time in understanding and mastering the material. There is more to the study of calculus than just being able to do some mathematics; you must understand the concepts and how they fit together.

Daily Assignments

Students will be given daily assignments. Some assignments may consist of old AP exam questions or review questions from the text to prepare the student for the AP exam. They may be given as individual or group work. They will always be graded according to the scoring guide from the exam. These are one of the best tools to use in preparing for the AP exam and we will us them often. Please keep all homework assignments in a notebook that you will turn in for a homework grade on the day of the unit test.

Class Participation

Students are expected to participate in group work when provided in class. Most days class will includegroup work that creates a classroom atmosphere of experimentation and inquiry. Each section of the book that we will cover consists of group activities. Groups will be required to share their answers, findings, and/or struggles with the class. Each student is expected to present throughout the semester. Explaining solutions using correct mathematical language and notation is emphasized. This will teach students how to communicate math.

Projects

Projects for this course may come in the form of either written, laboratory, applied, discovery forms. An extended assignment gives students the chance to take a focused problem or project and explore it in depth making conjectures, discussing them, eventually drawing conclusions and writing them up in a clear, précis format. Projects will typically be done in small groups.

Outlines are provided at the start of a Chapter or Section to assist the student in noting essential information while reading, viewing lectures, screencasts, or videos, and studying. They are meant to GO ALONG with whatever form of notes are taken in class/at home.

Grading

HW = 10%, Quiz = 10%, Tests = 80%

Course Outline – Note: I will provide you with better outlines for each chapter.

(Please note that due to our block schedule 1-2 sections may be covered in one day. The # of classes listed below with chapters is a very ROUGH estimate)

Chapter 1 – Functions and Models

(5 classes)

Content covered in this chapter:

  • To represent functions (verbally, numerically, visually, and algebraically).
  • Produce graphs of functions with the aid of technology
  • (review) Domain and range, composite functions, piecewise defined functions, exponential and log functions, trig functions, inverse functions.
  • Compare relative magnitudes of functions and their rates of change.

Chapter 2 – Limits and Derivatives

(9 classes)

Content covered in this chapter:

  • An intuitive understanding of the limiting process
  • The tangent line as the limit of secant lines and obtained by “zooming” in on a smooth function; local linearity.
  • Approximate the slope of the tangent line using slopes of secant lines.
  • The precise definition of a limit.
  • Estimate limits from graphs or tables of data.
  • Understand asymptotes in terms of graphical behavior.
  • Asymptotic behavior in terms of limits involving infinite.
  • Calculating limits using algebra
  • Continuity in terms of limits.
  • The Intermediate Value Theorem.
  • Derivative presented graphically, numerically, and analytically.
  • Derivative interpreted as an instantaneous rate of change.
  • Derivative defines as the limit of the difference quotient.
  • Slope of a curve at a point, including points at which there are vertical tangents and points at which there is no tangent.
  • Tangent line to a curve at a point.
  • Instantaneous rate of change as the limit of average rate of change.
  • Approximate rate of change from graphs and tables of values.
  • Corresponding characteristics of f and f’ and the graphs of f, f’, and f’’.
  • Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration.

Chapter 3 – Differentiation Rules

(9 classes)

Content covered in this chapter:

  • Tangent line to a curve at a point and local linear approximation.
  • Knowledge of derivatives of power functions, trigonometric functions, inverse trigonometric functions, and logarithmic functions.
  • Derivative rules for sums, products, and quotients of functions.
  • Chain Rule
  • Implicit Differentiation and the use of it to find the derivative of an inverse functions.
  • Instantaneous rate of change as the limit of average rate of change.
  • Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration.
  • Equations involving derivatives.
  • Modeling rates of change, including related rates problems.
  • Tangent line to a curve at a point and local linear approximation.

Chapter 4 – Applications of Differentiation

(9 classes)

Content covered in this chapter:

  • Extreme Value Theorem: Geometric understanding of graphs of continuous functions.
  • Optimization, both absolute (global) and relative (local) extrema.
  • The Mean Value Theorem and its geometric interpretation.
  • Relationship between increasing and decreasing behavior of f and the sign of f’.
  • Relationship between the concavity of f and the sign of f’’.
  • Points of inflection as places where concavity changes.
  • Analysis of curves, including the notions of monotonicity and concavity.
  • The use of graphing calculators for estimation of local extrema and inflection points, contrasted with the use of calculus for précis computation of such points.
  • Corresponding characteristics of the graphs of f, f’, and f’’.
  • Antiderivatives following directly from derivatives of basic functions.
  • Finding specific antiderivatives using initial conditions, including applications to motion along a line.

Chapter 5 – Integrals [1-5]

(8 classes)

Content covered in this chapter:

  • Computation of Riemann sums using left, right, and midpoint evaluation points.
  • Integrals in applications to model physical, social, or economic situation. Specific application will include finding the area of a region and the distance traveled by a particle along a line.
  • Definite integral as a limit of Riemann sums.
  • Basic properties of definite integrals.
  • Use of Riemann sums and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values.
  • Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval.
  • Using the integral of a rate of change to give accumulate change.
  • Use of the Fundamental Theorem of Calculus to evaluate definite integrals.
  • Use of the Fundamental Theorem of Calculus to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined.
  • Antiderivatives following directly from derivatives of basic functions.
  • Finding specific antiderivatives using initial conditions.
  • Antiderivatives by substitution of variables.

Chapter 6 – Applications of Integration

(3-4 classes)

Content covered in this chapter:

  • Appropriate integrals are used in a variety of applications to model physical, biological, or economic situations. Specific applications include finding the area of a region and volume of a solid with known cross sections.
  • Areas between curves
  • Volume

Chapter 7/8 – Techniques of Integration and Further Applications of Integration

(8 classes)

Content covered in this chapter:

  • Integration by parts
  • Trigonometric integrals
  • Use of Riemann sums and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values.
  • Arc length.

Chapter 9 – Differential Equations

(4-5 classes)

Content covered in this chapter:

  • Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa.
  • Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations.
  • Euler’s method.
  • Solving separable differential equations and using them in modeling. In particular, studying the equation and exponential growth.

Begin AP Exam Review and Prep.