The Advanced Placement Course
MA 0162 Advanced Placement CalculusThis course is intended to give the necessary background so that students may take the A.P. test which grants college credit upon successful completion if they desire. Topics included will be derivatives and applications, the integral and applications, exponential, logarithmic, trigonometric functions and methods of integration.
Level: A.P. AB Calculus
Credit: 1
Course Length: Full Year
Prerequisite: Students must have passed Algebra 1, Geometry, Algebra 2, Pre-Calculus or Trigonometry and Functions, and have a serious interest in advanced mathematics.
Course Type: Elective
Learning Standard/Objectives:
The goals for this course parallel the Mathematics Standards for grade 11-12.
Students will know and be able to do the following:
Number Sense and Operations
1. Simplify numerical expressions with powers and roots, including fractional and negative exponents.
Patterns, Relations, and Algebra
2. Demonstrate an understanding of the trigonometric, exponential, and logarithmic functions.
3. Perform operations on functions including composition. Find inverses of functions.
4. Given algebraic, numeric and/or graphical representations, recognize functions as polynomial, rational, logarithmic, exponential, or trigonometric.
5. Find solutions to quadratic equations (with real coefficients and real or complex roots) and apply to the solutions of problems.
6. Solve a variety of equations and inequalities using algebraic, graphical, and numerical methods, including the quadratic formula; use technology where appropriate. Include polynomial, exponential, logarithmic, and trigonometric functions; expressions involving absolute values; trigonometric relations; and simple rational expressions.
7. Use symbolic, numeric, and graphical methods to solve systems of equations and/or inequalities involving algebraic, exponential, and logarithmic expressions. Also use technology where appropriate. Describe the relationships among the methods.
8. Solve everyday problems that can be modeled using polynomial, rational, exponential, logarithmic, trigonometric, and step functions, absolute values, and square roots. Apply appropriate graphical, tabular, or symbolic methods to the solution. Include growth and decay; joint (e.g., I=Prt, y= k(w1 + w2)) and combined (F = G(m1m2)/d2) variation, and periodic processes.
9. Relate the slope of a tangent line at a specific point on a curve to the instantaneous rate of change. Identify maximum and minimum values of functions in simple situations. Apply these concepts to the solution of problems.
10. Describe the transla5tions and scale changes of a given function f(x) resulting from substitutions for the various parameters a, b, c, and d in y = af(b(x+ c/b)) + d. In particular, describe the effect of such changes on polynomial, rational, exponential, logarithmic, and trigonometric functions.
Geometry
11. Derive and apply basic trigonometric identities (e.g., sin2θ + cos2θ = 1, tan2θ + 1 = sec2θ) and the laws of sines and cosines.
12. Use the notion of vector to solve problems. Describe addition of vectors and multiplication of a vector by a scalar, both symbolically and geometrically. Use vector methods to obtain geometric results.
13. Relate geometric and algebraic representations of lines, simple curves, and conic sections.
14. Apply properties of angles, parallel lines, arcs, radii, chords, tangents, and secants to solve problems.
Measurement
15. Describe the relationship between degree and radian measures, and use radian measure in the solution of problems, in particular, problems involving angular velocity and acceleration.
16. Use dimensional analysis for unit conversion and to confirm that expressions and equations make sense.
Learning Experiences: The development of a student's power to use mathematics also involves learning signs, symbols, and terms of mathematics. This is best accomplished in problem-solving situations in which students have an opportunity to read, write, and discuss ideas so that the use of the language of mathematics becomes a natural. As students communicate their ideas, they learn to clarify, redefine and consolidate their thinking.
1. Students read more effectively, so they can write, speak, and think mathematically.
2. Success in subsequent mathematics courses and on standardized tests depends greatly on a student's ability to communicate in mathematics.
3. Students learn to use a wealth of problem-solving strategies. They learn to choose, compare, and combine strategies to solve problems.
Course Outline: (a brief outline of major topics and units that are central to the course; the sequence of topics and units may be altered by the teacher based on the needs of the students.)
Tests are given in two week intervals and quizzes are given daily.
Chapters 1-3 of this textbook emphasizes an intuitive approach to the understanding of calculus through the use of a graphing calculator and the examination of data, and graphical windows of specific ideas and concepts. These chapters detail the use and application of a graphing calculator to solve problems and perform experiments. Interpreting the results of the graphical windows and data analysis is stressed so that students can verbally and correctly write good justifications for the conclusions drawn from these tools. Students learn to program the calculator, running programs for Riemann Sums, Newton's approximation of Irrational Roots, and Slope fields. These chapters also involve many experiments to enhance the students understanding of functions, calculus ideas and concepts. Calculators are required on tests and quizzes in these chapters. These chapters are supported and supplemented by references to a textbook AP Calculus with the TI-82 Graphics Calculator by George Best and Sally Fischbeck. This is published by Venture Publishing, Andover, Ma.
Chapter #1. Students are introduced to the basics of a graphing calculator in this chapter. They become familiar to its basic functions by exploring previously learned material such as linear, quadratic, polynomial, rational, exponential, and logarithmic functions. Students learn how easy the calculator can write linear equations, and do quadratic, exponential, logarithmic, and trigonometric regressions. They also discover the how to find roots, maximum and minimum points.
1. Functions and Graphs
a. What is a function?
b. Basic functions and transformations.
c. Linear functions and Mathematical modeling.
d. Exponential functions.
e. Compounding interest and the Number e.
f. Inverse functions.
g. Logarithms.
h. Combining functions; Polynomial and Rational functions.
i. Composition of functions.
j. Trigonometric functions.
Chapter #2. Students are introduced to the idea of limits and derivatives from slopes of secant lines and tangent lines by an averaging technique. Students learn about limits after they become proficient in finding the slope of smaller and smaller secant segments surrounding a point. The students then learn to translate this idea to tables of data and graphical windows of specific conditions involving functions and derivatives. The students learn the relationships between a function and its derivatives before they know how the find these functions algebraically.
2. Derivative Functions
a. Average and instantaneous velocity.
b. The derivative of a function at a point.
c. The derivative function.
d. Calculating the derivative numerically.
e. Critical numbers; Relative maximum and minimum points.
f. Inflection points and the second derivative.
g. The limit of a function.
h. Continuity.
Chapter #3. Students are introduced to the relationship between velocity and distance traveled by a summation technique. They learn to program their calculators in order to explore the summing technique fully. They become proficient learning about left and right sums, midpoint sums and trapezoidal sums. The students spend time communicating these ideas with analytic geometry and their programs as they work on and present projects. Students also learn how these ideas relate to integrals and the Fundamental Theorem of Calculus before they learn the algebra of these topics.
3. The Definite Integral
a. Calculating distance traveled.
b. Calculating sums; Riemann Sums.
c. Definite integrals.
d. The Fundamental Theorem of Calculus.
Chapters 4-8 emphasizes the traditional algebraic approach to calculus which reinforces the ideas and concepts learned in the previous chapters where calculators were the emphases in the instruction. Calculators may only be used to verify answers obtained algebraically or on certain calculator required problems for these topics. Calculators may not be used on tests and quizzes in these chapters.
4. Differentiation Rules
a. Derivative rules for basic functions.
b. Differentiating Exponential functions.
c. The Product and Quotient rules.
d. Derivatives of Composite functions; the Chain rule.
e. Functions defined implicitly.
f. Related rates.
g. Approximations.
5. More Applications of the Derivative
a. Increasing and decreasing functions.
b. Applications of the second derivative.
c. Limits involving infinity.
d. Using calculus to solve optimization problems.
e. The Mean value theorem.
f. Antiderivatives.
6. Integrals
a. The definite integral again.
b. Functions defined by integrals; Accumulation functions.
c. The fundamental theorem again.
d. Areas of plane regions.
7. Finding Antiderivatives
a. Antiderivatives.
b. Integration using the chain rule.
c. Integration by parts.
d. The trig functions and their inverses
e. Numerical integration.
8. Using the Definite Integral
a. Net and total distance traveled.
b. Volumes by slicing.
c. The shell method.
d. Average value of a function.
e. More applications of the definite integral.
Chapter #9 In this chapter, students return to an emphasis on calculators in order to explore differential equations. They program their calculators with a slope field program and a Euler's Method program to help them develop a better understanding of these ideas while they learn to apply numerical approximations and graphical situations to previously learned ideas.
9. Differential Equations
a. Introduction.
b. Solving differential equations graphically; Slope fields.
c. Solving differential equations numerically; Euler's method.
d. Solving differential equations symbolically; Separation of variables.
Course Materials:
Calculus: Concepts and Calculators. Best, Carter, Crabtree. Venture Publishing. 2003
AP Calculus with the TI-82 Graphics Calculator. George Best, Sally Fishbeck. Venture Publishing. 1995
Technology: TI 83 and TI Presenter.
Formative Assessment:
Tests (2 week intervals)
Quizzes (daily)
Type Writings 1-3, Colin's Method (regularly)
Summative Assessments:
AB Calculus A.P. Exam in May (required)
