Primary Text

 

  • Stewart, James. Single Variable Calculus: Concepts and Contexts. Belmont, CA: Brookes/Cole.

 

Supplementary Materials

 

  • Larson, R, Hostetler, R, & Edwards, B (2006). Calculus of a single variable. Boston, MA: Houghton Mifflin.
  • Finney, R, Demana, F, Waits, B, & Kennedy, D (2007). Calculus: Graphical, numerical, and algebraic. Boston, MA: Pearson Prentice Hall.
  • Ebersole, D, Schattschneider, D, Sevilla, A, & Somers, K (1995). A companion to Calculus. Pacific Grove, CA: Brooks Cole.
  • Polya, G (1988). How to solve it. Princeton, NJ: Princeton.
  • Dunham, William (2005). The calculus gallery: Masterpieces from Newton to Lebesgue. Princeton, NJ: Princeton.
  • Maor, E (1998). e: The story of a number. Princeton, NJ: Princeton.

 

Technology

 

  • Maple V is used by the teacher to provide various visualizations and demonstrations to contextualize course concepts
  • All students are required to have an appropriate TI Graphing Calculator. The TI-89 is suggested; the TI-83 is the most popular model used.
  • Through the year students practice performing the following calculator tasks:
    • Graphing functions and adjusting viewing window
    • Finding Functional zeroes and roots
    • Finding a numeric derivative
    • Finding a numeric integral

 

Assessment

 

·         Tests: Each 9 week period includes 3 tests which count as 60% of the students’ average. Each test is comprised of AP-style problems (multiple choice and free response). Some tests require the use of a calculator; others exclude it.

·         Daily Grades: Homework is required and graded

·         Quizzes: Quizzes are given once or twice per week and typically spotlight a specific base skill required for the AP Test (e.g. calculator competencies, basic integral/derivative rules)

·         Semester Exams: These cumulative examinations count 20% towards a student’s semester grade. They are essentially mini-AP tests consisting of multiple choice and free response questions, timed in accordance with standard AP allowances.

·         Projects: During the fall semester, students engage in (both individually and as a group) in various minor projects.

§  Minor Project Example: Optimizing the shape of a vegetable can

§  Minor Project Example: Modeling the disappearance of a tootsie roll pop with related rates


 

Course Prospectus

 

This year-long course is equivalent to a first college course in calculus. Functions, Limits, Continuity, Derivatives, Integrals, and Differential Equations will be covered. The advanced placement curriculum is designed to mirror the progressive approach to mathematics now employed by universities. Unlike traditional math curricula in which answer-focused algorithmic mechanisms are emphasized, the College Board course is designed to provide a conceptually deep understanding of the included concepts, in addition to the traditional algorithmic proficiencies. As a result, answers per se are deemphasized; the cognitive approach used to achieve those answers is emphasized. All topics will be examined in a rich context; each major theme is approached numerically, graphically, analytically, and verbally.

In accordance with the policies of the College Board this course will provide students not only with the generic algorithms utilized in calculus, but empower them to communicate mathematics both orally and in well-written sentences. Students will be able to explain solutions to problems, model physical situations with functions or differential equations, use technology to help solve problems, experiment, interpret results, verify conclusions, and determine the reasonableness of solutions including sign, size, relative accuracy, and units of measurement.

 

Communications

 

The ability to communicate mechanisms of mathematical reasoning efficiently via writing is the most important skill students will develop in this course. Written problem solutions must be written up neatly and completely in pencil. Students will be required to argue theoretically from the evidence which they have been given. Students must be able to communicate clearly (with words, numbers, mathematical symbols, and graphs) their understanding of both a particular problem and the answers they find. Please note, “The calculator did it” is not a viable method of explaining your understanding. Failure to communicate mathematics neatly and in pencil will result in point reductions.

 

Problem Solving

 

Problem solving is the essence of mathematics. Throughout this calculus course students are expected to problem solve. Student will be introduced to a problem solving heuristic at the beginning of the course and for each topic will be expected to complete a problem solving exercise.

 

The importance of problem solving in this course is as the unifier of the all the course aspects described above and below. The students will learn all the specific topics (with accompanying theorems and procedures) listed below. In the problem solving exercises they will be challenged to apply their learning into a new context. In course of problem solving students will begin to work in groups, in which students can enhance their communication skills. Not only are students responsible for communicating via writing the problem approach, but they will also have to negotiate verbally, exchange suggestions for approaching a difficult problem, and most importantly verbalize their conceptual understanding.

 

As the second semester begins, students are challenged to heighten the level of their problem solving as they become accustomed to the AP-type problems. One such activity is based on the 2002’s Amusement Park Free Response Item. We use Play-Doh® and cut-out shapes to help student cognitively construct the notion of volumes with fixed cross sections. Finally the culminating project is a Clue®-inspired calculus murder mystery which I have designed. The students work in groups of two to determine who killed our victim by applying differential equations for Newton’s laws of motion and cooling, and the decay of poison. In part to its open-ended nature, it is the hardest problem which they will encounter, but it is also a fine approximation of an authentic “real-world” problem solving situation.


 

First Semester Outline:

 

Topics

Book Sections

Activities

I. Pre-Calculus Materials [2 weeks]

A.       Four ways to represent a function:
Verbally, Graphically, Numerically, Algebraically

B.       Introduction to Program Solving

(Polya’s 4 steps)

C.       Parent Functions Review

D.       Parametric Functions

E.        Geometric Function Behavior

a.        Extreme Value Theorem

b.       Intermediate Value Theorem

c.        Fundamental Theorem of Algebra

F.        Function Growth

a.        Polynomial vs. Exponential vs. Logarithmic

 

1.1-1.7

 

 

First Problem Solving
(p. 86)

II. Limits [4 weeks]

A.       Asymptotic Behavior

a.        Horizontal

b.       Vertical

B.       Finding limits

a.        Numerically

b.       Graphically

C.       Limit Notation & Laws

D.       Finding Limits algebraically

E.        Continuity & One Sided Limits

F.        Infinite Limits

 

2.2-2.5

Problem Solving (179)

III. Conceptual Derivatives [4 weeks]

A.       Secant approach & Average Rate of Change

B.       Local Linearity

C.       Tangent Approach & Instantaneous Rate of Change

D.       The Definition of the Derivative as a Numerical limit (of a difference quotient)

a.        Epsilon – Delta Definition

E.        Continuity & Graphical Interpretation of the Derivative

a.        Does Differentiability Imply Continuity?

b.       Does Continuity Imply Differentiability?

2.1, 2.3-2.9

Finding the derivative  graphically

 

Finding the derivative numerically

 

IV. Algebraic Derivatives [4 weeks]

A.       Polynomial Derivatives

a.        Power Rule

b.       The derivative as a linear operator (Scalar & Addition Rules)

B.       The Product & Quotient Rules

C.       Derivatives of Exponential, Logarithmic, Trigonometric, and Inverse Functions

D.       The Chain Rule

E.        Implicit Derivatives

 

3.1-3.8

Minor Project: Taylor Polynomials (254)

V. Derivative Applications [2 weeks]

A.       Connecting the Graphical and Algebraic

a.        Increasing vs. Decreasing (First Derivative Test)

b.       Concavity (Second Derivative Test)

B.       What does f tells us about f’

C.        Related Rates

D.       Minimums & Maximums (Optimization)

E.        Geometric Consequences of Mean Value Theorem

F.        Indeterminate Forms and L’Hopital’s Rule

 

4.1-4.6

Minor Project: Optimizing the Shape of a Can (316)

 

Problem Solving (339)

 

 

Second Semester Outline:

 

Topics

Book Sections

Activities & Assignments

VI. Basic Integrals [3 weeks]

  1. Anti-derivatives
    1. With Initial Conditions
    2. Without Initial Conditions
  2. Motion (Position, Velocity, Acceleration)
    1. Finding distance from a velocity function
  3. Area and Accumulated Change
    1. Naïve Fundamental Theorem of Calculus
    2. What is the meaning of a particular integral function

                                                               i.      Biological, Physical, Economic

  1. Riemann Sums
    1. Left, Right, Midpoint, Trapezoid
  2. Average Value

 

 

4.9-5.3, 6.4

Problem Solving (437)

VII. Integrals [4 weeks]

  1. Definite Integrals
  2. The Integral as a Linear Operator (Scalar & Addition Rules)
  3. Formal Fundamental Theorem of Calculus
    1. 1st FTC
    2. 2nd FTC
  4. Substitution
  5. Parts
  6. Improper

 

5.4-5.6, 5.10

Minor Project: Integrals at the Amusement Park

VIII. Differential Equations [3 weeks]

  1. Slope Fields
  2. Separable Equations
  3. Exponential Growth and Decay
  4. Graphical Implications of Differential Equations

 

7.1-7.4

Calculus Murder Mystery

IX. Integral Applications [3 weeks]

  1. Area between Curves
  2. Volumes of Revolution
    1. Disk/Washer Methods
    2. Shell Method
    3. Defined Cross-Sections

 

6.1-6.2

Play-Doh Discovery

 

Problem Solving (496)

X. Test Review

Assorted

Various Problem Solving

XI. After the AP Test

e: The story of a number

Various Problem Solving