Instructor: Mr. Geanuracos (Mr. G)

School: Columbia High School

Room #: 4

Phone: (440) 236-5001

E-mail: jgeanuracos@columbia.k12.oh.us

**Course Description:**

The course aims at helping students understand the fundamental concepts of Calculus: limits, derivatives, and integrals. Multiple representations of functions (graphical, numerical, algebraic and verbal) deepen these concepts by allowing students various ways to access and coordinate knowledge. .

**Materials Needed:**

· Three ring binder ( I suggest at least 1 inch) A folder will also be helpful

· Loose Leaf paper

· Pencils

· Text Book: Finney, Ross L., Franklin D. Demana, Bert K. Waits, and Daniel Kennedy. *Calculus: Graphical, Numerical, Algebraic: AP Edition*. Boston: Pearson Prentice Hall

**Technology Component**

** **I will use a Texas Instruments 84 + C Silver Edition graphing calculator in class regularly. Students will need to have a graphing calculator as well. I recommend the TI-84+ C Silver Edition. We will use the calculator in a variety of ways, including:

• Conducting explorations

• Graphing functions within arbitrary windows

• Solving equations numerically

• Analyzing and interpreting results

• Justifying and explaining results of graphs and equations

**Earning your Grade:**

Grades for each 9 weeks are determined by the following percentages:

· Class work & Homework 20%

· Quizzes 30%

· Tests 50%

***Students can expect a test or quiz usually once a week. Plenty of advance warning and review will be given.

Final Grades will be calculated with each 9 weeks counting 40% each and the Exam worth 20%.

**Grading System:**

A 100 – 90

B 89 – 80

C 79 – 70

D 69 – 60

F 59 and below

** **

**Homework/Class Assignments/Teaching strategies:**

· Throughout the course, students work together on a regular basis, both formally and informally. At times, I set up groups to work on a particular activity, but students do not need to be told to work together. Our classroom has tables instead of desks to make it more conducive to group work. When students are working on a problem, they will often work alone initially but then turn to their partners to collaborate. This course teaches students how to use graphing calculators to aid in solving problems. I provide students with the opportunity to work with functions represented verbally. The course teaches all topics associated with integrals as delineated in the Calculus AB Topic outline in the AP Calculus course description. In discovering new concepts, the class works as a whole. It is not necessary for students to raise their hands. If students have a thought to share, they are welcome to make a contribution. If they are so inclined, students will go up to the board to illustrate a point. At times, I am able to step back and just listen to the interaction among my students as they explore a topic. Technology can be used to help make calculus concepts come alive and it enables students to “see” what is being discussed. Students are issued TI-84+ calculators that they will use to interpret Our classroom also an interactive white board as well as enough Ti-83+ /84+ calculators for every student to have their own. Topics are presented using the “rule of four”: graphically, numerically, algebraically, and orally. Through this multifaceted approach, students gain an in-depth understanding of the material.

** **

· Starting in October, I assign six AP free-response questions for students to work on for two weeks. Students may work on these questions with one other person and come to me for extra help. At the end of the two weeks, I randomly select one of these questions for a quiz. Students are graded as they would be graded on an AP Exam. Students are expected to explain the solutions to problems using written sentences.

**Tests and Quizzes:**

You will have a major test or quiz **at least** once every two weeks. Tests and quizzes will be a mix of short answer and multiple choice questions. I want to see your thought process, even if you get the final answer wrong, you may be able to get some points for your work. Don’t ever give up!

**The Syllabus **

**Fall Semester **

**Preliminaries** **4 days**

Introducing hidden behaviors

The calculator can mislead as well as help. *y = *cos(95*x*) and *y = *int(*x*), for example, do not graph properly in a standard viewing window and *y = *(1 + 1/*x*)^{x} will not graph properly for *x* > 10^{12}.

Review piecewise functions

Introduce *y = fpart*(*x*), the "fractional part of *x*," a built-in TI function, as a non-standard function requiring careful analysis.

Test on summer review

A test on some Precalculus concepts to determine both readiness for AP Calculus and motivation. (Students are supposed to complete this review on their own over the summer.)

**Limits and Continuity ** **7 days**

Finding limits numerically, graphically, algebraically.

For example, students use the calculator to compute values of *y = x*sin(*x*) for a sequence of values approaching *x* = π/2 then trace these values on the graph. Later they will use the properties of continuous functions to compute the limit exactly. Similar work is repeated for other functions, including polynomials.

One-sided limits, limits at infinity. Removable and essential discontinuities. Formal Limits: Target Values, epsilon-delta as domain/range restrictions. *Exploring Calculus with the Geometer Sketchpad* is very helpful here. Students will be skilled with finding and interpreting limits by the end of this unit.

**Differentiation Preliminaries** ** 17 days**

Essay 1: Analyze the function *y = *int(sin(*x*))

Developing the concept of the derivative

We follow a similar multiple representation approach here as for examining limits. Students approximate the derivative of a quadratic polynomial (at a point) by finding the average rate of change of the function over smaller and smaller intervals. These limits are visualized on a graph and, finally, computed algebraically using the definition of the derivative. The process is repeated with many functions. When the processes have been established, we introduce NDeriv, the calculator derivative approximator.

Local linearity

Discovering derivative formulas for power functions, *A* sin(*Bx*), *A* cos(*Bx*), ln(*x*), *a*^{x}

The students use the calculator's ability to graph an approximate derivative using NDeriv to graph derivatives of these functions and guess formulas for the graphs. Later we derive the formulas.

Classifying functions as differentiable or not

We examine the two ways in which a function can fail to have a derivative at a point, by not being continuous there (hole, jump, vertical asymptote or infinite oscillation) or by having different values for a left-handed and right-handed derivative (corner.)

The Product and Quotient Rules

Tangent lines and tangent linear approximations

Higher order derivatives

Turning points, Concavity, Points of inflection

First and Second Derivative Tests

The Mean Value Theorem

**Some Advanced Techniques 13 days**

Essay 2: Analyze the function *y = *sin(*x*) + *kx*

The Chain Rule

Derivatives of Inverse Functions

Implicit Functions, Implicit Differentiation

We do a lot of work with *Converge* in the computer lab to graph relations and to then use those visualizations to support the algebraic techniques of implicit differentiation.

Using Local Linearity to Find Certain Limits

L'Hopital's Rule and Indeterminate Forms

**Integration Preliminaries 13 days**

Recovering distance from velocity

Left- and right- rectangle approximations

Sigma notation and Riemann sums

The definite integral

Numerical methods (Rectangles, Trapezoids)

Properties of the integral

Average value of a function

Integrals as accumulators

Integrals and signed areas

The First Fundamental Theorem of Calculus

Midterm

**Spring Semester **

**Applications of Differentiation 10 days**

Related Rates

Optimization

**Differential Equations 10 days**

Initial value problems

IVP Lab, a *Converge* based lab

Visualizing solutions with slope fields

The Second Fundamental Theorem of Calculus

Separation of Variables

**Integration Techniques 8 days**

Integration by substitution

Integration by Parts

Supporting both methods with our calculator for accuracy

**Applications of Integration 15 days**

Finding areas

Finding volumes of uniform cross-section

Finding volumes of solids of revolution

Density

Work

**AP Exam Review 15 days**

The AP Exam

**Introduction to Multivariable Calculus 12 days**

Visualizing surfaces

Quadratic surfaces

Finding volumes of solids (informal double integration)

Final Exam