## Calculus AB AP

- Course ID:MATH 507/508
- Semesters:2
- Department:Mathematics
- Course Rank:College Level
- Teachers:Miguel Hernandez

## Description and Objectives

The course AP Calculus (AB) is a one year course which serves to teach the student basic calculus at the university level while at the same time preparing him for the AP exam given by the College Board in May. When successfully completed, this course should replace a one-semester course in calculus given at an engineering school (e.g. Math 140 at University of Maryland). The fact that we take a full year to do a semester course provides us with the opportunity to solidify certain concepts and prevents us from rushing through the material.

The course can be broken down into three (3) basic segments of unequal lengths: Limit theory, Differential Calculus and Integral Calculus. The time allocation for these blocks is as follows: Approximately four (4) weeks are spent on limit theory, twelve (12) weeks on Differential Calculus and ten (10) weeks on Integration. Interspersed throughout will be review of and further development of Precalculus concepts already introduced in previous courses as well as student activities using Matlab and Mathematica on machines in the Computer Lab.

Homework is given daily and comprises 20% of the overall grade. Quizzes are given without warning but usually weekly and make-up 20% of the grade. There are typically six (6) full-period exams per semester (three per quarter) and a two (2)-hour midyear and Final exam.

As we approach the May AP exam date, our classes will increasingly concentrate on review of the material for the exam. We will have mini-mockAP exams daily in the final weeks before the real thing – seven multiple choice questions in 15 minutes followed by an immediate discussion of the solutions. Additionally there will be after-school reviews.

Students are required to obtain (whether by purchase or borrow) a graphing calculator. We will use the TI-83 plus in class for overhead presentations, but many others are suitable. Please consult with the instructor prior to purchasing a different machine.

## Textbooks

Our textbook is *Calculus of a Single Variable 8 ^{th}ed.*, Larson, Hostetler and Edwards. Additionally we will make use of the

*Princeton Review Cracking the AP Calculus*test preparation book.

## Course Requirements

#### Limit Theory

Objective: To introduce the student to the concept of a limit of a function. Here the interplay between domain and range are explored so that the student has a conceptual understanding of the symbols:

. We use this as a springboard to discuss what is meant by continuous function, a “right” continuous function. In addition we discuss what is meant by essential singularities of functions. All of these notions are taught with the understanding that ultimately we would like to describe the derivative of a function (i.e. the “slope” of a curve at a given point.)

A TI-84 plus connected to a ViewScreen provides quick and concrete examples for the students to see and consider. Additionally, Mathematica on a machine connected to a projector provides pages of viewable examples for the student. A brief review of rational functions as well as trigonometric functions is introduced. Lastly we look at the idea of a discrete function in the form of “sampled” functions. Using the TI-84 Regression subroutines we show how a sample can be modeled as linear, quadratic, cubic, quartic, exponential, periodic.

Text: Chapter 1

#### Differential Calculus

Objective: Here is the meat of the course. We introduce the concept of a derivative: the instantaneous velocity of a position function. Although we begin with the theoretical, we quickly move into the practical by discussing techniques for calculating derivatives of many functions. Specifically, we look at derivatives of polynomials, rational functions, exponential and trigonometric functions.

After the basics we look at the Algebra of Differentiation. Derivatives of sums/differences, products and composition of functions are explored. Next the derivative of the inverse of a function is discussed. We use this as a springboard to the discussion of derivatives of Logarithms and of inverse trigonometric functions.

The TI-84Plus with the ViewScreen overhead projector allows us to view the graphs of these derivatives.

The use of books I have created in Mathematica using the projector and my laptop allow the student to see the comparison of a function and it’s derivative in two different graphs vertically. They soon see the relationship between the critical numbers and “important” things happening on the graphs of the functions.

Second derivatives are discussed as “acceleration” and higher order derivatives are mentioned but not thoroughly explored. Here we make use of the First and Second Derivative tests and discuss the concepts of increasing/decreasing functions in terms of the first derivative, as well as concavity in terms of the second derivative.

We also discuss the very important Mean Value Theorem, tipping our “chapeau” to Michel Rolle. Students graph the secant line as well as the tangent line of a “smooth” function over a closed interval on a TI-84Plus and get accustomed to the idea of “instantaneous” vs. “average” velocity.

The derivative of a relation is discussed as the student learns to differentiate implicitly.

Next we look at applications of the derivative by turning our attention to Max/Min problems, curve sketching, and problems involving related rates.

Rounding things out is a brief discussion of L’Hopital’s rule and why we couldn’t use it to solve .

Text: Chapters 2, 3 & 5

#### Integral Calculus

Objective: This last segment of the curriculum serves to bridge the gap between function and velocity of function. Here we introduce the idea of “antidifferentiating”. That is, doing a derivative backwards. If we can find the instantaneous velocity of a function by knowing its position, can we also know its position by knowing its velocity? While Heisenberg (and Schroedinger’s cat) would have much to say on the subject, for us the answer will be yes! Kind of: It depends if the “conditions” are initially right! Here we will discuss the genesis of integration through Riemann sums.

The student will get comfortable with his calculator by learning to program it to calculate Riemann Sums of uniform width. Left/Right and Mid – point rules as well as trapezoidal rules are discussed. The interplay between “area” under a curve and integration are discussed.

Later we will discuss basic integration techniques: anti-power rule, -substitution and easy integration by parts. Next we look at integrals of exponential and logarithmic functions. Finally we discuss the integrals of trigonometric functions and inverse trigonometric functions.

The Fundamental Theorem of Calculus helps tie everything together and we discuss the “moving-area” function: . Differentiating such a function provides the springboard for the discussion of the Second Fundamental Theorem of Calculus.

We then turn our attention to applications of integration beginning with Volume of Solids of Revolution. The student gets to confirm some of the volume formulas learned so long ago for solids such as the right circular cone, right circular cylinder and sphere.

Lastly we look at differential equations. Although our treatment is cursory, the student learns to separate variables (for separable are the only type of differential equations we explore) and write equations in differential form. We use Matlab on an overhead projector to demonstrate the slope fields and how to generate them to aid in the solution of differential equations.

Text: Chapters 4, 5, 6 & 7

## Successful Students

Students may ask questions any time. Office hours are plentiful and include before school every day, lunch hour every day and after school by appointment. Students are encouraged to make use of the instructor’s email address: mhernandez@heights.edu or his extension on campus 301.365.0227×216