Advanced Placement Calculus

I think that calculus defines more unequivocally than anything else, the inception of modern mathematics; and the system of mathematical analysis still constitutes the greatest technical advance in exact thinking.

- John von Newman

Textbooks: 1. CALCULUS Concepts and Applications, Paul A. Forester.

2. CALCULUS, Finney, Thomas, Demana & Waits.

Instructor: Judson Swets

676-6481 x5220

jswets@bham.wednet.edu

Course Overview:

Welcome to Advanced Placement Calculus.

Calculus is the reformulation of elementary mathematics through the use of the limit. Of course, you have yet to learn about the limit, so this explanation may be less than illuminating. However, to give you a sense of the power of calculus, I will contrast it with the mathematics that you already know.

Without Calculus	With Calculus
You can find the slope of a line.	You can find the slope at any point on a curve.
You can find the equation of a secant line.	You can find the equation of a tangent line.
You can calculate average speed.	You can calculate instantaneous speed.
You can find the height of f(x) when x = c.	You can find the maximum or minimum height of f(x) on a given interval.
You can describe the direction of motion along a straight line.	You can describe the direction of motion along a curve.
You can find the area of a rectangle.	You can find the area of a curved region.
You can calculate work done by a constant force. Work = Force x Distance	You can calculate work done by a variable force. (i.e. friction)
You can find the length of a line segment.	You can find the length of a curved segment.
You can find the surface area of a cylinder.	You can find the surface area of a solid of revolution.
You can find the mass of a solid with constant density.	You can find the mass of a solid with variable density.
You can find the volume of a rectangular solid.	You can find the volume of a region under a surface.
You can find the sum of a finite number of terms.	You can find the sum of an infinite number of terms.

Almost every professional field uses calculus in some way. Economists use calculus to forecast global market trends. Meteorologists use calculus to describe the flow of air in the upper atmosphere. Biologists use calculus to forecast population size. Medical researchers use calculus to design ultrasound and x-ray equipment for scanning the internal organs of the body. Astrophysicists use calculus to study distant objects and to contemplate the nature of time. Hydraulic engineers use calculus to find safe closure patterns for valves in pipelines. Large companies use calculus to determine profitable inventory levels. Timber companies use calculus to decide the most profitable time to harvest.

Now, it is true that you will be hard pressed to find any engineer, scientist, or economist that does any calculus by hand. Nope. They use computers, and why not? They understand the ideas of changing rates of change, and they are not troubled by the idea of a sum of an infinite number of infinitely small quantities. These ideas are still new to you, and we must become comfortable with them. Remember that it will be you that poses the question to the computer and not the other way around. It’s not an equal relationship. You can be buddies with your calculator, but only one of you will understand what’s going on. Kinda sad, isn’t it? I always end up feeling bad for the calculator. Smart, but clueless.

First Semester Grading:

0% Homework

Homework will be given daily. It is imperative that you work the problems out completely. You cannot learn calculus by simply watching it done or reading about it. The homework will give you the experiences you need to be successful.

20% Formal Sets

Formal Sets are assignments or handouts that will be graded for correctness. You will receive about one Formal Set per week.

20% Quizzes

You will have a quiz on most Fridays. Quizzes can be made up for excused absences and your lowest two quizzes will be dropped.

20% Test I

Test I will be given at midterm. It will be a comprehensive multi-chapter test. If unavoidable circumstances will cause you to miss the test, you must notify me in advance. If you are out of school, use e-mail or the telephone.

20% Test II

Test II will be given in early December. It will be a comprehensive multi-chapter test. If unavoidable circumstances will cause you to miss the test, you must notify me in advance. If you are out of school, use e-mail or the telephone.

20% Semester Exam

A cumulative exam will be given during Finals schedule.

The AP Exam:

Every student is strongly encouraged to take the AP Calculus exam in May. The exam will require that you learn the coursework in depth, develop your analytical reasoning skills, and form disciplined study habits. Some of the rewards for putting forth your best efforts are

1. College credit for as much as one semester or two quarters depending on your score

2. A demonstration of college readiness that will prepare you for continued success at the college level

3. A selection advantage for the student who wishes to attend a highly competitive college.

Students who take the AP Exam are among the most academically able students of the college-bound group. The exam is challenging, but you will be well prepared.

www.collegeboard.org/ap/calculus/

A.P. Calculus Checklist

Course Content:

I. Functions, Graphs, and Limits

A. Graph Analysis: With the aid of technology, graphs of functions are often easy to produce. The emphasis is on the interplay between the geometric and analytic information and on the use of calculus both to predict and to explain the observed local and global behavior of a function.

B. Limits of Functions (including one-sided limits).

1. An intuitive understanding of the limiting process.

2. Calculating limits using algebra.

3. Estimating limits from graphs or tables of data.

C. Asymptotic and unbounded behavior

1. Understanding asymptotes in terms of graphical behavior.

2. Describing asymptotic behavior in terms of limits involving infinity.

3. Comparing relative magnitudes of functions and their rates of change. (For example, contrasting exponential growth, polynomial growth, and logarithmic growth.)

D. Continuity as a property of functions

1. An intuitive understanding of continuity (Close values of the domain lead to close values of the range.)

2. Understanding continuity in terms of limits

3. A geometric understanding of the graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem.)

II. Derivatives

A. Concept of the derivative

1. Derivative presented geometrically, numerically, and analytically

2. Derivative interpreted as an instantaneous rate of change

3. Derivative defined as the limit of a quotient

4. Relationship between differentiability and continuity

B. Derivative at a point

1. Slope of a curve at a point (Examples include points at which there are vertical tangents and points at which there are no tangents.)

2. Tangent line to a curve at a point and local linear approximation

3. Instantaneous rates of change as the limit of average rates of change

4. Approximate rate of change from graphs and tables of values

C. Derivative as a function

1. Corresponding characteristics of graphs of f and f’

2. Relationship between the increasing and decreasing behavior of f and the sign of f’

3. The Mean Value Theorem and its geometric consequences

4. Equations involving derivatives: Verbal descriptions are translated into equations involving derivatives and vice versa.

D. Second derivatives

1. Corresponding characteristics of the graphs of f, f’ and f”

2. Relationship between the concavity of f and the sign of f”

3. Points of inflection as places where concavity changes

E. Applications of derivatives

1. Analysis of curves, including the notions of monotonicity and concavity

2. Optimization of absolute and relative extrema

3. Modeling rates of change (Related rate problems)

4. Use of implicit differentiation to find the derivative of an inverse function

5. Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed and acceleration

F. Computation of derivatives

1. Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions

2. Basic rules for the derivative of sums, products, and quotients of functions

3. Chain rule and implicit differentiation

III. Integrals

A. Interpretations and properties of definite integrals

1. Computation of Riemann sums using left, right, and midpoint evaluation points

2. Definite integral as a limit of Riemann sums over equal subdivisions

3.

Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval:

4. Basic properties of definite integrals, including additivity and linearity

B. Applications of integrals: The integral will be used in applications to find an accumulated change or to approximate the limit of a Riemann sum. To provide a foundation, students should be familiar with applications such as finding the area of a region under a curve, the volume of a solid with known cross sections, the average value of a function, and the distance traveled by a particle along a line.

C. Fundamental Theorem of Calculus

1. Use of the Fundamental Theorem to evaluate definite integrals

2. Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined

D. Techniques of antidifferentiation

1. Antiderivatives following directly from derivatives of basic functions

2. Antiderivatives by substitution of variables (including change of limits for definite integrals)

E. Applications of antidifferentiation

1. Finding specific antiderivatives using initial conditions, including applications to motion along a line

2. Solving separable differential equations and using them in modeling, including the study of y’=ky and exponential growth

F. Numerical approximations for definite integrals: Use Riemann and trapezoidal sums to approximate definite integrals of functions represented algebraically, geometrically, and by tables of values.