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The Calculus™

Description

The Calculus sequence is designed to be understood by every high schooler. This course can be used to prepare students for the AP BC Calculus examination.

Students will learn the ancient origins of the infinitesimal and they will see how such an idea can have a profound impact on the understanding of the world around us.

Students will be amazed at how calculus helps explain phenomena involving movement, population growth, financial markets, capital investment value, atmospheric temperature and pressure, planetary motion, and electromagnetic radiation velocity.

To many American high school students, calculus is a set of tools and rules that do nothing, leaving the student to never comprehend the innate beauty and powerful practicality that is calculus.

In this sequence, we set out to recapture the intuitive essence of the mathematics presented to us by Newton and Liebniz.

Sequence

The Infinitesimal™
This is the opening round of what will amaze students. The idea of the infinitesimal was borne from ancient times, and students will explore its roots and development over a period of 2000 years. Students will apply this concept to solve a range of practical mathematical problems involving areas and volumes.

topics include: xenon's paradoxes, derivation of pi, squaring curves, method of exhaustion, Cavalieri method, Wallis method, Fermat method, center of mass, volume of solids, limits, Riemann sums

The Fundamental Theorem™
In this chapter, students discover the ideas of incremental change, the derivative, and they discover that there exists a beautiful relationship between the integral and derivative. We call it... The Fundamental Theorem of Calculus.

topics include:  instantaneous velocity, limits, infinitesimal change, derivative definition, application of derivative, optimization, least path, linear regression, fundamental theorem of calculus, series

The Science of Fluxions™
Isaac Newton was the first scientist/mathematician to codify the ideas of the calculus. He used it solve a range of amazing physical problems, and with it he was able to show unknown mathematical relationships present in the very laws of the universe. Students will spend their time making these same discoveries.

topics include: Kepler's Laws, Newton's Laws, optimization, power series expansions of functions, Gauss' Law, chain rule, L'Hopital's rule, local extremums, Newton's method, Euler's method, indefinite integrals, growth and decay problems
Information

The Calculus sequence consists of three 30-hour workshops for a total of 90 hours of in-class material. Academic year sequences are held in 10 week classes each 3 hours long. Summer camps are held in a single week and cover 30 hours of material.

Classes consist of hands-on activities, practice problems, and concept synthesis. Academic year students are required to complete a minimal amount of practice problems.

Prerequisites

Students should have completed our Advanced Math sequence or Pre-Calculus before taking any of The Calculus courses.

CDE AP Calculus Standards

1.0 Students demonstrate knowledge of both the formal definition and the graphical interpretation of limit of values of functions.
2.0 Students demonstrate knowledge of both the formal definition and the graphical interpretation of continuity of a function.
3.0 Students demonstrate an understanding and the application of the intermediate value theorem and the extreme value theorem.
4.0 Students demonstrate an understanding of the formal definition of the derivative of a function at a point and the notion of differentiability.
5.0 Students know the chain rule and its proof and applications to the calculation of the derivative of a variety of composite functions.
6.0 Students find the derivatives of parametrically defined functions and use implicit differentiation in a wide variety of problems in physics, chemistry, economics, and so forth.
7.0 Students compute derivatives of higher orders.
8.0 Students know and can apply Rolle's theorem, the mean value theorem, and L'Hôpital's rule.
9.0 Students use differentiation to sketch, by hand, graphs of functions. They can identify maxima, minima, inflection points, and intervals in which the function is increasing and decreasing.
10.0 Students know Newton's method for approximating the zeros of a function.  11.0 Students use differentiation to solve optimization (maximum-minimum problems) in a variety of pure and applied contexts.
12.0 Students use differentiation to solve related rate problems in a variety of pure and applied contexts.
13.0 Students know the definition of the definite integral by using Riemann sums. They use this definition to approximate integrals.
14.0 Students apply the definition of the integral to model problems in physics, economics, and so forth, obtaining results in terms of integrals.
15.0 Students demonstrate knowledge and proof of the fundamental theorem of calculus and use it to interpret integrals as antiderivatives.
16.0 Students use definite integrals in problems involving area, velocity, acceleration, volume of a solid, area of a surface of revolution, length of a curve, and work.
17.0 Students compute, by hand, the integrals of a wide variety of functions by using techniques of integration, such as substitution, integration by parts, and trigonometric substitution. They can also combine these techniques when appropriate.
18.0 Students know the definitions and properties of inverse trigonometric functions and the expression of these functions as indefinite integrals.
19.0 Students compute, by hand, the integrals of rational functions by combining the techniques in standard 17.0 with the algebraic techniques of partial fractions and completing the square.
20.0 Students compute the integrals of trigonometric functions by using the techniques noted above.
21.0 Students understand the algorithms involved in Simpson's rule and Newton's method. They use calculators or computers or both to approximate integrals numerically.
22.0 Students understand improper integrals as limits of definite integrals.
23.0 Students demonstrate an understanding of the definitions of convergence and divergence of sequences and series of real numbers. By using such tests as the comparison test, ratio test, and alternate series test, they can determine whether a series converges.
24.0 Students understand and can compute the radius (interval) of the convergence of power series.
25.0 Students differentiate and integrate the terms of a power series in order to form new series from known ones.
26.0 Students calculate Taylor polynomials and Taylor series of basic functions, including the remainder term.
27.0 Students know the techniques of solution of selected elementary differential equations and their applications to a wide variety of situations, including growth-and-decay problems.