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Pre-Calculus

The Physics of Spinning Objects

150 150 Deborah

Overview

Any spinning object rotates around a central point called an axis. Tops, dreidels, gyroscopes, and spinning eggs rotate, rise, and seem to defy gravity, as long as they are moving. When they stop moving, they fall.

Angular Momentum

Angular momentum refers to movement in a circle rather than a straight line. Imagine the movement of one dot on the spinning object as it twirls (such as one dot above the right of the Hebrew letter shin). Its path could be described by angular momentum around the axis through the point of support. The equation that describes angular momentum L takes into account its mass, shape, and rate of speed. In this equation, L = Iω, means that angular momentum is the product of an object’s inertia, or resistance to change in its velocity, times its angular (or circular) velocity ω.

Conservation of Angular Momentum

Many spinning objects tend to conserve angular momentum. Imagine a figure skater entering a spin. She brings her arms close to her body, spinning faster and faster, and is able to keep balance partly because of angular momentum (and a lot of practice). A rigid object, such as a sphere, will remain upright as it spins on an axis because angular momentum is conserved as long as it is spinning. The system is effectively closed due to Newtonian laws of motion, and can be described in terms of relationships between derivatives.

Precession

As a top, planet, or gyroscope spins, its axis does not stay at a fixed point. It tilts and rolls. Precession depends upon whether a force, known as torque, is applied to the spinning object, and how much force is applied. Similar to the equations describing angular momentum, the equations that describe precession also use inertia, rate of speed around the axis, and angle between the symmetry axis and direction of inertia.

Gyroscopes, Tops, and Spinning Eggs

Gyroscopes are self-balancing, and have been used since the 18th century in compasses, navigational tools, and in toys. They use general principles of three axes: pitch, roll, and yaw in order to plot precise position in space. Tops have been used as toys in cultures from ancient China and India to Greece and Rome. Spinning eggs and some types of tops actually change axes as they swirl and slow down. All spinning objects rely on similar principles, involving angular momentum, precession, and inertia.

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Math Review of Leibniz, Newton, and the Development of Calculus

150 150 Deborah

Overview

One of the biggest controversies in science in the early 18th century was around the development of a new mathematical tool called calculus. In Europe, the mathematician, philosopher, and scientist Gottfried Leibniz held the attention of the scientific community. The most famous scientist of the day, Sir Isaac Newton, was the champion of Great Britain. According to scientific history, both invented calculus by working independently on different aspects.

Background

Many mathematical and geometric ideas were already known before calculus was formulated. Archimedes and other Greek geometers, mathematicians in China and India, and thinkers in the Middle East used methods of calculating area and volume, work with infinite series, and other formulas. However, they did not put all those parts together into a system of thought.

Leibniz

Gottfried Leibniz (1646 -1716) was a German mathematician, philosopher, and scientist who may or may not have been a nobleman. He developed a modern calculating machine, and was an advisor to many political figures in Germany, France, and Austria. One of his many innovations was a version of calculus. He published a paper using the new methods in 1684.

Newton

Sir Isaac Newton (1642 – 1727) was one of the most famous scientists and mathematicians of his day. He was President of the Royal Society in Great Britain from 1703 to 1727. Newton’s Three Laws are the basic of classical mechanics and physics, especially gravitation. He developed a mathematical “method of fluxions” which was his form of calculus. He described the geometric background in the Principia Mathematica in 1687, in 1693, and in 1704.

Controversy

In 1711, some of Newton’s partisans in the Royal Society accused Leibniz of plagiarizing Newton’s system. The controversy escalated between the European scientists and the British scientists, fueled by Newton’s supporters in the Royal Society and the political climate of the time. It wasn’t until the 1800’s that British mathematicians began using the notation that Leibniz developed for calculus concepts such as ∫ for integral, and dx and dy for infinitesimal parts of x and y. Historians of science generally regard both Leibniz and Newton as the inventors of calculus, coming at its mathematical concepts from different directions.

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Math Review of Roulettes and Cycloids

150 150 Deborah

Overview

Roulettes are special types of curves that are made when one curve is rotated around another by a fixed point. They are in Euclidean space, and can be described by calculus equations. Roulettes, cycloids, and other variations of circular movement have applications to pendulums, gears and subatomic particles.

General Definition of a Roulette

A roulette is a type of curve. It is the path of a fixed point in relation to one curve as that curve travels along a straight line or another curve. The simplest form of a roulette is a cycloid. Imagine a circle that looks like a wheel with one spoke from the center to the edge of the wheel, one radius long. The path traced as that circle travels along the line is a cycloid.

The Cycloid Family

If the roulette is formed by a circle travelling outside the circumference of another circle it is called an epicycloid, and if the circle is traveling inside the circumference it is called a hypocycloid. Bernoulli in the late 17th century proposed a challenge, which was solved by him, Newton, and Leibniz. The challenge was to find the fastest path a point will take to travel moving without friction, using gravity.

Playing with Cycloids

Mathematicians play with cycloids using differential calculus equations. The path a cycloid travels can be measured by an equation involving the differential of x divided by the differential of y all squared in relation to the radius of the circle.  Many children of all ages have played with a toy called a Spirograph that essentially generates cycloids of various types, using pens of different colors.

Applications

Huygens, living in the late 17th century, used the cycloid to make a pendulum that would be the most efficient type ever devised, in order to improve timekeeping and measurements aboard ship. The motions of gears also follow paths described by the cycloid family. In order for gears to regulate machinery, they need to touch and move freely, along cycloid paths. Engineers calculate the speed of gears; the location, number, and size of teeth (fixed points); and the size of the gears themselves. The path of electrons in electric and magnetic fields also follow cycloid movements.

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Math Introduction to Infinitesimals

150 150 Deborah

Overview

Using infinitesimal quantities to approximate measurement of any item is an ancient way to determine the size and shape of irregular objects. Although it was very controversial in the 17th and 18th century Europe, the practical aspects of using infinitesimal quantities in calculations led to advances in science, engineering, and technology, along with the development of calculus.

Geometric Infinitesimals

Greek mathematicians such as Archimedes used the smallest possible indivisibles to find areas of solids. In the 17th century, the astronomer and mathematician Johannes Kepler looked at a different way to compute the area of a circle or curved figure. Suppose a circle or curve were made up of infinitely many polygons, like thinner and thinner slices of pie. The sum of all those slices would equal the area of the circle or curved figure, even though the area of one slice was infinitesimal.

The Controversy

Slicing a figure into infinitely many thin fragments was very attractive to many mathematicians and scientists, because it solved a number of practical problems. However, many philosophers hated the idea. Using infinitesimals in mathematical calculations was banned in Rome in the 1600’s, and denounced from pulpits and in books.

Practical Use of Infinitesimals

Infinitesimals are close to zero and retain properties such as angles or slopes. Before calculus, mathematicians, scientists, and engineers could use infinitesimal quantities in calculations such as finding the area under a curve, or approximating the rate of change. Suppose that a large tank holds 1000 gallons of water. The instantaneous rate that water drains from the tank can be calculated using infinitesimal approximations.

Leibniz, Newton, and Calculus

Infinitely small building blocks (such as 1/∞) add up to something if enough of them are used. The mathematician and philosopher Gottfried Leibniz used those and other mathematical observations to promote a new system of mathematics to calculate areas under curves called calculus. At the same time, the more famous Sir Isaac Newton developed a similar system of calculus, to be applied to many aspects of mathematical physics. Although it was very controversial in the 1700s, both Leibniz and Newton made independent contributions to a new method using mathematics to describe the natural world.

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Math Introduction to Integrals

150 150 Deborah

Overview

Finding integrals, or integration, is the opposite of finding the derivative in calculus. The concept is closely connected with functions, and was independently discussed by both Newton and Leibniz in the Fundamental Theorem of Calculus.

Review of Functions

Suppose the values of x are {1, 2, 3, 4, 5} using set notation, and the function rule is [x +1 =y]. The values of y are {2, 3, 4, 5, 6}, because f(1) is 2, f(2) is 3, f(3) is 4, f(4) is 5, and f(5) is 6. In math language f(x) = x +1. Every value of x (the independent variable) has a corresponding value of y (the dependent variable). That simple function is linear, but functions can take many shapes, depending on the equation or inequality that describes them.  A graphed curve is a function if and only if a vertical line intersects it in only one place.

Infinitesimals

Mathematicians used the theory of infinitesimals and the vertical line test to find approximate values of the area under a curve before calculus was invented. (Derivatives measure the rate of change of the curve itself). The closer the approximation, the smaller the slices used. Suppose the area under the curve were measured by slices of 1 unit wide. Some of the units would overshoot the curve and add more area, while others would undershoot the curve and leave the area not measured. If the slices were made smaller, ½ unit, the measurement would be more accurate. The slices could be made smaller still until there were an infinite amount of slices that almost touched the curve. Then the measurement of the area under the curve would be the most accurate possible.

Integrals and Integral Notation

Newton and Leibniz arrived at a theory that brought together all of the rules and systems to deal with systems that were not linear. They laid the foundations for a new mathematical theory called calculus. The Fundamental Theorem of Calculus shows how integrals and differentials are related. Sir Isaac Newton developed calculus to account for the laws of motion and physics, while Gottfried Leibniz developed calculus to account for observations of how integrals and differentials worked. He paid close attention to symbolic meaning, and the symbols he defined are in common use.

Applications of Integrals

As scientists and engineers required more precise calculations, the methods of calculus became more useful. The continuous variation of light intensity in optics, the energy output of a steam engine, the reaction rate of chemicals all required calculus during the Industrial Revolution. Integrals figure in atomic reactions and the behavior of distant galaxies, the trajectory of a thrown ball and of a rocket carrying an orbital satellite.

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Math Introduction to Derivatives

150 150 Deborah

Overview

A derivative of a function describes its rate of change at a particular point on the function. The rate of change doesn’t have to be constant, so it can be approximated along any point of a curve. Derivatives in calculus have many applications in quantitative sciences such as physics and chemistry.

Geometric Definition

Not all functions are linear. If a function is continuous, so that very small changes in input result in changes in output, the shape of the graph is a curve. In order to approximate the amount of change at any point on the curve, a tangent line can be dropped. The derivative is the measurement of the slope of the line at that point.

Differentiation

Differentiation is the process of finding derivatives. Both Newton and Leibniz used differentiation in the process of developing calculus. The differential is an infinitesimal change in a varying quantity, and can be related to all other changes in a function. Even though the change is infinitely small, it can still be measured by an approximation.

Derivatives in Space and Time

In order to measure changes in space and time, derivatives are used in differential equations. The time derivative, or rate of change over time, is significant to concepts such as velocity and acceleration. (It can be said that Newton developed calculus to quantify his observations in classical mechanics.) For example, velocity is the rate of change in position with respect to time. Acceleration is the rate of change of velocity over time. It is not necessarily constant, and can involve minute adjustments of speed.

Other Applications

Derivatives and differential equations are used in quantitative sciences and modeling. For example, the reaction rate in chemistry is a rate of change, measured by differential equations. Most measures of behavior (such as in psychology, sociology, and economics) can be approximated by the normal curve, which represents change that is continuous but not linear. Many high school students take standardized tests such as the SAT and the ACT. Changes in scores over time can be modeled using advanced statistical techniques, based upon derivatives and differential equations.

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Math Review of Real, Complex, and Hyperreal Numbers

150 150 Deborah

Overview

Real numbers are the rational and irrational numbers that people deal with in everyday life. Hyperreal numbers include numbers that are infinitely large, infinitely small, or infinitesimal, along with the reals. Surreal numbers include the reals, the hyperreals, and other constructs in advanced mathematics that sometimes behave like numbers and sometimes do not.

Rational Numbers

Real numbers include rational numbers, any number that can be expressed as a ratio, whether positive or negative. Rational numbers include the natural or counting numbers, such as 1, 2, 3, 4, and so on.  The integers include negative numbers and zero, which is neither positive nor negative. Rational numbers can be plotted on a number line with the origin at zero.

Irrational Numbers

Irrational numbers are types of real numbers that cannot be expressed as an exact ratio. When they are represented by decimals, the decimal digits do not repeat, nor do they terminate in zeros. They are some of the most useful numbers, including pi π, tau τ, the natural logarithm e, and any root that is not an exact number, such as √2. The irrational numbers pi, tau, e, and others are transcendental and more numerous than rational numbers.

Complex Numbers

Rational and irrational numbers can be shown on a number line. Complex numbers can be shown on a number plane, at angles to the number line. They are used in special algebraic situations and take the form a + bi, where a and b are real parts of the number and the construct i equals the square root of -1. Since the imaginary part of a complex number can be equal to 0, all numbers are complex numbers. They are used in many applications, including chemistry, physics, and electrical engineering.

Hyperreal and Surreal Numbers

Hyperreal and surreal numbers are relatively new concepts mathematically. Hyperreal numbers include all the real numbers, the various transfinite numbers, as well as infinitesimal numbers, as close to zero as possible without being zero. They have applications in calculus. Surreal numbers include all the real, complex, hyperreal, transfinite, and infinitesimal numbers as well as some constructs that have applications in game theory. They may also have applications in computer processing.

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Math Review of More Properties of Circular Functions

150 150 Deborah

Overview

The unit circle has as its center the origin point of the Cartesian coordinates x and y, and has a radius 1. The trigonometric functions are also called circular functions, because they describe relationships between angles on the unit circle. We use different ways to describe trigonometry in order to see how the relationships and equations apply to different situations.

Special Values of the Trigonometric Functions

When the trigonometric functions are defined as circular, the special values of trigonometric functions can be seen on the unit circle. If t, the length of the arc from the coordinates (1, 0) equals 0, then sin t equals 0, cos t equals 1, and tangent t equals 0. This is because sin t equals y, cos t equals x, and tan t equals y/x. When t equals 0, y/x equals 0/1. If t equals π/4, then its coordinates are ([√2]/2, [√2]/2). Therefore, sin t is (√2)/2, cos t is (√2)/2, and tangent t is 1.

Domains of the Trigonometric Functions

If a circular function results in division by zero, it is undefined. Therefore, each function has a specific domain. Sines and cosines are defined for all real values of t. Tangents and secants are defined for all real numbers, except when the value of x is 0. This happens when the value of t equals (π/2) + nπ. Similarly, cosecants (csc) and cotangents (cot), are defined for all real numbers other than nπ, which happens when x equals 0.

Sign of the Trigonometric Functions

All other values of the trigonometric functions can also be seen on the unit circle. The unit circle is divided into quadrants by the Cartesian coordinates, so the signs of each circular function can be determined by the value of t. If the value of t has a positive value for both x and y, then it lies in Quadrant I. In Quadrant II, only the value of y is positive, and x has a negative value, so sin t and cosecant t will be positive. All other functions will be negative. In Quadrant III, both coordinates for x and y are negative. This means that tangents and their reciprocal cotangents will be positive. In Quadrant 4, x is positive and y is negative, so cosines and their reciprocal, secants, will be positive. There are several ways to remember the direction of the signs.

Fundamental Identities

The trigonometric functions are related to one another because the formula for the unit circle is x2+ y2=1. Therefore, the cosecant of t (csc t) is defined as 1/sin t; the secant of t (sec t) as 1/cos t, and the cotangent of t (cot t) as 1/tan t. Similarly, sin2t + cos2t = 1, tan2t +1 = sec2t, and 1 + cot2t = csc2t.

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Math Review of Applications of Circular Functions

150 150 Deborah

Overview

Circular trigonometric functions can be applied to situations in physical, biological, and social sciences involving data that follows a pattern that is not linear. Many of those patterns are periodic, and can be modeled by approximations of sine, cosine, or other functions.

The Sine and Cosine Waves

The numerical value of t around the unit circle has the same value whether it is represented by 2π or τ, 4π or 2τ, and so on. Therefore, a sine function can be graphed as sin (t +2nπ) = sin t and a cosine function can be graphed as cos (t +2nπ) = cos t, for any real integer n. Both the sine wave and the cosine wave are periodic and repeat their graphs symmetrically.

Amplitude, Period, and Phase Shift

The period is the time it takes for a complete cycle through all values, from the beginning through the highest point to the baseline down to the lowest point, then back to baseline again. The amplitude is the highest point of the t curve and the absolute value of the lowest point, often represented by the letter a. A curve can be shifted horizontally by moving it left or right, called the phase shift. In trigonometric language, a sine wave can be represented by the expression y = a sin kx, and a cosine wave can be represented by the expression y = a cos kx.

Oscillations

Many things are variable in nature, and follow a sine wave, a cosine wave, or a combination of different types of waves. Electromagnetic energy, whether infrared, visible light, UV light, X-rays, microwaves, or sound waves follow periodic oscillations and can be modeled by periodic oscillations. The period of variable stars can also be measured and calculated. The motion of a pendulum or electrical currents also can be measured using periodic functions.

Other Periodic Functions

Behavior that can be modeled by sine, cosine, and combinations of waves occurs in biology, physiology, and other social sciences. Populations of predators and prey tend to vary with a cyclical change that can be approximated by sine or cosine functions. The population of predators increase when more prey are available and decrease when fewer prey are available, so the populations mirror one another.

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Math Review of Angles and Circular Motion

150 150 Deborah

Overview

Trigonometry is defined by the measurement of angles and their relationships. One of the ways that trigonometry can be applied is in the measurement of angles and circular motion.

Angle Measure

Angles in trigonometry and calculus can be measured in radians, which is a relationship of the measurement of an angle by the arc it makes on a circle of radius 1. If the initial side of the ray is measured counterclockwise from the terminal side, it is a positive angle, and if the initial side of the ray is measured clockwise from the terminal side, it is a negative angle.

Standard Position

The standard position for measuring an angle θ is to measure it with the vertex at the origin at the xy axis and the initial side of the angle along the x axis. This is the same concept as the angle at the unit circle. If two angles are both in the standard position, and both sides coincide, they are coterminal. The initial side of one angle is the terminal side of the other and the terminal side of one angle is the initial side of the other.

Arcs and Sectors

The radius of the circle is one of the factors that determines the length s of the arc. Therefore, the radian measure θ of an angle equals the length of the arc s/radius r. Similarly, the area of a circular sector equals ½ r2θ. These values are only true when the angle θ is measured in radians, as those measurements are standardized for the radius of a circle.

Circular Motion

Suppose a point is moving around the outside of a circle, for example, if an object is in a circular orbit in space. There are two ways to describe the rate that it is moving. The linear speed is the distance traveled divided by the elapsed time. The angular speed is the change in the measure of the central angle in radians divided by the elapsed time. In math language, the linear speed v equals the distance s/time t, or v=s/t. The angular speed is represented by the Greek letter omega ω, which equals the angle θ/time t, or ω =θ/t. Linear and angular speed are related such that linear speed v =radius r times angular speed ω, or v= rω. This is a very useful concept in circular mechanics and in orbital motion.

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