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Calculus

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 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 Introduction to Sets and Logic

150 150 Deborah

Overview

Thinking about sets of objects and numbers is as simple as counting and as complex as infinity and transfinite numbers. Sets are an essential underlying concept of mathematics and logic.

Roster or Description

Mathematicians use the language of set theory to describe sets. For example, small, finite sets can be defined using the roster method, which is exactly like a team roster. Each player on the team roster can be listed as a member of a set, such as {Russell Wilson, Marshawn Lynch, Derrick Coleman, Percy Harvin, Luke Wilson …Jimmy Staten, Julius Warmsley}. The other term for sets is the description method, such as {players on the active list for the Seattle Seahawks}. Using math language, Set A = {3, 4, 5, 6} using the roster method, or Set A = {x |x is an integer ≥3 and ≤6} using the description method. In symbol form {x| x I ≥3  ≤6}.

One-to-One Correspondence and Cardinality

Sets are equal if they contain exactly the same elements. They are equivalent if they have the same number of elements. If each member of one set can be paired with only one member of another set, they are in one-to-one correspondence. The members do not have to equal one another, but the sets have to be equivalent, the same size. This is the basic idea behind counting, as one and only one number is paired with exactly one object. The sets have the same cardinality. Some sets are finite, such as Set A or the players on the active list of the Seattle Seahawks. Other sets are infinite, such as the set of integers I or Z, the set of whole numbers W, the set of rational numbers Q, real numbers R, and complex numbers C. A mathematician named Georg Cantor showed using one-to-one correspondence that the cardinality of infinite sets differed, so that there are levels of infinity. He designated the lowest level of infinity as א0, a level that is shared by the sets of integers, the sets of odd numbers, and the sets of even numbers. There are even more real numbers, and even more complex numbers, in higher levels of infinity, א1 and beyond.

Figure 1: Georg Cantor theorized that there were different levels of infinity.

Subsets

If an element is a member of a set, the symbol for that relationship is ∈. For example, {Marshawn Lynch ∈ Seattle Seahawks}. If an element is not a member of a set, the symbol for that relationship is ∉, so that {Peyton Manning ∉ Seattle Seahawks}. Suppose set B consists of the numbers {1, 2, 3, 4, 5, 6}. Set C consists of the numbers {2, 4, 6}. In this case, C is a subset of B, because all the numbers in Set C are in set B. In symbol form, C⊂ B. If Set D contains the number {7}, it is nowhere in set B, so D⊄B. The empty set has no elements in it. By definition, the empty set is a subset of all sets.

Figure 2: Using set language to describe a relationship between an element of a set and the entire set.

Venn Diagrams

Relationships between different sets are shown with Venn Diagrams. If one set is a complete subset of another, so that all the members of one set are also members of another, the circle that represents the subset will be inside the circle that represents the larger set. If circles intersect, that means that both sets have some elements in common. If the sets have no common elements, the circles will be independent from each other.

Figure 3: The Venn diagram of the types of real numbers shows their relationship.

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Solving Integrals Through the Use of Integration by Parts

150 150 oren

Overview

Suppose we have an integral which appears unsolvable by ordinary means, such as
∫xsin(x)dx In many cases, we can do what is called integration by parts, which is where we split the equation we are taking the integral of into two parts, with the goal of simplifying the equation such that we can reduce it down to an equation we know how to solve.

Formula

Figure 1: General formula for Integration by Parts

IntByParts

In essence, what this means is that we are going to attempt to split up the equation in a way such that we can eliminate one of the equations inside the integral, allowing us to end up with a form that we know how to solve.

Examples

∫xsin(x)dx
Let u(x)=x and v’(x)=sin(x)
Then we need to find u’(x) and v(x), so u’(x) = d/dx(x) = 1
Next, we need to find v(x), which we do by doing ∫v’(x)dx, or in this case, ∫sin(x)dx = -cos(x)
Now that we have u(x), v(x), u’(x), and v’(x), we can plug everything in, resulting in the following:
∫xsin(x)dx = x*(-cos(x)) – ∫1*(-cos(x))dx
= -xcos(x)+∫cos(x)dx
= -xcos(x)+sin(x)+C

In general, we will want to treat the polynomial as u(x) and the other team as v(x) where possible. In addition, Integration by Parts can be done multiple times. Suppose we have the equation

∫x3exdx
Then we will let u(x) = x3 and v’(x) = ex, then u’(x) = 3x2 and v(x)=ex, and we find that
∫x3exdx = x3ex – ∫3x2exdx
Then we can do the same procedure with the slightly easier integral ∫3x2exdx
Let u(x) = 3x2 and v’(x) = ex, then u’(x) = 6x and v(x) = ex
Then ∫x3exdx = x3ex – ∫3x2exdx = x3ex – (3x2ex – ∫6xexdx)
And then with ∫6xexdx, we split it one last time into u(x) = 6x, v’(x) = ex, and therefore u’(x) = 6 and v(x) = ex
Then we can substitute in one last time in order to get ∫6xexdx
∫x3exdx = x3ex – ∫3x2exdx = x3ex – (3x2ex – (6xex – ∫6exdx)) =
x3ex – 3x2ex + 6xex – 6ex + C. Whew!

 

 

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How to Use the Chain Rule to Find Derivatives

150 150 oren

Overview

Occasionally, you will have what are called composite functions; that is, functions that are composed of multiple functions and thus cannot be differentiated easily. In reality, although these may look tricky, they are actually fairly straightforward.

Function composed of two functions

Suppose we have an equation written as f(g(x)). Then
d/dx f(g(x)) = f’(g(x))g’(x)

If the equation is more complicated, such as
d/dx(f(g(h(x))),
then it can be broken thought of as doing the chain rule multiple times.
Think of it instead as f(j(x)) where j(x) = g(h(x)), and as we know the derivative of f(j(x)) is f'(j(x))j'(x). So since j(x)=g(h(x)), we can rewrite this as f'(g(h(x)))*(g(h(x)))’, or f'(g(h(x)))g'(h(x))h'(x).

It can be thought of as a chain, in that first we take the derivative of the outer equation and then move steadily inward.

Examples

d/dx(sin(2x))
Then f(y)=sin(y) and g(x)=2x, so sin(2x) can be written as f(g(x)).
Then as explained before, d/dx(f(g(x)) = f’(g(x))g’(x)
=(sin(g(x)))’(2x)’
=cos(g(x))*2
=2cos(2x)

d/dx(3(cos(2x3))4)
=3d/dx(cos(2x3))4)
Let f(z)=z4, g(y)=cos(y), and h(x)=2x3
Then f’(z)=4z3
g’(y)=-sin(y)
h’(x)=6x2
And since the equation can be written as f(g(h(x))), then
3*d/dx(f(g(h(x))))=3*f’(g(h(x)))*g’(h(x))*h’(x)
Or, by substituting the values in,
3*4(cos(2x3))3(-sin(2x3)*6x2
And, by combining like terms and
=-72x2sin(2x3)(cos(2x3))3

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U Substitution as a Method of Solving Integrals

150 150 oren

Overview

Sometimes, a derivative is done using the chain rule, and it leaves with an equation that, at first glance, can look intractable when we are attempting to integrate it. In those cases, although the problem may look difficult or often will look like something that can only be solved through the use of Integration by Parts, there often exists a much simpler solution: that of U-Substitution.

Idea

Suppose we have an equation in the form ∫u’(x)f(u(x))dx. Then we can do a substitution in order to make the equation easier to integrate in a way that we will see shortly.
Although the way it was phrased above may make it sound intimidating, in reality the process is very straightforward.

Example

Suppose we wish to find ∫x3sin(x4)dx Although this may look tricky, once we notice that the derivative of x4 includes x3, it becomes much more straightforward.
We let u=x4
Then in that case, du/dx=4x3, or du/4 = x3dx
If we move things around in the initial integral, we can note that
∫x3sin(x4)dxdx = ∫sin(x4)x3dx
Now, since x4 = u, and x3dx = du/4, we can substitute in for the x as such

∫sin(u)du/4 = 1/4 ∫sin(u)du
Which is ¼(-cos(u))+C = -¼cos(u)+C
And now we need to substitute back for u, so
-¼cos(u)+C = -¼cos(x4)+C

Common Mistakes

If all of the xs have not been eliminated, then we cannot go through with the substitution. A good choice of u will leave no x or dx in the equation at all, only u and du.

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Distance, Velocity, and Acceleration

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There is a relationship between distance, velocity and acceleration. When you understand how they interact, and their equations; they are much easier to grasp. The relationship between distance and velocity is proportional.
Distance = velocity x time
If acceleration is involved in the question, the equation becomes
Distance = v0 x t + 0.5 a t^2
Where v0 is original velocity.
With these two equations, you can solve almost every question.

Example

If a car goes 3m/s and it accelerates 2m/s^2 for 5 seconds. What is the distance the car went?
First, identify the values. a = 2,v0=3, and t=5
Now, substitute the value in the equation

D = 3 x 5 + 0.5 x 2 x 5^2
= 15 + 25
= 40

Like this you can simply apply the values in the equation.

Example 2

A car was going X m/s. It accelerates 4m/s^2 for 4 seconds. The total distance it went was 40m. What is X?
First, identify the values. a = 4,d=40, and t=4
Now, substitute the value in the equation
40 = v0 x 4 + 0.5 x 4 x 4^2
40=4v0 + 32
8 = 4v0
v0 = 2
=> X = 2m/s

 

Like this you can apply the equation and solve the question.

 

Example 3: An object falling.

Newton dropped an apple from 10m high. How long would it take for it to touch the ground if the acceleration is 9.81m/s^2?
First, identify the values. We know that v0 = 0 since it’s falling.

Also, a = 9.81m/s^2, d=10
Now, substitute the value in the equation

D = 0.5at^2
We know D = 10m, and a = 9.81m/s^2

10 = 0.5*9.81*t^2
t^2 = 5.095
t = 2.247seconds

 

Example 4:

Galileo dropped a feather from 20m high. The acceleration it falls is 4m/s^2. How long would it take for the feather to touch the ground?

First, identify the values. We know that v0 = 0 since it’s falling.
Also, a = 4m/s^2, d=20

Now, substitute the value in the equation
D = 0.5a*t^2
20=0.5*4*t^2
20=2t^2
t^2=10
t= 3.16s

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This article was written for you by Edmond, one of the tutors with SchoolTutoring Academy.

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