René Descartes introduced innovative algebraic techniques for analyzing geometric problems and understanding the connection between a curve's construction and algebraic equation in his work La Géométrie

1. Diophantusof Alexandria , sometimes called “the father of algebra”, was an
Alexandrian Greek mathematician and the author of a series of books called
Arithmetica, many of which are now lost. These texts deal with solving algebraic
equations,. While reading Claude Gaspard Bachet de Mxziriac‟s edition of
Diophantus„ Arithmetica, Pierre de Fermat concluded that a certain equation
considered by Diophantus had no solutions, and noted in the margin without
elaboration that he had found “a truly marvelous proof of this proposition,” now
referred to as Fermat‟s Last Theorem. This led to tremendous advances in
number theory, and the study of Diophantine equations (“Diophantine
geometry”) and of Diophantine approximations remain important areas of
mathematical research. Diophantus coined the term παρισὀ της to refer to an
approximate equality. This term was rendered as adaequalitat in Latin, and
became the technique of adequality developed by Pierre de Fermat to find
maxima for functions and tangent lines to curves. Diophantus was the 1st Greek
mathematician who recognized fractions as numbers; thus he allowed positive
rational numbers for the coefficients and solutions. In modern use, Diophantine
equations are usually algebraic equations with integer coefficients, for which
integer solutions are sought. Diophantus also made advances in mathematical
notation.
René Descartes'contributions to the history of functions is to speak of his La
Géométrie (1637), a short tract included with the anonymously published
Discourse on Method. In La Géométrie, Descartes details a groundbreaking
program for geometrical problem-solving—what he refers to as a “geometrical
calculus” (calculgéométrique)—that rests on a distinctive approach to the
relationship between algebra and geometry. Specifically, Descartes offers
innovative algebraic techniques for analyzing geometrical problems, a novel
way of understanding the connection between a curve's construction and its
algebraic equation, and an algebraic classification of curves that is based on
the degree of the equations used to represent these curves. Examining the main
questions and issues that shaped Descartes' early mathematical researches
sheds light on how Descartes attained the results presented in La Géométrie
and also helps reveal the significance of this work for the debates surrounding
early modern mathematics.

2. Leonard Euler made important discoveries in fields as diverse as calculus and
graph theory. He introduced much of the modern mathematical terminology
and notation, particularly for mathematical analysis, and defined the modern
notion of a mathematical function. Euler is also renowned for his work in
mechanics, fluid dynamics, optics, and astronomy.
Euler worked in almost all areas of mathematics: geometry, calculus,
trigonometry, algebra, and number theory.
Leonard Euler discovered ways to express various logarithmic functions using
power series, and he successfully defined logarithms for negative and complex
numbers, thus greatly expanding the scope of mathematical applications of
logarithms.
He is famous for:
e(i π) + 1 = 0
called "the most remarkable formula in mathematics" by Richard Feynman, for
its single uses of the notions of addition, multiplication, exponentiation, and
equality, and the single uses of the important constants 0, 1, e, i and π (pi).
Apollonius of PergaConics is his most famous book which introduced many
important functions used in mathematics today such as parabolas, hyperbolas,
and ellipses. He gave these functions their names. His methodology and
terminology in conics influenced many other scholars who came after him
including Ptolemy, FrencesoMaurolisco, Isaac Newton, and Rene Descartes.
In mathematics, the parabola (pronounced /pəˈræbələ/, from the Greek
παραβολή) is a conic section, the intersection of a right circular conical surface
and a plane parallel to a generating straight line of that surface. Given a point
(the focus) and a line (the directrix) that lie in a plane, the locus of points in that
plane that are equidistant to them is a "In mathematics a hyperbola is a smooth
planar curve having two connected components or branches, each a mirror
image of the other and resembling two infinite bows. The hyperbola is
traditionally described as one of the kinds of conic section or intersection of a
plane and a cone, namely when the plane makes a smaller angle with the axis
of the
"In mathematics, an ellipse (from Greek ἔ λλειψιςelleipsis, a "falling short") is the
bounded case of a conic section, the geometric shape that results from cutting
a circular conical or cylindrical surface with an oblique plane (the two
unbounded cases being the parabola and the hyperbola). It is also the locus of
all points of the plane whose distances to two fixed points (the foci) add to the
same constant." cone than does the cone itself parabola.

3. Applications of linear and quadratic function in real
life situation
In Quadratic Function:
a. When working with area, if both dimensions are written in terms
of the same variable, we use a quadratic function. Because
the quantity of a product sold often depends on the price, we
sometimes use a quadratic function to represent revenue as a
product of the price and the quantity sold. Quadratic functions
are also used when gravity is involved, such as the path of a
ball or the shape of cables in a suspension bridge.
b. A very common and easy-to-understand application of a
quadratic function is the trajectory followed by objects thrown
upward at an angle. In these cases, the parabola represents
the path of the ball (or rock, or arrow, or whatever is tossed). If
we plot distance on the x-axis and height on the y-axis, the
distance of the throw will be the x value when y is zero. This
value is one of the roots of a quadratic equation, or x-
intercepts, of the parabola. We know how to find the roots of a
quadratic equation—by either factoring, completing the
square, or by applying the quadratic formula.
c. A baseball is hit at a point 3 feet above the ground at a
velocity of 100 feet per second and at an angle of 45 respect
to the ground.
d. For a parabolic mirror, a reflecting telescope or a satellite dish,
the shape is defined by a quadratic equation. Quadratic
equations are also needed when studying lenses and curved
mirrors.
e. When a company is going to make a frame as a part of a new
product they are launching. The frame will be cut of a piece of
steel and to keep the steel down, the final area should be 28
cm2.

4. Linear Function:
a. One of the paradoxes is that just about every linear system is also a
nonlinear system. Thinking you can make one giant cake by
quadrupling a recipe will probably not work. If there's a really
heavy snowfall year and snow gets pushed up against the walls of
the valley, the water company's estimate of available water will be
off. After the pool is full and starts washing over the edge, the
water won't get any deeper. So most linear systems have a "linear
regime" --- a region over which the linear rules apply--- and a
"nonlinear regime" --- where they don't. As long as you're in the
linear regime, the linear equations hold true.
b. If you've ever doubled a favorite recipe, you've applied a linear
equation. If one cake equals 1/2 cup of butter, 2 cups of flour, 3/4
tsp. of baking powder, three eggs and 1 cup of sugar and milk,
then two cakes equal 1 cup of butter, 4 cups of flour, 1 1/2 tsp. of
baking powder, six eggs and 2 cups of sugar and milk. To get twice
the output, you put in twice the input. You might not have known
you were using a linear function.
c. Suppose a water district wants to know how much snowmelt runoff
it can expect this year. The melt comes from a big valley, and
every year the district measures the snowpack and the water
supply. It gets 60 acre-feet from every 6 inches of snowpack. This
year surveyors measure 6 feet and 4 inches of snow. The district put
that in the linear expression (60 acre-feet/6 inches) * 76 inches.
Water officials can expect 760 acre-feet of snowmelt from the
water.
d. It's springtime and Irene wants to fill her swimming pool. She doesn't
want to stand there all day, but she doesn't want to waste water
over the edge of the pool, either. She sees that it takes 25 minutes
to raise the pool level by 4 inches. She needs to fill the pool to a
depth of 4 feet; she has 44 more inches to go. She figures out her
linear equation: 44 inches * (25 minutes/4 inches) is 275 minutes, so
she knows she has four hours and 35 minutes more to wait.
e. Ralph has also noticed that it's springtime. The grass has been
growing. It grew 2 inches in two weeks. He doesn't like the grass to
be taller than 2 1/2 inches, but he doesn't like to cut it shorter than
1 3/4 inches. How often does he need to cut the lawn? He just puts
that calculation in his linear expression, where (14 days/2 inches) *
3/4 inch tells hims he needs to cut his lawn every 5 1/4 days. He just
ignores the 1/4 and figures he'll cut the lawn every five days.

5. Project in Math
Submitted by:
Rhea Ensendencia
Submitted to:
Ms.CieloMorgia