Polar representation a complex number z is represented by two parameters r and Θ. Parameter r is the modulus of complex number and parameter Θ is the angle with the positive direction of x-axis. The polar form of a complex number is: This representation is very useful when we
multiply or divide complex numbers.
A polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. An example of a polynomial of a single indeterminate, x, is x²
− 4x + 7.
The degree of a polynomial is the highest power of x in its expression. Constant
(non-zero) polynomials, linear polynomials, quadratics, cubic and quartic are polynomials of degree 0, 1, 2, 3 and 4 respectively. The function f(x)=0 is also a
polynomial, but we say that its degree is ‘undefined’.
Biological exponential growth is the exponential growth of biological organisms. When the resources availability is unlimited in the habitat, the population of an organism living in the habitat grows in an exponential or geometric fashion.
Position vector is the vector which represents the position of a point in a space with respect to the origin O. It also represents the distance and direction of the points from the origin. If P is the point, then the position vector of P is represented by.
A power function is in the form of f(x) = kx^n, where k = all real numbers and n = all real numbers. You can change the way the graph of a power function looks by changing the values of k and n.
A power series is an infinite series of the form where aₙ represents the coefficient of the nth term and c is a constant. Aₙ is independent of x and may be expressed as a function of n. Power series are useful in
analysis since they arise as Taylor series of infinitely differentiable functions.
Principal Values of Inverse Trigonometric Functions. The principal value of sin x for x
> 0, is the length of the arc of a unit circle centred at the origin which subtends an
angle at the centre whose sine is x.
A product is the result of multiplying, or an expression that identifies factors to be multiplied. Thus, for instance, 6 is the product of 2 and 3 (the result of multiplication) and is the product of and (indicating that the two factors should be
multiplied together).
Product rule tells us that, when multiplying two powers that have the same base, you can add the exponents. In this example, you can see how it works. Adding the exponents is just a short cut! The “power rule” tells us that to raise a power to a power, just multiply the exponents.
Projective geometry is a topic in mathematics. It is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts.
They are often numbered using Roman numerals: I II III IV. Pages referring to ‘quadrant’ The Coordinate Plane. The coordinate plane defined with description of x,y axis, quadrants, origin.
A quadratic equation is an equation of the second degree, meaning it contains at least one term that is squared. The standard form is ax² + bx + c = 0 with a, b, and c being constants, or numerical coefficients, and x is an unknown variable.
Suppose that Xt is a real-valued stochastic process defined on a probability space
{\displaystyle (\Omega ,{\mathcal
{F}},\mathbb {P} )} and with time index t ranging over the non-negative real numbers. Its quadratic variation is the process, written as [X]t, defined as {\displaystyle [X]_{t}=\lim _{\Vert P\Vert \rightarrow 0}\sum
_{k=1}^{n}(X_{t_{k}}-X_{t_{k-1}})^{2}} where P ranges over partitions of the interval [0,t] and the norm of the partition P is the mesh. This limit, if it exists, is defined using convergence in probability. Note that a process may be of finite quadratic variation in the sense of the definition given here and its paths be nonetheless almost surely of infinite 2-variation for every t>0 in the classical sense of taking the supremum of the sum over all partitions / this is in particular the case for Brownian Motion.
The radian is a unit of measure for angles used mainly in trigonometry. It is used instead of degrees. Whereas a full circle is 360 degrees, a full circle is just over 6 radians. A full circle has 2π radians (Roughly 6.28)
Radians are the standard mathematical way to measure angles. One radians equal to the angle created by taking the radius of a circle and stretching it along the edge of the circle. The radian is a pure mathematical measurement and, therefore, is preferred by
mathematicians over degree measures.
A radius of a circle or sphere is any of the line segments from its centre to its perimeter, and in more modern usage, it is also their length. The name comes from the Latin radius, meaning ray but also the spoke
of a chariot wheel.
The radius of convergence of a power series is the radius of the largest disk in which the series converges. It is either a non-negative real number or ∞.