# The Basics of the Mathematical Subject Calculus

Calculus is the branch of mathematics that is a combination of algebra and geometry. It was discovered at the same time by the English mathematician and physicist Isaac Newton (1642-1727) and German mathematician and philosopher Gottfried Leibniz (1646-1716) in the 17th century. It is one of the most important branches of mathematics. The integral and derivative are two very important calculus terms.

The definition of the integral is the Fundamental Theorem of Calculus. Integral calculus has to do with the integral. The integral is the area under a curve. It is found using the maximum and minimum values of the curve. The integral is the opposite of the derivative. It is also called the antiderivative. The derivative is found using the formula limit as h approaches infinity of [f(x+h)-f(x)]/h. The limit is an important concept in mathematics. It determines how a function

(explained below) will behave up to a specified value called the limit. For example, the limit

of (x^2-1)/3 as x approaches 2 is 1 because x will not exceed that value. When using limits,

a zero in the denominator has a limit of infinity. Also if the denominator is infinity, it has a limit of zero. A shortcut is to subtract one from the exponent, then multiply the constant times the previous exponent. Examples are the derivative of x^2 is 2x, the derivative of 5x^3 is 15x^2. Derivatives can be used to find average velocity. Therefore, since the integral is the opposite of the derivative, the integral of 2x is x^2 and the integral of 15x^2 is 5x^3. It is found by adding one to the exponent, then dividing by the exponent plus one again. These are simple integrals, they can get more difficult. The derivative is used for rate of change problems. Examples for definite integrals: the integral of 2x=x^2 for x=5 to 7 is 7^2-5^2=49-25=24, the integral of15x^2= 5x^3 for x=3 to x=4 is 5(4)^3-5(3)^3=5*64-27*5=320-13 5=185. The indefinite integral does not have any values.

The function is another important part of calculus. It is described using the argument of x, denoted f(x). F(x) always equals an equation in terms of x like f(x)= x^2+2x+3, f(x)=7x+9, f(x)=9, etc. Functions can also be any desired variable (x, y, z, etc.). If every value in the domain has exactly one value in the range, then it is called a one-to-one function. Functions can be exponential, logarithmic, etc. There can be more than one variable in a function, called a multivariable function. Multivariable functions are part of multivariable calculus.

Calculus is used to find the absolute convergence, convergence, or divergence of an infinite series. Some types of infinite series are geometric, alternating, and harmonic. Tests used include the ratio test, integral test, ratio test, root test, and comparison test.