Calculus is the study of mathematical functions. It involves calculating the derivative of a function and finding its critical points. A critical point is where the derivative is either zero or undefined. These points are unique in the domain of the derivative. To find a critical point, students must first calculate the function’s derivative, then set it to zero. They can then plug in the critical numbers to solve for x or y values.

**Approximations**

Calculus approximations involve general functions being approximated in a certain way. One common way is the method of finite differences. This method is widely used in solving equations. The method aims to produce a first-order method for solving equations. In addition to the method of finite differences, many other types of approximations are used in math.

Calculus approximations are an integral part of the subject. These methods include integration, differentiation, and limits and are used in various economic and scientific applications. Calculus is a collection of techniques that approximate flat and curved things. Standard calculus texts tend to focus on the limiting process. However, Calculus Approximations emphasize approximating processes and their theoretical foundations.

**Integrals**

Integrals are numbers that describe functions. They can be used to explain concepts like volume, area, or displacement. These numbers are obtained by combining infinitesimal data and combining them into a single number. The process of finding integrals is known as integration. This concept is beneficial in many areas of science.

In calculus, there are two kinds of integrals: definite and indefinite. Definite integrals calculate the numerical value of a function, whereas indefinite integrals compute the inverse function. General equations represent both types of integrals. The constant of an integral is called C.

Integrals are used extensively in calculus. For example, they are used to find the areas and volumes of solid bodies and to solve differential equations.

**Derivatives**

Calculus derivatives are a formal way to discuss rates of change. Derivatives are often used to find limits and maxima on a function. Derivatives are also valuable for graphing a function. The most straightforward derivative is f(x). A derivative of a function is a slope that is dependent on some variable, usually x.

Students study derivatives by examining the symmetric difference quotient at many points on a graph. They also form conjectures based on numerical investigations. In addition, they investigate information about graphs based on first and second derivatives. For example, they learn that the second derivative is positive when a function increases and negative when it decreases. Additionally, they learn that the second derivative is negative when a graph has a concave upward or downward shape.

Another way to represent derivatives is to write them mathematically. Differentiation is the process of taking the derivative of a function and comparing its value to the original value. This can be done using the slope of a line or calculating the slope of a function.

**Extreme value theorem**

The Extreme Value Theorem is a concept in calculus that states that a real-valued function must reach its maximum and minimum at least once. This is a critical concept in mathematical analysis. Understanding this concept is crucial to solving many problems in the field.

The first step in applying the Extreme Value Theorem is to identify the limits of a function. For example, a continuous function at one end of the interval must also be continuous at the other. A second step in proving the Extreme Value Theorem is using the discontinuity principle.

When analyzing a function, the Extreme Value Theorem can help us understand why specific values are higher or lower than others. For example, if the function f(x) is a constant, the minimum value of a function equals the maximum value of the function. But if the function is variable, then the interval is not continuous.

**Newton and Leibniz**

Both Newton and Leibniz were instrumental in the development of calculus. Although they worked in opposite directions, they reached the same conclusions in their respective fields. Newton’s calculus was based on the principle of differentiation, while Leibniz’s was based on the principle of integration.

In Newton’s calculus, the primary derivative is the measure of velocity; Leibniz’s approach was to apply calculus to geometry. He developed a notation involving undefined, discrete variables and used the notion of integrals to describe areas under curves.

Newton and Leibniz’s calculus theory was a significant breakthrough in mathematics. Earlier, the concept of calculus was limited to studying geometrical problems. For example, until the work of Newton and Leibniz, no one had considered the existence of infinitely small numbers. However, they were able to develop a way to solve such problems and make them accessible to everyone. They were also able to simplify the language and terminology of calculus.