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Separation of variables method

Separation of variables is a method used to solve certain types of differential equations. The method involves assuming that the solution to the differential equation can be expressed as a product of two functions, each of which depends on only one of the variables in the equation. For example, consider the partial differential equation: ∂u/∂t…

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Formation of ordinary differential equations

Ordinary differential equations (ODEs) describe the relationships between a function and its derivatives with respect to a single independent variable. There are several ways to form ODEs, but some of the most common are: In general, the formation of ODEs requires an understanding of the underlying physical or mathematical principles that govern the system being…

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Application of definite integrals to the determination of areas bounded by simple curves

Definite integrals can be used to determine the area bounded by a simple curve and the x-axis, or by a simple curve and the y-axis. The area can be found by dividing it into small rectangles, finding the area of each rectangle, and then adding up the areas of all the rectangles. This process is…

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Fundamental theorem of integral calculus

The fundamental theorem of calculus is a theorem that links the concept of differentiation with that of integration, and it has two parts. Part 1: Let f be a continuous function on the interval [a,b], and let F be a function defined by F(x) = ∫a^x f(t) dt where a ≤ x ≤ b. Then…

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Definite integral and their properties

A definite integral is a mathematical concept that represents the area under a curve between two specified points on the x-axis. It is denoted by the symbol ∫, and is defined as: ∫[a,b] f(x) dx = F(b) – F(a) where f(x) is a function, F(x) is its antiderivative or indefinite integral, and a and b…

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Indefinite integrals of standard functions

Sure, here are some common indefinite integrals of standard functions: Note that there are many more integrals of standard functions, but these are some of the most common ones. Also, the notation $\int f(x) dx$ represents the indefinite integral of the function $f(x)$ with respect to $x$, and $C$ is the constant of integration. What…

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Integration as the inverse process of differentiation

Integration is the inverse process of differentiation. Differentiation is a process by which we find the rate of change of a function with respect to one of its variables. The inverse process of differentiation is integration, which is used to find the function that gives rise to a given rate of change. If we have…

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Geometric interpretation of derivatives

The derivative of a function can be interpreted geometrically in several ways. One common interpretation is that the derivative represents the slope of the tangent line to the graph of the function at a given point. More specifically, if we consider a function f(x) and a point (a, f(a)) on its graph, the slope of…

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Derivatives up to order two of implicit functions

When dealing with implicit functions, we often have equations of the form: F(x, y) = 0 where y is an implicit function of x. That is, we can’t solve for y explicitly in terms of x. However, we can still differentiate the equation with respect to x to find the derivative of y with respect…

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Rolle’s theorem and Lagrange’s mean value theorem

Rolle’s Theorem and Lagrange’s Mean Value Theorem are two important results in calculus that relate to the behavior of functions on a given interval. Rolle’s Theorem states that if a function f(x) is continuous on the closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists at…

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