The absolute scale of temperature is a temperature scale that starts at absolute zero, the theoretical lowest possible temperature where all matter would have zero thermal energy. The most commonly used absolute temperature scale is the Kelvin (K) scale, which is defined such that 0 K is equal to -273.15°C, the temperature at which all…
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…
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…
Integration by parts is a technique used in calculus to find the integral of a product of two functions. The general formula for integration by parts is: ∫ u dv = u v – ∫ v du where u and v are functions of x, and dv and du are their differentials. To use this…
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…
Definite integrals are a way to find the area under a curve between two points. One way to think about definite integrals is as the limit of a sum of rectangles. Suppose we want to find the area under the curve of a function f(x) between x=a and x=b. We can start by dividing the…
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…
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…
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…
Differential calculus is a branch of calculus that deals with the study of rates of change and slopes of curves. Trigonometry is a branch of mathematics that deals with the study of triangles and the relationships between their sides and angles. The two subjects are related in that trigonometric functions, such as sine, cosine, and…
