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Introduction to differentiation
Published in John Bird, Bird's Basic Engineering Mathematics, 2021
Calculus is a branch of mathematics involving or leading to calculations dealing with continuously varying functions such as velocity and acceleration, rates of change and maximum and minimum values of curves. Calculus has widespread applications in science and engineering and is used to solve complicated problems for which algebra alone is insufficient.
Segmentation and Edge/Line Detection
Published in Scott E. Umbaugh, Digital Image Processing and Analysis, 2017
Gradient operators are based on the idea of using the first or second derivative of the gray-level function as an edge detector. Remember from calculus that the derivative measures the rate of change of a line or the slope of the line. If we model the gray-level transition of an edge by a ramp function, which is a reasonable approximation to a real edge, we can see what the first and second derivatives look like in Figure 4.2-5. When the gray level is constant, the first derivative is zero, and when it is linear, it is equal to the slope of the line. With the following operators we will see that this is approximated with a difference operator, similar to the methods used to derive the definition of the derivative. The second derivative is positive at the change on the dark side of the edge, negative at the change on the light side, and zero elsewhere.
Differentiation of functions of one or more real variables
Published in Alan Jeffrey, Mathematics, 2004
The calculus has two interrelated branches, called the differential and integral calculus. The fundamental concept in the differential calculus that forms the subject of study in the first part of this chapter is the definition and interpretation of the derivative df/dx of a function f(x) of a single real variable x. The derivative is constructed and interpreted using the notions of a continuous function and a limit. The rules for performing differentiation on combinations of functions are then derived in order to enable the derivatives of more complicated functions to be determined in terms of a table of derivatives of elementary functions.
A resource for introducing students to the integral concept
Published in International Journal of Mathematical Education in Science and Technology, 2019
The integral, the accumulation part of the Fundamental Theorem of Calculus, and the realization that the rate of change of this accumulation is related to the integrand, is considered a landmark in the development of calculus [1]. Historically, the idea shed light on many problems in mathematics, physics and astronomy [1]. Some important uses for the integral in a physics context involve using it to derive quantities such as work, impulse, centre of mass, moment of inertia, electric flux and magnetic flux; while in a chemistry context, it can also be used to determine the rate law for a chemical reaction. Yet, despite the integral’s importance, it is a problematic idea for students to master in a mathematics context. Furthermore, it is difficult for students to transfer to different contexts [2–5]. For the purposes of this article, a mathematics context is defined as a situation in which the relevant variables are not assigned a particular meaning, such as time, distance, chemical concentration, etc.
Differential and integral proportional calculus: how to find a primitive for
Published in International Journal of Mathematical Education in Science and Technology, 2021
William Campillay-Llanos, Felipe Guevara, Manuel Pinto, Ricardo Torres
Let where the symbol represents the integral used in the traditional courses of calculus. From elementary calculus, we know that it is impossible to find an antiderivative g that satisfies (6).