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Differentiation of Functions of a Complex Variable
Published in Vladimir Eiderman, An Introduction to Complex Analysis and the Laplace Transform, 2021
The limit of the secant line through the point z(t0) is called the tangent line to the curve at that point. So the vector z′(t0) is in the direction of the tangent line at z(t0).
Function Approximation
Published in John D. Ross, Kendall C. Richards, Introductory Analysis, 2020
John D. Ross, Kendall C. Richards
In building the above Taylor polynomials, we imposed several conditions at a single point x0. Suppose instead that we want to impose a single condition at each of several distinct points (often referred as “nodes” in this context). For example, as an alternative to a tangent line approximation (a first-degree polynomial that agrees with the function value and its derivative value at a single point), we could consider a secant line approximation (which is constructed so that the polynomial and the function have the same value at two distinct points, as in Figure E1.4). It is worth observing that both the secant line and the tangent line have required levels of agreement of m = 2, but the kind of agreement is different. Similarly, if we want to build a polynomial that agrees with our function at three or more (non co-linear) points, then a higher order polynomial will be necessary (see Figures E1.5 and E1.6). A polynomial that agrees with a function at multiple points will be said to interpolatef at these values.
Systems Biology
Published in Lawrence S. Chan, William C. Tang, Engineering-Medicine, 2019
Lawrence S. Chan, William C. Tang
While non-calculus method cannot accurately determine the rate of change at point A (slope 1), we could estimate the rate of change by using other mathematical rule, for example, by drawing a line that touches point A and intersects the curve (secant line, slope 2). We now can estimate the rate of change (average rate of change) at point A by using the same formula: {RateofChange}=f(x2)−f(x1)x2−x1=ΔyΔx
Enhancing shear behaviour of the exterior beam-column joint using sufficient reinforcement details or ultra-high-performance fiber-reinforced concrete
Published in European Journal of Environmental and Civil Engineering, 2023
Sayed Ahmed, Heba A. Mohamed, Talaat Ryad, Ayman Abdo
The stiffness of a cycle is defined as the slope of the secant line for that cycle’s curve. The specimens’ decreased stiffness is caused by flexural and shear cracking, joint zone deformation, nonlinear deformation of concrete, loss of concrete covering, and reinforcing slip, among many other factors. The secant stiffness for all specimens is shown in Figure 8b. The secant stiffness reduces as the displacement increases. Specimens C1, C2, and C3 have secant stiffness greater than the control specimen, with values of 12%, 10%, and 22%, respectively. The secant stiffness of the C4 specimen was roughly 20% greater than the control specimen due to UHPFRC's superior mechanical behavior. The stiffness performance of a C4 specimen without a shear link in the joint core is better than that of a C1 specimen with shear reinforcement that meets seismic requirements. So, the UHPFRC materials give the joint core enough shear strength that it doesn’t need shear reinforcement.
Exploring mathematical connections of pre-university students through tasks involving rates of change
Published in International Journal of Mathematical Education in Science and Technology, 2019
Crisólogo Dolores-Flores, Martha Iris Rivera-López, Javier García-García
Stewart [28] defines the average rate of change of y with respect to x as the quotient of their differences: in the interval . This expression is interpreted in graphical terms as the slope of the secant line of the curve f. However, the measure of change in real-world situations is the most meaningful interpretation. Real-world representations of slope exist in two different forms: physical situations, such as mountain pathways, ski slope and wheelchair ramps; and functional situations, such as distance versus time or quantity versus costs [9]. The most elemental change in real-world situations happens when a changing magnitude increases or decreases in a constant way, this quantity is calculated by the quotient of the differences of and , so the expression is a ratio or rate between changes. According to Stump [9], in both physical situations and functional situations, the slope can be thought of as either a ratio or a rate, depending on the level of reflective abstraction. The measuring units highlight the idea that the rate connects with the real-world situation [6]; we consider that they give context and specificity to the rates of change: m/s in the case of velocity and m/s2 in the case of acceleration.
Mechanical behaviour of femoral prosthesis with various cross-section shape, implant configurations and material properties
Published in Journal of Medical Engineering & Technology, 2021
Hesam Homaei, Javad Jafari Fesharaki
By drawing a line between the two endpoints of the stress-length curve, there should be at least one point on the interval, whose instantaneous slope is equal to the average slope of the secant line shown in Figure 5. Therefore, through calculating The maximum distance between the tangential lines or between the tangential line and the secant line (either showing maximum intercepts’ difference value), as indicated in Figure 5, the maximum deviation from the secant line can be calculated as “H”. Since the most favoured SD appears nearer to the secant line for a certain slope, it represents a contributing parameter in determining the stress-shielding effect.