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The slope field can be defined for the following type of differential equations y ′ = f ( x , y ) , {\displaystyle y'=f(x,y),} which can be interpreted geometrically as giving the slope of the tangent to the graph of the differential equation's solution ( integral curve ) at each point ( x , y ) as a function of the point coordinates.
Fig. 1: Isoclines (blue), slope field (black), and some solution curves (red) of y' = xy. The solution curves are y = C e x 2 / 2 {\displaystyle y=Ce^{x^{2}/2}} . Given a family of curves , assumed to be differentiable , an isocline for that family is formed by the set of points at which some member of the family attains a given slope .
Symbolab is an answer engine [1] that provides step-by-step solutions to mathematical problems in a range of subjects. [2] It was originally developed by Israeli start-up company EqsQuest Ltd., under whom it was released for public use in 2011. In 2020, the company was acquired by American educational technology website Course Hero. [3] [4]
Microsoft Math contains features that are designed to assist in solving mathematics, science, and tech-related problems, as well as to educate the user. The application features such tools as a graphing calculator and a unit converter. It also includes a triangle solver and an equation solver that provides step-by-step solutions to each problem.
By doing the above step, we have found the slope of the line that is tangent to the solution curve at the point (,). Recall that the slope is defined as the change in y {\displaystyle y} divided by the change in t {\displaystyle t} , or Δ y Δ t {\textstyle {\frac {\Delta y}{\Delta t}}} .
The geometric interpretation of Newton's method is that at each iteration, it amounts to the fitting of a parabola to the graph of () at the trial value , having the same slope and curvature as the graph at that point, and then proceeding to the maximum or minimum of that parabola (in higher dimensions, this may also be a saddle point), see below.
Photomath is an educational technology mobile app, owned by Google.It features a computer algebra system with an augmented optical character recognition system, designed for use with a smartphone's camera to scan and recognize mathematical equations; the app then displays step-by-step explanations onscreen.
The solution is the weighted average of six increments, where each increment is the product of the size of the interval, , and an estimated slope specified by function f on the right-hand side of the differential equation.