Glossary
Approximation
A value that is close to the true value but not necessarily exact, often obtained through numerical methods like Euler's.
Example:
Euler's Method provides an approximation of y(2) rather than its exact value, especially with larger step sizes.
Change in y (Δy)
The estimated vertical displacement for a given step, calculated as the product of the slope at the current point and the step size.
Example:
If the slope is 5 and the step size is 0.1, the change in y for that step is 0.5.
Differential Equation
An equation that relates a function with its derivatives, describing the rate of change of a quantity.
Example:
The differential equation dy/dx = 2x + y describes how the slope of a function changes based on its x and y values.
Euler's Method
A numerical procedure used to approximate the solution to a differential equation by constructing a sequence of line segments.
Example:
Using Euler's Method to estimate the population of bacteria at a future time, given their growth rate and initial count.
Initial Condition
A specific point (x₀, y₀) that provides a starting value for the solution of a differential equation.
Example:
For a falling object, the initial condition might be (t=0, v=0), meaning at time zero, its velocity is zero.
Slope at a given point (dy/dx or f(x,y))
The instantaneous rate of change of a function at a specific point, determined by the differential equation.
Example:
If dy/dx = xy, then at the point (2, 3), the slope at a given point is 23 = 6.
Step Size (h or Δx)
The constant increment in the x-value used in each step of Euler's Method to move from one point to the next.
Example:
Choosing a step size of 0.1 means you'll calculate approximations at x = 0.1, 0.2, 0.3, and so on.