An Euler approximation is a numerical method of solving differential equations when the solution to a differential equation cannot be found analytically. Euler approximations are found using a recursive formula (repeated several times) that uses slope information (the derivative) to approximate a value on a solution curve close to an initial point.

Euler approximations begin with a given differential equation dy/dx=F(x_o,y_o) and a point (x_o,y_o) of the solution y=F(x). This information can be used to find an approximation using the point-slope form of a tangent line: y=y_o + F(x_o,y_o)•(x-x_o)

Note: The solution to y is only an approximation of f(x). y≠f(x)

The equation can be further simplified to y=y_o+h•F(x_o,y_o) with h=(x-x_o)

1. The initial point is (x_o,y_o)
2. The slope is F(x_o,y_o)
3. The step size is h
4. The approximated value is y

If the function is concave down, the approximation will be an overestimate, and if the function is concave up, the approximation will be an underestimate. (Concavity is found using the second derivative test.) These approximations are more accurate with smaller step sizes.