Linear approximation is crucial to many well-known numerical techniques, such as Euler’s Method, Ranga Kutta methods, etc. to approximate solutions to ordinary differential equations. The calculation is based on the closeness of the tangent line to the graph of the function around a point.

For a given point x0, if x0 is in the domain of the function f(x) i.e. the function is differentiable at that point, the equation of the graph of f(x) at the point (x0,y0) where y0=f(x0) becomes:

y-y0=f’(x0)(x-x0)

For any value x1 that is sufficiently close to x0, it can be said that x1=x0+Δx, where Δx is the difference between x1 and x0.

Consequently, the value of y1 is approximated by f(x0+Δx) on the tangent line, given by:

y1=y0+Δx*f’(x0)

where f’(x0) is the derivative of the function f(x) at the point x0.

For Δy=y1-y0, we obtain Δy=Δx*f’(x0). In the limit that x approaches x0, the final equation becomes:

L(x)≈f(x0)+f’(x)(x-x0)

where L(x) represents the linear approximation to the function f(x) at the point x0. The calculator will calculate linear approximation to the explicit curve at any given point.