Linear approximation

This calculator can derive linear approximation formula for the given function, and you can use this formula to compute approximate values. You can use linear approximation if your function is differentiable at the point of approximation (more theory can be found below the calculator).

When you enter a function you can use constants: pi, e, operation signs: + - addition, - - subtraction, * - multiplication, / - division, ^ - power, and functions: sqrt - square root, rootN - N th root, e.g. root3(x) - cube root, exp - exponential function, lb - binary logarithm ( base 2 ), lg - decimal logarithm ( base 10 ), ln - natural logarithm ( base e), logB - logarithm to the base B , e.g. log7(x) - logarithm to the base 7, sin - sine, cos - cosine, tan - tangent, cot - cotangent, sec - secant, cosec - cosecant, arcsin - arcsine, arccos - arccosine, arctan - arctangent, arccotan - arccotangent, arcsec - arcsecant, arccosec - arccosecant, versin - versine, vercos - coversine, haversin - haversine, exsec - exsecant, excsc - excosecant, sh - hyperbolic sine, ch - hyperbolic cosine, tanh - hyperbolic tangent, coth - hyperbolic cotangent, sech - hyperbolic secant, csch - hyperbolic cosecant.

Linear approximation

Function f(x)

Center of the approximation

Point of the approximation

Calculation precision

Digits after the decimal point: 2

Calculate

Function f(x)

 

Function value at the point of the approximation

 

Linear approximation at a given point

 

Approximate value

 

Graph

The file is very large. Browser slowdown may occur during loading and creation.

Download 

 Link  Save  Widget

Linear approximation

Taylor's theorem gives an approximation of a k-times differentiable function around a given point by a k-th order Taylor polynomial.

Linear approximation is just a case for k=1. For k=1 the theorem states that there exists a function h1 such that

where

is the linear approximation of f at the point a.

Thus, by dropping the remainder h1, you can approximate some general function using a linear function, the resulting graph is the tangent line to the graph of a general function at the point of approximation a. This is a good approximation for x when it is close enough to a, since a closely observed curve resembles a straight line. But, of course, Taylor's theorem also ensures that the quadratic approximation (and other higher degree approximations) is, in a sufficiently small neighborhood of the point a, a better approximation than the linear approximation.