Linear approximation

This online calculator derives the formula for the linear approximation of a function near the given point, calculates approximated value and plots both the function and its approximation on the graph

Karen Luckhurst

Created: 2019-09-12 13:28:22, Last updated: 2020-11-09 11:52:07

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

Digits after the decimal point: 2
Function f(x)

Function value at the point of the approximation

Linear approximation at a given point

Approximate value

Graph
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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

${\displaystyle f(x)=f(a)+f'(a)(x-a)+h_{1}(x)(x-a),\quad \lim _{x\to a}h_{1}(x)=0.}$

where

${\displaystyle P_{1}(x)=f(a)+f'(a)(x-a)}$

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.

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PLANETCALC, Linear approximation