Linear Approximation | Formula & Example
Linear Approximation | Formula & Example
In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function). They are widely used in the method of finite differences to produce first-order methods for solving or approximating solutions to equations.
Given a twice continuously differentiable function {\displaystyle f} of one real variable, Taylor’s theorem for the case {\displaystyle n=1} states that
- {\displaystyle f(x)=f(a)+f'(a)(x-a)+R_{2}\ }
where {\displaystyle R_{2}} is the remainder term. The linear approximation is obtained by dropping the remainder:
- {\displaystyle f(x)\approx f(a)+f'(a)(x-a)}.
This is a good approximation for {\displaystyle x} when it is close enough to {\displaystyle a}; since a curve, when closely observed, will begin to resemble a straight line. Therefore, the expression on the right-hand side is just the equation for the tangent line to the graph of {\displaystyle f} at {\displaystyle (a,f(a))}. For this reason, this process is also called the tangent line approximation.
If {\displaystyle f} is concave down in the interval between {\displaystyle x} and {\displaystyle a}, the approximation will be an overestimate (since the derivative is decreasing in that interval). If {\displaystyle f} is concave up, the approximation will be an underestimate.
Linear approximations for vector functions of a vector variable are obtained in the same way, with the derivative at a point replaced by the Jacobian matrix. For example, given a differentiable function {\displaystyle f(x,y)} with real values, one can approximate {\displaystyle f(x,y)} for {\displaystyle (x,y)} close to {\displaystyle (a,b)} by the formula
- {\displaystyle f\left(x,y\right)\approx f\left(a,b\right)+{\frac {\partial f}{\partial x}}\left(a,b\right)\left(x-a\right)+{\frac {\partial f}{\partial y}}\left(a,b\right)\left(y-b\right).}
The right-hand side is the equation of the plane tangent to the graph of {\displaystyle z=f(x,y)} at {\displaystyle (a,b).}
In the more general case of Banach spaces, one has
- {\displaystyle f(x)\approx f(a)+Df(a)(x-a)}
where {\displaystyle Df(a)} is the Fréchet derivative of {\displaystyle f} at {\displaystyle a}.
Linear Approximation Calculator
Linear approximation is a method of estimating the value of a function, f(x), near a point, x = a, using the following formula:
This lesson shows how to find a linearization of a function and how to use it to make a linear approximation. This method is used quite often in many fields of science, and it requires knowing a bit about calculus, specifically, how to find a derivative. The formula we’re looking at is known as the linearization of f at x = a, but this formula is identical to the equation of the tangent line to f at x = a.
Linear Approximation Formula
So, how do you find the linearization of a function f at a point x = a? Remember that the equation of a line can be determined if you know two things:
- The slope of the line, m
- Any single point that the line goes through, (a, b).
We plug these pieces of info into the point-slope form, and this gives us the equation of the line. (This is just algebra, folks; no calculus yet.)
y – b = m(x–a)
But, in problems like these, you will not be given values for b or m. Instead, you have to find them yourself. Firstly m = f ‘(a), because of the derivative measures the slope, and secondly, b = f(a), because of the original function measures y-values.
Putting it all together and solving for y:
