Linearizations of a function are lines—usually lines that can be used for purposes of calculation. Linearization is an effective method for approximating the output of a function {\displaystyle y=f(x)}y=f(x) at any {\displaystyle x=a}x=a based on the value and slope of the function at {\displaystyle x=b}x=b, given that {\displaystyle f(x)}f(x) is differentiable on {\displaystyle [a,b]}[a,b] (or {\displaystyle [b,a]}[b,a]) and that {\displaystyle a}a is close to {\displaystyle b}b. In short, linearization approximates the output of a function near {\displaystyle x=a}x=a.

For example, {\displaystyle {\sqrt {4}}=2}{\sqrt {4}}=2. However, what would be a good approximation of {\displaystyle {\sqrt {4.001}}={\sqrt {4+.001}}}{\sqrt {4.001}}={\sqrt {4+.001}}?

For any given function {\displaystyle y=f(x)}y=f(x), {\displaystyle f(x)}f(x) can be approximated if it is near a known differentiable point. The most basic requisite is that {\displaystyle L_{a}(a)=f(a)}L_{a}(a)=f(a), where {\displaystyle L_{a}(x)}L_{a}(x) is the linearization of {\displaystyle f(x)}f(x) at {\displaystyle x=a}x=a. The point-slope form of an equation forms an equation of a line, given a point {\displaystyle (H,K)}(H,K) and slope {\displaystyle M}M. The general form of this equation is: {\displaystyle y-K=M(x-H)}y-K=M(x-H).

Using the point {\displaystyle (a,f(a))}(a,f(a)), {\displaystyle L_{a}(x)}L_{a}(x) becomes {\displaystyle y=f(a)+M(x-a)}y=f(a)+M(x-a). Because differentiable functions are locally linear, the best slope to substitute in would be the slope of the line tangent to {\displaystyle f(x)}f(x) at {\displaystyle x=a}x=a.

While the concept of local linearity applies the most to points arbitrarily close to {\displaystyle x=a}x=a, those relatively close work relatively well for linear approximations. The slope {\displaystyle M}M should be, most accurately, the slope of the tangent line at {\displaystyle x=a}x=a.

An approximation of f(x)=x^2 at (x, f(x))
Visually, the accompanying diagram shows the tangent line of {\displaystyle f(x)}f(x) at {\displaystyle x}x. At {\displaystyle f(x+h)}f(x+h), where {\displaystyle h}h is any small positive or negative value, {\displaystyle f(x+h)}f(x+h) is very nearly the value of the tangent line at the point {\displaystyle (x+h,L(x+h))}(x+h,L(x+h)).

The final equation for the linearization of a function at {\displaystyle x=a}x=a is:

{\displaystyle y=(f(a)+f'(a)(x-a))}{\displaystyle y=(f(a)+f'(a)(x-a))}

For {\displaystyle x=a}x=a, {\displaystyle f(a)=f(x)}f(a)=f(x). The derivative of {\displaystyle f(x)}f(x) is {\displaystyle f'(x)}f'(x), and the slope of {\displaystyle f(x)}f(x) at {\displaystyle a}a is {\displaystyle f'(a)}f'(a).


solving problems in geometric optics
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