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Linearization of a function

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 y = f (x) {\displaystyle y=f(x)} at any x = a {\displaystyle x=a} based on the value and slope of the function at x = b {\displaystyle x=b}, given that f (x) {\displaystyle f(x)} is differentiable on {\displaystyle } (or {\displaystyle }) and that a {\displaystyle a} is close to b {\displaystyle b}. In short, linearization approximates the output of a function near x = a {\displaystyle x=a}.

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

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

Using the point (a, f (a)) {\displaystyle (a,f(a))}, L a (x) {\displaystyle L_{a}(x)} becomes y = f (a) + M (x − a) {\displaystyle 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 f (x) {\displaystyle f(x)} at x = a {\displaystyle x=a}.

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

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

The final equation for the linearization of a function at x = a {\displaystyle x=a} is: y = (f (a) + f ′ (a) (x − a)) {\displaystyle y=(f(a)+f'(a)(x-a))}

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