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jk_16
 3 years ago
Suppose that y=f(x) is differentiable at x=a and that g(x)=m(xa) + c is a linear function in which m and c are constants. If there error E(x)=f(x)g(x) were small enough near x=a, we might think of using g as a linear approximation of f instead of the linearization L(x)=f(a)+f'(a)(xa).
Show that if we impose on g the conditions:
1. E(a)=0
2. lim as x>a E(x)/(xa)=0
then g(x)=f(a)+f'(a)(xa). Thus, the linearization L(x) gives the only linear approximation whose error is both zero at x=a and negligible in comparison to xa.
jk_16
 3 years ago
Suppose that y=f(x) is differentiable at x=a and that g(x)=m(xa) + c is a linear function in which m and c are constants. If there error E(x)=f(x)g(x) were small enough near x=a, we might think of using g as a linear approximation of f instead of the linearization L(x)=f(a)+f'(a)(xa). Show that if we impose on g the conditions: 1. E(a)=0 2. lim as x>a E(x)/(xa)=0 then g(x)=f(a)+f'(a)(xa). Thus, the linearization L(x) gives the only linear approximation whose error is both zero at x=a and negligible in comparison to xa.

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SomeBloke
 3 years ago
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