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Suppose that y = f(x) is differentiable at x = a and that g(x) = m(x - a) +c is a linear function in which m and c are constants. If the error E(x) = f(x)– g(x) were small enough near x = a, it might be considered to use g as a linear approximation of f instead of the linearization L(x) = f(a) + f'(x – a). Show that if accompanying conditions were imposed on g then g(x) = f(a) +f'(a)(x– a). Thus, the linearization L(x) gives the only linear approximation whose error is both zero at x= a and negligible in comparison with x - a.

Suppose that y = f(x) is differentiable at x = a and that g(x) = m(x - a) +c is a linear function in which m and c are constants. If the error E(x) = f(x)– g(x) were small enough near x = a, it might be considered to use g as a linear approximation of f instead of the linearization L(x) = f(a) + f'(x – a). Show that if accompanying conditions were imposed on g then g(x) = f(a) +f'(a)(x– a). Thus, the linearization L(x) gives the only linear approximation whose error is both zero at x= a and negligible in comparison with x - a.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 69EQ: Let x=x(t) be a twice-differentiable function and consider the second order differential equation...
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Suppose that y = f(x) is differentiable at x = a and that g(x) = m(x - a) +c is a linear function in which m and c are constants. If the error E(x) = f(x)– g(x) were small enough near x = a, it might be considered to use g as a linear approximation of f instead of the linearization L(x) = f(a) + f'(x – a). Show that if accompanying conditions were imposed on g then g(x) = f(a) +f'(a)(x– a). Thus, the linearization L(x) gives the only linear approximation whose error is both zero at x= a and negligible in comparison with x - a.
Transcribed Image Text:Suppose that y = f(x) is differentiable at x = a and that g(x) = m(x - a) +c is a linear function in which m and c are constants. If the error E(x) = f(x)– g(x) were small enough near x = a, it might be considered to use g as a linear approximation of f instead of the linearization L(x) = f(a) + f'(x – a). Show that if accompanying conditions were imposed on g then g(x) = f(a) +f'(a)(x– a). Thus, the linearization L(x) gives the only linear approximation whose error is both zero at x= a and negligible in comparison with x - a.
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