Problem 1: Consider some arbitrary function f(x), whose value and whose derivative values of every order are already known at some point ro. We can approximate the function value f(x) with f(xo)+f'(xo)(x–x0) for points a that are very close to ro- (a) Justify this "linear" approximation using the graphical interpretation of the derivative. Because this is an approximation, there will be a discrepancy between the actual f(x) and our approximation – call this g(x), and write down an expression for f(x) including g(x). (The approximation is called linear due to how the (æ – xo) term is first-order.) You may use the graph of f (x) = e* , with your own choice of ro, as an illustrative example.
Problem 1: Consider some arbitrary function f(x), whose value and whose derivative values of every order are already known at some point ro. We can approximate the function value f(x) with f(xo)+f'(xo)(x–x0) for points a that are very close to ro- (a) Justify this "linear" approximation using the graphical interpretation of the derivative. Because this is an approximation, there will be a discrepancy between the actual f(x) and our approximation – call this g(x), and write down an expression for f(x) including g(x). (The approximation is called linear due to how the (æ – xo) term is first-order.) You may use the graph of f (x) = e* , with your own choice of ro, as an illustrative example.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Problem 1: Consider some arbitrary function f(x), whose value and whose derivative values of every order
are already known at some point ro. We can approximate the function value f (x) with f(xo)+f'(ro)(x– xo)
for points a that are very close to xo.
(a) Justify this "linear" approximation using the graphical interpretation of the derivative. Because this
is an approximation, there will be a discrepancy between the actual f(x) and our approximation – call
this g(x), and write down an expression for f(x) including g(x). (The approximation is called linear
due to how the (x – xo) term is first-order.) You may use the graph of f(x) = e", with your own choice
of ro, as an illustrative example.
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