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Problem 1: Consider some arbitrary function f(x), whose value and whose derivative values of every order are already known at some point xo. We can approximate the function value f(x) with f(xo)+f'(xo)(x– xo) for points x 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) = e², with your own choice of xo, as an illustrative example. (b) Noting that f'(x) is also another function of x, what is the linear approximation of f'(x) in terms of g(x), its derivatives, x, and xo? Hint: Consider how f(x), g(x), and their (possibly higher-than-first- order) derivatives at xo are related. (c) f(x) can be found, of course, by integration of f'(x). Using this insight and our linear approximation of f'(x) in (b), g(x) in our f(x) from (a) can be re-written as some term of some order in (x – xo) plus another term h(x). What is f(x) with this re-written g(x)?

Problem 1: Consider some arbitrary function f(x), whose value and whose derivative values of every order are already known at some point xo. We can approximate the function value f(x) with f(xo)+f'(xo)(x– xo) for points x 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) = e², with your own choice of xo, as an illustrative example. (b) Noting that f'(x) is also another function of x, what is the linear approximation of f'(x) in terms of g(x), its derivatives, x, and xo? Hint: Consider how f(x), g(x), and their (possibly higher-than-first- order) derivatives at xo are related. (c) f(x) can be found, of course, by integration of f'(x). Using this insight and our linear approximation of f'(x) in (b), g(x) in our f(x) from (a) can be re-written as some term of some order in (x – xo) plus another term h(x). What is f(x) with this re-written g(x)?

Advanced Engineering Mathematics
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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Problem 1: Consider some arbitrary function f(x), whose value and whose derivative values of every order are already known at some point xo. We can approximate the function value f(x) with f(xo)+f'(xo)(x-xo) for points x 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 xo, as an illustrative example. (b) Noting that f'(x) is also another function of x, what is the linear approximation of f'(x) in terms of g(x), its derivatives, x, and xo? Hint: Consider how f(x), g(x), and their (possibly higher-than-first- order) derivatives at xo are related. (c) f(x) can be found, of course, by integration of f'(x). Using this insight and our linear approximation of f'(x) in (b), g(x) in our f(x) from (a) can be re-written as some term of some order in (x – xo) plus another term h(x). What is f(x) with this re-written g(x)?
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 xo. We can approximate the function value f(x) with f(xo)+f'(xo)(x-xo) for points x 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 xo, as an illustrative example. (b) Noting that f'(x) is also another function of x, what is the linear approximation of f'(x) in terms of g(x), its derivatives, x, and xo? Hint: Consider how f(x), g(x), and their (possibly higher-than-first- order) derivatives at xo are related. (c) f(x) can be found, of course, by integration of f'(x). Using this insight and our linear approximation of f'(x) in (b), g(x) in our f(x) from (a) can be re-written as some term of some order in (x – xo) plus another term h(x). What is f(x) with this re-written g(x)?
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