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3. Consider the function f : R \ {-3} → R : x + x +3° (a) Determine the general expression for f(")(x), i.e., the formula of the n-th derivative of f, for n > 1. Prove that this expression is correct. (b) Provide the Taylor series of f about 0 and compute the convergence radius. You do not have to examine the convergence of the remainder. (c) Approximate f(0.5) using a Taylor approximation of f of order 3.

3. Consider the function f : R \ {-3} → R : x + x +3° (a) Determine the general expression for f(")(x), i.e., the formula of the n-th derivative of f, for n > 1. Prove that this expression is correct. (b) Provide the Taylor series of f about 0 and compute the convergence radius. You do not have to examine the convergence of the remainder. (c) Approximate f(0.5) using a Taylor approximation of f of order 3.

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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3. Consider the function f :R\ {-3} → R : x → x +3* (a) Determine the general expression for f(")(x), i.e., the formula of the n-th derivative of f, for n > 1. Prove that this expression is correct. (b) Provide the Taylor series of f about 0 and compute the convergence radius. You do not have to examine the convergence of the remainder. (c) Approximate f(0.5) using a Taylor approximation of f of order 3. (d) Use Taylor's mean value theorem to show that the maximum error on the approxi- mation made in (c) is less than 0.0008. Explain clearly. (e) Sketch the graph of f. For this, determine: the intersections points with the axes, the local extrema, the intervals over which the function is increasing/decreasing, the intervals over which the function is convex/concave, and all asymptotes. We gone veo differont hronda goll:, :1l, Lou...olle Camp ond Inor Tho nuna
Transcribed Image Text:3. Consider the function f :R\ {-3} → R : x → x +3* (a) Determine the general expression for f(")(x), i.e., the formula of the n-th derivative of f, for n > 1. Prove that this expression is correct. (b) Provide the Taylor series of f about 0 and compute the convergence radius. You do not have to examine the convergence of the remainder. (c) Approximate f(0.5) using a Taylor approximation of f of order 3. (d) Use Taylor's mean value theorem to show that the maximum error on the approxi- mation made in (c) is less than 0.0008. Explain clearly. (e) Sketch the graph of f. For this, determine: the intersections points with the axes, the local extrema, the intervals over which the function is increasing/decreasing, the intervals over which the function is convex/concave, and all asymptotes. We gone veo differont hronda goll:, :1l, Lou...olle Camp ond Inor Tho nuna
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