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3. In this problem we will look at higher degree Taylor approximations at 0: (a) Suppose f(x) is a polynomial, with constant term ao, linear term a,x, quadratic term az and so on, ie. Then f(0) = ag. Show that a = f'(0) and az =F(0)- (b) Show by induction that in any polynomial f(1), the coefficient of x* is given by f=) (0) (recall, n! =n x (n – 1) x --- x 1, so that n! = n x (n – 1)! and O! = 1). (Hint: if the coefficient of * in f(x) is a, what is the coefficient of in f(x)?]

3. In this problem we will look at higher degree Taylor approximations at 0: (a) Suppose f(x) is a polynomial, with constant term ao, linear term a,x, quadratic term az and so on, ie. Then f(0) = ag. Show that a = f'(0) and az =F(0)- (b) Show by induction that in any polynomial f(1), the coefficient of x* is given by f=) (0) (recall, n! =n x (n – 1) x --- x 1, so that n! = n x (n – 1)! and O! = 1). (Hint: if the coefficient of * in f(x) is a, what is the coefficient of in f(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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3. In this problem we will look at higher degree Taylor approximations at 0: (a) Suppose f(x) is a polynomial, with constant term ao, linear term ajx, quadratic term azr and so on, i.e. f(x) = agx" + a4-1x +azx² + a,r+ ap. Then f(0) = ap. Show that a = f'(0) and az = }f"(0). (b) Show by induction that in any polynomial f(x), the coefficient of x" is given by f(m) (0) (recall, n! = n x (n – 1) x ... x1, so that n! = n x (n – 1)! and 0! = 1). [Hint: if the coefficient of x" in f(x) is a, what is the coefficient ofr" i
Transcribed Image Text:3. In this problem we will look at higher degree Taylor approximations at 0: (a) Suppose f(x) is a polynomial, with constant term ao, linear term ajx, quadratic term azr and so on, i.e. f(x) = agx" + a4-1x +azx² + a,r+ ap. Then f(0) = ap. Show that a = f'(0) and az = }f"(0). (b) Show by induction that in any polynomial f(x), the coefficient of x" is given by f(m) (0) (recall, n! = n x (n – 1) x ... x1, so that n! = n x (n – 1)! and 0! = 1). [Hint: if the coefficient of x" in f(x) is a, what is the coefficient ofr" i
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