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For non-negative integer n, let the function fn be given by (picture) (where n! = 1 · 2 ··· n, with the convention that 0! = 1, so that f0 (x) = 1, f1 (x) = 1 + x etc.).   a) Show by induction that for non-negative integers n then f2n (x) is a positive number for all real numbers x. (hint: you can use that fn + 1 (x) = fn (x) + (xn + 1/(n+1)!) and that f′n+1 (x) = fn (x) (why ?!) (b) Using the result in part (a) (incl. hint); why can we conclude that f9: R → R has an inverse function? Does f10 have an inverse function? (c) Given n ≥ 0; justify that limx→ ∞

For non-negative integer n, let the function fn be given by (picture) (where n! = 1 · 2 ··· n, with the convention that 0! = 1, so that f0 (x) = 1, f1 (x) = 1 + x etc.).   a) Show by induction that for non-negative integers n then f2n (x) is a positive number for all real numbers x. (hint: you can use that fn + 1 (x) = fn (x) + (xn + 1/(n+1)!) and that f′n+1 (x) = fn (x) (why ?!) (b) Using the result in part (a) (incl. hint); why can we conclude that f9: R → R has an inverse function? Does f10 have an inverse function? (c) Given n ≥ 0; justify that limx→ ∞

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section: Chapter Questions
Problem 43RE
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For non-negative integer n, let the function fn be given by

(picture)

(where n! = 1 · 2 ··· n, with the convention that 0! = 1, so that f0 (x) = 1, f1 (x) = 1 + x etc.).

 

a) Show by induction that for non-negative integers n then f2n (x) is a positive number for all real numbers x.

(hint: you can use that fn + 1 (x) = fn (x) + (xn + 1/(n+1)!) and that f′n+1 (x) = fn (x) (why ?!)

(b) Using the result in part (a) (incl. hint); why can we conclude that

f9: R → R has an inverse function? Does f10 have an inverse function?

(c) Given n ≥ 0; justify that limx→ ∞ fn (x) / ex = 0.

(hint: Fun fact: limx → ∞ limn → ∞ fn (x) / ex = 1, while limn → ∞ limx → ∞ fn (x) / ex = 0. Is this a cause for concern?

n x2 = 1+x + 2 x3 fn(x) Σ k! k=0 6 n!
Transcribed Image Text:n x2 = 1+x + 2 x3 fn(x) Σ k! k=0 6 n!
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