Suppose that is a function such that (0) = 1 and for all x, '(x) =

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Suppose that ƒ is a function such that ƒ(0) = 1 and for all x, ƒ'(x) = ƒ(x) and ƒ(x) > 0 (in Chapter 7, we will see that ƒ(x) is the exponential function ex). Prove that for all x ≥ 0 (each assertion follows from the previous one),

(a) f(x)  1 (b) f(x) > 1 + x 1 (c) f(x)  1 + x + x

Then prove by induction that for every whole number n and all x ≥ 0,

f(x)>1+x+ 1 PM 2! x + 1 + ? n!

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Calculus

ISBN: 9781319055844

4th Edition

Authors: Jon Rogawski, Colin Adams, Robert Franzosa

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