   Chapter 4.3, Problem 84E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# (a) Show that ex ≥ 1 + x for x ≥ 0.(b) Deduce that e x ≥ 1 + x + 1 2 x 2   for   x ≥ 0 .(c) Use mathematical induction to prove that for x ≥ 0 and any positive integer n, e x ≥ 1 + x + x 2 2 ! + ⋯ + x n n !

(a)

To determine

To show: The inequality ex1+x is true for all x0 .

Explanation

Proof:

Consider the function, f(x)=ex(1+x) .

Obtain the derivative of f(x) .

f(x)=ddx(ex(1+x))=ex(0+1)=ex1

Since ex is an increasing function, ex1 for all x0 ,

Substitute 0 in f(x)

(b)

To determine

To deduce: The inequality ex1+x+12x2 for x0 .

(c)

To determine

To prove: For x0 and any positive integer n, ex1+x+x22!+......+xnn! by mathematical induction.

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Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 