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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

(a) Prove that the equation has at least one real root. (b) Use your graphing device to find the root. correct to three decimal places.

100e–x/100 = 0.0 1x2

(a)

To determine

To prove: The equation 100ex100=0.01x2 has at least one real root.

Explanation

Theorem used: The Intermediate value Theorem:

Suppose that if f is continuous on the closed interval [a, b] and let N be any number between f(a) and f(b), where f(a)f(b). Then there exists a number c in (a, b) such that f(c)=N.

Proof:

Rewrite the equation as follows, 100ex1000.01x2=0.

Consider the function f(x)=100ex1000.01x2.

The exponential function defined on and the polynomial function defined on .

The function f(x) is a combination of polynomial and exponential function and it is defined on the interval (,). So it is continuous everywhere on its domain (,).

Without loss of generality, take the sub-interval [70,71].

In order to show that there is at least one root of the equation 100ex1000.01x2=0 in the interval (70,71), it is enough to show that there is a number c between 70 and 71 for which f(c)=0.

Take a=70, b=71 and N=0.

Substitute 70 for x in f(x),

f(70)=100e701000.01(70)2=100e7100

(b)

To determine

To find: The root correct to three decimal places by using the graph of the function.

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