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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.

arctan x = 1 – x

(a)

To determine

To prove: The equation arctanx=1x 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, arctanx1+x=0.

Consider the function f(x)=arctanx1+x.

The inverse tangent function defined in and the polynomial function also defined in .

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

Without loss of generality, take the sub-interval [0,1].

In order to show that there is at least one root of the equation arctanx1+x=0 in the interval (0,1), it is enough to show that there is a number c between 0 and 1 for which f(c)=0.

Take a=0, b=1 and N=0

(b)

To determine

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

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