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Chap4_Sec2

# Chap4_Sec2 - x How large can f(2 possibly be Example 5(4.2...

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Prove that the equation x 3 + x – 1 = 0 has exactly one real root. First, we use the Intermediate Value Theorem to show that a root exists. To show that the equation has no other real root, we use Rolle’s Theorem and argue by contradiction. Example 2 (4.2) ROLLE’S THEOREM

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MEAN VALUE THEOREM To illustrate the Mean Value Theorem with a specific function, let’s consider f ( x ) = x 3 x , a = 0, b = 2. Example 3 (4.2)
MEAN VALUE THEOREM The figure illustrates this calculation. square4 The tangent line at this value of c is parallel to the secant line OB . Example 3

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MEAN VALUE THEOREM The main significance of the Mean Value Theorem is that it enables us to obtain information about a function from information about its derivative. Suppose that f (0) = -3 and f ’( x ) 5 for all values of x .

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Unformatted text preview: x . How large can f (2) possibly be? Example 5 (4.2) NOTE Care must be taken in applying the Theorem “If f ’( x ) = 0 for all x in an interval ( a , b ), then f is constant on ( a , b )”. s Let s The domain of f is D = { x | x ≠ 0} and f ’ ( x ) = 0 for all x in D . However, f is obviously not a constant function. This does not contradict the Theorem because D is not an interval. s Notice that f is constant on the interval (0, ∞ ) and also on the interval (-∞ , 0). 1 if ( ) 1 if | | x x f x x x > = = -< MEAN VALUE THEOREM Prove the identity tan-1 x + cot -1 x = π /2. s Although calculus isn’t needed to prove this identity, the proof using calculus is quite simple. Example 6 (4.2)...
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Chap4_Sec2 - x How large can f(2 possibly be Example 5(4.2...

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