Final 3 Solutions

Final 3 Solutions - Create assignment 57950 Final 1 Dec 03...

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Unformatted text preview: Create assignment, 57950, Final 1, Dec 03 at 4:37 pm 1 This print-out should have 106 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. The due time is Central time. CalC4b24s 50:02, calculus3, multiple choice, > 1 min, wording-variable. 001 If f is differentiable on (- 2 , 2) and f ( x ) ≤ 3 , (- 2 < x < 2) , f (- 1) = 7 , find the smallest value of M so that the in- equality f (1) ≤ M holds for all such f . 1. M = 13 correct 2. M = 16 3. M = 15 4. M = 14 5. M = 12 6. no such M exists Explanation: Since f is differentiable on (- 2 , 2), it is continuous on [- 1 , 1] and differentiable on (- 1 , 1). The MVT applies thus to f on the interval [- 1 , 1]: there exists some c,- 1 < c < 1 such that f (1)- f (- 1) 2 = f ( c ) . Consequently, if f ( x ) ≤ 3 on (- 1 , 1) and f (- 1) = 7, we see that f (1) = 2 f ( c ) + f (- 1) ≤ 13 = M . StewartC5 04 02 17 50:02, calculus3, multiple choice, > 1 min, wording-variable. 002 How many real roots does the equation x 5 + 3 x + 2 = 0 have? 1. exactly one real root correct 2. exactly two real roots 3. exactly three real roots 4. exactly four real roots 5. no real roots Explanation: Define a function f by f ( x ) = x 5 + 3 x + 2 . Then the roots of the equation x 5 + 3 x + 2 = 0 are the x-intercepts of the graph of f . Now f ( x ) > 0 for x very large, while f ( x ) < 0 for- x very large. Thus the graph of f must cross the x-axis at least once. Suppose the graph crosses the x at values x = a and x = b with a < b , i.e. , f ( a ) = f ( b ) = 0 , ( a < b ) . On the other hand, f is a polynomial function, it is continuous and differentiable for all x . Hence Rolle’s Theorem applies, so there exists some c , a < c < b , at which f ( c ) = 0. But f ( x ) = 5 x 4 + 3 > for all x which isn’t consistent with f ( c ) = 0 for some c . Consequently, the graph of f can’t have x-intercepts at both x = a and x = b , so the equation has exactly one root . Create assignment, 57950, Final 1, Dec 03 at 4:37 pm 2 Conceivably, the equation could have a re- peated root at x = d , say. But then f ( x ) = ( x- d ) 2 g ( x ) for some polynomial g in which case f ( d ) = ‡ 2( x- d ) g ( x ) + ( x- d ) 2 g ( x ) ·fl fl fl x = d = 0 . This again is inconsistent with f ( x ) > 0 for all x , so the equation cannot have repeated roots either. CalC4c05b 50:03, calculus3, multiple choice, < 1 min, wording-variable. 003 When Sue uses first and second derivatives to analyze a particular continuous function y = f ( x ) she obtains the chart y y y 00 x <- 3 +- x =- 3 4- 3 < x <-- x = 0 1- 1 < x < 2- + x = 2- 1 DNE x > 2 + + Which of the following can she conclude from her chart?...
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This note was uploaded on 02/28/2010 for the course M 56200 taught by Professor Radin during the Fall '09 term at University of Texas.

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Final 3 Solutions - Create assignment 57950 Final 1 Dec 03...

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