Rolle Problems - Contents 5 Rolles Theorem the Mean Value...

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Contents 5 Rolle’s Theorem, the Mean Value Theorem, and L’Hˆopital’s Rule 80 5.1 Rolle’s Theorem ........................................... 80 5.2 Mean Value Theorem ......................................... 81 5.3 L’Hospital’s Rule 84 Close the Window to Return to the Table of Contents
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Chapter 5 Rolle’s Theorem, the Mean Value Theorem, and L’Hˆopital’s Rule 5.1 Rolle’s Theorem In the following problems (a) Verify that the three conditions of Rolle’s theorem have been met. (b) Find all values z that satisfy the conclusion of the theorem. 1. (a) f ( x )= x 2 - 7 x +10 , on [2 , 5] (b) f ( x x 2 - 7 x , on [0 , 7] (c) f ( x x 2 - 7 x , on [ - 1 , 8] 2. (a) f ( x x 2 - 4 x, on [0 , 4] (b) f ( x x 2 - 4 x, on [ - 1 , 5] (c) f ( x x 2 - 4 x, on [ - 4 , 8] 3. (a) f ( x x 3 - 5 x 2 - 17 x +21 , on - 3 x 7 (b) f ( x x 3 - 16 x, on - 4 x 4 (c) f ( x x 3 +2 x 2 - x - 2 , on - 2 x 1 4. (a) f ( x x 2 - 6 x - 7 , on [ - 1 , 7] (b) f ( x x 3 + x 2 - 6 x, on [0 , 2] (c) f ( x x 2 - x - 2 , on [ - 1 , 2] (d) g ( x x 3 +5 x 2 +6 , on [ - 3 , 0] 5. (a) f ( x )=s in2 x, on [0 ] (b) f ( x ) = cos ( x 2 ) , on [ π, 3 π ] (c) f ( x in x + cos x on £ - π 4 , 3 π 4 / ( Hint: 3 π 4 = π 4 + π 2 ) Close the Window to Return to the Table of Contents
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81 5.2 Mean Value Theorem In problems 6 to 11 determine in what way the function fails to meet the conditions of Rolle’s theorem. 6. f ( x )=4 x 2 - 8 x for 0 x 3. 7. f ( x )=1 - cos 4 x + 2 cos x for 3 π 2 x 7 π 2 . £ f ( 3 π 2 ) 6 =0 / 8. f ( x )= 3 x 2 - 2 x +4 x - 2 for 1 x 5. 9. f ( x )=s in2 x + cos 2 x for 0 x 3 π 8 . 10. f ( x 3 x +2 ( x - 1) 2 for 1 2 x 2. 11. f ( x - x 2 3 for - 1 x 1. [ f 0 (0) is undefined] 12. Find a function f ( x ) on the interval [ - 1 , 1] such that f ( x ) fails to meet the conditions of Rolle’s theorem, but [ f ( x )] 2 does meet them. In what way does f ( x ) fails to meet the conditions? 13. Use Rolle’s theorem to prove that, regardless of the value of b , there is at most one point x in the interval - 1 x 1 for which x 3 - 3 x + b . 5.2 Mean Value Theorem For problems numbered 14 to 22, (a) Verify that the conditions of the mean value theorem have been met. (b) Find all numbers z that satisfy the theorem. Example : f ( x x 3 - 2 x 2 for 1 x 3. Solution. (a) f ( x ) is continuous (all polynomials are continuous) f 0 ( x )=3 x 2 - 4 x exists for x (1 , 3). Thus the conditions of the mean value theorem have been met. (b) f (3)=27 - 18 = 9. f (1)=1 - 2= - 1. Therefore, f ( b ) - f ( a ) b - a = f (3) - f (1) 3 - 1 = 9 - ( - 1) 3 - 1 = 10 2 =5 f 0 ( z f ( b ) - f ( a ) b - a (M.V.T.) 3 z 2 - 4 z 3 z 2 - 4 z - 5=0 z = 4 ± p 16 - 4( - 5)(3) 6 = 4 ± 76 6 2 .
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This note was uploaded on 09/22/2011 for the course ECON 101 taught by Professor Mr.tull during the Spring '11 term at De La Salle University.

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Rolle Problems - Contents 5 Rolles Theorem the Mean Value...

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