m180.hw.section4.3

m180.hw.section4.3 - 230 CHAPTER 4. APPLICATIONS OF...

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Unformatted text preview: 230 CHAPTER 4. APPLICATIONS OF DIFFERENTIATION Homework (a) Critical Numbers: Find the deriva- tive of f and the critical num- bers of f . (b) Increasing/Decreasing Chart: Make a chart showing where f is pos- itive or negative and f is in- creasing or decreasing. (c) For large x , f behaves like its highest degree term. Evaluate the following. lim x f ( x ) lim x - f ( x ) (d) Inflection Points: Find the sec- ond derivative of f and the in- flection points of f . (e) Concavity Chart: Make a chart showing where f 00 is positive or negative and where f is concave up or concave down. (f) Plot Points: Give the y-coordinates for the local max, local min, and inflection points. (g) Sketch and label the graph of f . 1. f ( x ) = x 3- 12 x + 1 2. f ( x ) = x 4- 4 x- 1 3. f ( x ) = x 3 + x 4. f ( x ) = 2 x 3- 3 x 2- 12 x 5. f ( x ) = 2 + 3 x- x 3 6. f ( x ) = 200 + 8 x 3 + x 4 7. f ( x ) = 3 x 5- 5 x 3 + 3 8. f ( x ) = x 4- x 2 9. f ( x ) = x 3 + 6 x 2 + 9 x 10. f ( x ) = 2- 15 x + 9 x 2- x 3 11. f ( x ) = 8 x 2- x 4 12. f ( x ) = x 4 + 4 x 3 13. f ( x ) = x ( x + 2) 3 14. f ( x ) = 2 x 5- 5 x 2 + 1 15. f ( x ) = 20 x 3- 3 x 5 4.3. DERIVATIVES AND THE SHAPE OF GRAPHS 231 Solutions 1. f ( x ) = x 3- 12 x + 1 (a) Critical Numbers. f ( x ) = 3 x 2- 12 = 3( x- 2)( x + 2) = 0 x = 2 (b) Increasing/Decreasing Chart x =- 2 x = 2 max min f | | f pos | neg | pos (c) For large x , f behaves like its highest degree term, y = x 3 lim x f ( x ) = lim x x 3 = lim x - f ( x ) = lim x - x 3 =- (d) Inflection Points f ( x ) = 3 x 2- 12 f 00 ( x ) = 6 x = 0 x = 0 (e) Concavity Chart x = 0 I.P. f | f 00 neg | pos 232 CHAPTER 4. APPLICATIONS OF DIFFERENTIATION (f) Plot Points: Give the y-coordinates for the local max, local min, and inflection points. x f ( x ) max- 2 17 IP 1 min 2- 15 (g) Sketch and label the graph of f . 2. f ( x ) = x 4- 4 x- 1 (a) Critical Numbers. f ( x ) = x 4- 4 x- 1 f ( x ) = 4 x 3- 4 = 4( x 3- 1) = 4( x- 1)( x 2 + x + 1) = 0 x 3- 1 = 0; x 3 = 1; x = 1 (b) Increasing/Decreasing Chart x = 1 min f | f neg | pos 4.3. DERIVATIVES AND THE SHAPE OF GRAPHS 233 (c) For large x , f behaves like its highest degree term, y = x 4 lim x f ( x ) = lim x x 4 = lim x - f ( x ) = lim x - x 4 = (d) Inflection Points f ( x ) = 4 x 3- 4 f 00 ( x ) = 12 x 2 = 0 x = 0 (e) Concavity Chart not IP x = 0 f | f 00 ( x ) = 12 x 2 pos | pos (f) Plot Points: Give the y-coordinates for the local max, local min, and inflection points for f ( x ) = x 4- 4 x- 1. x f ( x ) min 1- 4- 1 (g) Sketch and label the graph of f . 234 CHAPTER 4. APPLICATIONS OF DIFFERENTIATION 3. f ( x ) = x 3 + x (a) Critical Numbers....
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This note was uploaded on 10/14/2010 for the course MATH 180 taught by Professor Koines during the Spring '10 term at Orange Coast College.

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m180.hw.section4.3 - 230 CHAPTER 4. APPLICATIONS OF...

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