This preview shows pages 1–6. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ycoordinates 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 ycoordinates 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 ycoordinates 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....
View
Full
Document
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.
 Spring '10
 Koines
 Math, Derivative

Click to edit the document details