m180.hw.section4.3

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

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 Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the 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 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....
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.

### Page1 / 25

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

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online