The ebook Elementary Calculus is based on material originally written by H.J. Keisler. For more information please read the copyright pages.

Home Continuous Functions Derivatives and Curve Sketching Examples Example 3: Sketching a Curve - Step by Step
 
del.icio.us Reddit Facebook StumbleUpon MySpace Twitter Search the VIAS Library  |  Index

Example 3: Sketching a Curve - Step by Step

y = x4/2 - x2, -2 ≤ x ≤ 2.

Step 1

dy/dx = 2x3 - 2x. d2y/dx2 = 6x2 - 2.

Step 2

dy/dx = 0 at x = -1,0,1.
Step 3 d2y/dx2 = 0 at x ± √⅓. -2, -1, -√⅓, 0, √⅓, 1, 2.

Step 4

x

y

03_continuous_functions-252.gif

03_continuous_functions-253.gif
-2 4 -12 +

-1

-½

0

+

-√⅓

-5/18

4/(3√3)

0

0

0

0

-

√⅓

-5/18

-4/(3√3)

0

1

-½

0

+

2

4

12

+

Step 5

We indicate the conclusions schematically in Figure 3.7.10.

03_continuous_functions-254.gif

Figure 3.7.10

Step 6

The curve is W-shaped, as shown in Figure 3.7.11.

03_continuous_functions-255.gif

Figure 3.7.11

 
Home Continuous Functions Derivatives and Curve Sketching Examples Example 3: Sketching a Curve - Step by Step

Last Update: 2006-11-15