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


Example 6

f(x) = x4/5.

Then

f'(x) = 4/5x-1/5, f"(x) = -4/25x-6/5

f'(x) is undefined at x = 0. We make the table:

f(x)

f'(x)

f"(x)

Comments

limx→-∞

0

horizontal

x= -1

1

-4/5

-4/25

decreasing, ∩

limx→0-

0

- ∞

vertical

x = 0

0

undef.

limx→0+

0

vertical

x= 1

1

4/5

-4/25

increasing, ∩

limx→∞

0

horizontal

With this information we can sketch the curve in Figure 5.3.4.

05_limits_g_approx-187.gif

Figure 5.3.4

This time the limits of the derivative as x approaches zero show that there is a cusp at x = 0, with the curve decreasing when x < 0 and increasing when x > 0.


Last Update: 2006-11-15