Math 125 (CRN 10240,11371,15269)
Fall 2013
Quiz 22 (10-25-13)
Carter
Name
1. Find the linearization L(x) of f (x) at the point x = a for the function
f (x) = x1/3 ,
a = −8.
Solution. We evaluate,
f (−8) = −2.
Also,
f 0 (x) = (1/3)x−2/3 .
Hence,
f 0 (−2) =
1
1
= .
2
3 · (−8) /3
12
This gives
L(x) = −2 + (x + 2)/12.
2. A girl flies a kite at a height of 300 feet, the wind carrying the kite horizontally away
from her at a rate of 25 feet/second. How fast must she let out the string when the
kite is 500 feet away from her? Solution.
• Let x denote the horizontal displacement.
• Let z denote the length string.
• Given
dx
dt
• Find dz
dt = 25 feet per second.
.
z=500
• By the Pythagorean Theorem,
x2 + 3002 = z 2 .
• Thus
2x
•
dx
dz
= 2z .
dt
dt
dz
x dx
= · .
dt
z dt
At the time in question, x2 + 3002 = 5002 . So x = 400.
dz 4
=
· 25 = 20 feet/sec.
dt z=500 5