Math 125-103
Fall 2012
Quiz 24, Oct. 29
Carter
Name
1. Compute the derivative:
y = ln
1
x x+1
√
Solution. First rewrite,
y = ln (1) − ln (x) −
1
1
ln (x + 1) = 0 − ln (x) − ln (x + 1).
2
2
y0 = −
1
1
−
.
x 2(x + 1)
2. Compute the derivative:
y = arcsin (1 − t)
Solution.
−1
y0 = p
1 − (1 − t)2
3. A girl flies a kite at a height of 300 feet, the wind carrying the kite away from her at a rate
of 25 feet per second. How fast must she let out the string when the kite is 500 feet away?
Solution.
• Let z denote the length of the kite line.
• Let x denote the horizontal displacement of the kite.
dx
dt = 25
dz
dt |z=500 .
• Given
• Find
feet per second.
(Remark. There are other ways to interpret the question, but this one gives the most reasonable (no radicals and non-trivial) solution.
Evidently,
x2 + 3002 = z 2 .
Thus
dx
dz
= 2z ,
dt
dt
dz
x dx
=
dt
z dt
2
When z = 500, x = 250, 000 − 90, 000 = 160, 000. So x = 400.
2x
dz
4
|z=500 = · 25 = 20.
dt
5
4. Find the absolute maximum and absolute minimum for the function over the given interval:
f (x) = 4 − x2 ,
for − 3 ≤ x ≤ 1.
Solution.
• f (−3) = 4 − 9 = −5.
• f (1) = 4 − 1 = 3.
• f 0 (x) = −2x.
Thus to compute the critical points, set f 0 (x) = 0.
−2x = 0.
Hence,
x = 0.
At x = 0, one has f (0) = 4. The Maximum value of f on this interval is 4 and occurs at
x = 0. The minimum value of f on this interval is −5 which occurs at x = −3.