No.33 The length l, width w, and height h of a box change with time. At a certain
instant the dimensions are l＝1m and w＝h＝2m, and l and w are increasing at a rate of
2 m/s while h is decreasing at a rate of 3 m/s. At that instant find the rates at which the
following quantities are changing.
(a) The volume
(b) The surface area
(c) The length of a diagonal
(a)Volume V  l w h
dV
dt
 l w h  w l h  h l w
t
t
t
And l  1 m w  h  2 m
l
t
w
t
 2 m/s
h
t
 2 m/s
 3 m/s
 dV  2  2  2  2  1  2  3  1  2
dt
 8  4  6  6 m 3 /s
(b)Surface
t
S  2l w  lh  wh
 2 l w  w l  l h  h l  w h  h w
dS
dt
t
t
t
t
t

 22  2  2 1  2  2  3 1  2  2  3  2
 24  2  4  3  4  6
 2  5  10 m 2 /s


1
(c)Length of diagonal D  l 2  w2  h 2  l 2  w 2  h 2 2






1
1
1
2
2 2
2
2
2 2
2
2
2 2
dD 1 2
dl
dw
1
1
2l  dt  2 l  w  h
2w dt  2 l  w  h
2h dh
 l w h
dt
2
dt

 


1
 l 2  w 2  h 2 2 l dl
 w dw
 h dh
dt
dt
dt

1
 12  2 2  2 2 2 1  2  2  2  2  3
 13 2  4  6  0 m/s Length of diagonal do not changing.
No.36 If a sound with frequency fs is produced by a source traveling along a line with
speed vo along the same line from the opposite direction toward the source, then the
c  vo 
frequency of the sound heard by the observer is f o  
 f s , where c is the speed
 c vs 
of the sound, about 332m/s.(This is the Doppler effect.) Suppose that, at a particular
moment, you are in a train traveling at 34 m/s and accelerating at 1.2 m/s. A train is
approaching you from the opposite direction on the other track at 40 m/s, accelerating at
1.4 m/s, and sounds its whistle, which has a frequency of 460 Hz. At that instant, what is
the perceived frequency that you hear and how fast is it changing?
c  vo 
(a) f o  
 fs
 c vs 
332  40 
 332  34  460  576.6
The perceived frequency is 576.6 Hz.
c  vo 
(b) Since f o  
 f s , in this case c and f s are constants.
 c vs 
dvo
dv
c  v s    s c  v o 
dt
df o
 dt


fs
dt
2
c  v s 

dvo
dv
c  v s  s c  vo 
dt
dt
fs
c  v s 2
and
dvo
2
 1.2 m/s
dt
df o
1.2 332  40 1.4 332  34 

 460
dt
332  40 2
 350 .4  512 .4  460  4.66 Hz/s
85264
No.39 If z  f x  y , show that z  z  0 .
x
then u  1
設 x y u
x
z  f x  y   f u 
z
x
 z u  z
z
y
 z u   z
u x
u y
u
u  x y
Chain rule
u
 z  z  z  z  0
x
y
u
u
u
y
x
 1
dv s
2
 1.4 m/s
dt