Name: __________________
1
Class:
Date: _____________
dz if z = x ln x + 5y , x = sin t , and y = cos t .
(
)
dt
Use the Chain Rule to find
a.
dz =
dt
x
+ ln ( x + 5y )
x + 5y
cos t x
( sin t )
x + 5y
b.
dz =
dt
x
+ ln ( x + 5y )
x + 5y
sin t +
x
( cos t )
x + 5y
c.
dz =
dt
x
+ ln ( x + 5y )
x + 5y
sin t +
5x
( cos t )
x + 5y
d.
dz =
dt
x
+ ln ( x + 5y )
x + 5y
cos t +
5x
( sin t )
x + 5y
e.
dz =
dt
x
+ ln ( x + 5y )
x + 5y
cos t 5x
( sin t )
x + 5y
Problem
code: stet.
14.05.04m
2
Use the Chain Rule to find
a.
b.
c.
Problem
code: stet.
14.05.05m
PAGE 1
dw if w = x e y /z , x = t 2 , y = 8 dt
dw = e y /z
dt
t + x + 3xy
2
z
z
dw = e y /z
dt
2t dw = e y /z
dt
2t d.
x + 3xy
2
z
z
x z
3xy
z
2
e.
t , and z = 4 + 3t .
dw = e y /z
dt
t x z
3xy
dw = e y /z
dt
2t + x z
3xy
z
z
2
2
Name: __________________
3
Use the Chain Rule to find
Class:
Date: _____________
z and z if z = x , x = s e 5t , and y = 2 + s e s
t
y
a.
z = 1 e 5t s
y
b.
z = 5s e 5t + 3sx e 2
s
y
y
c.
z = 5s e 5t y
s
d.
z = 5s e 5t + 3sx e 2
s
y
y
e.
z = 1 e 5t y
s
x e
2
y
3t
3t
3sx e 2
y
x e
2
y
z = 5s e 5t t
y
,
3t
3t
3t
,
z = 5 e 5t +
t
y
,
,
,
3sx e 2
y
x e
2
y
3t
.
3t
3t
z = 1 e 5t y
t
x e
2
y
z = 5 e 5t t
y
x e
2
y
3t
z = 5s e 5t + 3sx e 2
y
t
y
3t
3t
Problem
code: stet.
14.05.08m
4
Let W ( s , t ) = F ( u ( s , t ) , v ( s , t )
)
, where F, u and v are differentiable, u (1, 0 ) = 2 , u
u (1, 0 ) = 1 , v (1, 0 ) = 3 , v (1, 0 ) = 1 , v (1, 0 ) = 9 , F
t
s
W (1, 0 ) and W (1, 0 ) .
s
W (1, 0 ) =
________
W (1, 0 ) =
________
s
t
Problem
code:
stet.
14.05.14
PAGE 2
t
t
u
( 2, 3 )
= s
(1, 0 )
5 , and F
v
= 7,
( 2, 3 )
= 18 . Find
Name: __________________
5
Date: _____________
Use a tree diagram to write out the Chain Rule for u = f ( x , z ) , where x = x ( r , b , c ) and z = z ( r , b , c ) . Assume
all functions are differentiable.
a.
b.
c.
d.
e.
Problem
code: stet.
14.05.17m
PAGE 3
Class:
u =
r
u =
c
u x
u x
u
z
u
z
x r
x c
z
r
z
c
,
u =
b
u x
u
z
x b
z
b
,
u = u x + u z , u = u x + u z , u = u x + u z
r
x r
z r b
x b
z b c
x c
z c
u =
r
u =
c
u +
x
u +
x
u
z
u
z
x +
r
x +
c
z
r
z
c
u =
r
u +
x
x +
r
u
z
u =
,
b
z
r
u +
x
x +
b
u =
r
u x
x r
u
z
u =
,
z
b
r
u x
x b
u =
b
u + u
x
z
u
z
u =
,
c
z
b
u +
x
x +
c
,
u
z
u =
,
z
c
b
u x
x c
x + z
b
b
u
z
z
c
u
z
z
c
,
Name: __________________
Class:
Date: _____________
6
dy = dx
F
x
F
y
= F
F
x
y
2
dy if y 6 + x 2 y 5 = 7 + y e x .
Use the given equation to find
dx
dy =
a.
dx
6y
dy =
b.
dx
5
5
5
4
e
2xy e
2
+ 5x y
x
5
2xy
2
2xy e
6y
+ 5x y
2xy
6y
dy =
c.
dx
5
2
x
2xy e
4
x
2
2
+ 5x y
2xy
4
x
4
x
5
4
x
2
2
dy = 6y + 5xy e
e.
2
dx
x
2 5
2xy e
2x y
2
e
2
5
dy = 6y + 5xy + e
d.
2
dx
x
2 5
2xy e
2x y
x
2
5
+ e
x
2
Problem
code: stet.
14.05.28m
7
(x,y)
is T ( x , y ) , measured in degrees Celsius. A bug crawls so that its position after t seconds is given
1
by x = 1 + t , y = 6 +
t , where x and y are measured in centimeters. The temperature function satisfies
3
T ( 2, 7 ) = 4 and T ( 2, 7 ) = 9 . How fast is the temperature rising on the bug's path after 3 seconds?
The temperature at a point
x
dT =
dt
y
________
C/s
Problem
code:
stet.
14.05.35
8
The pressure of 1 mole of an ideal gas is increasing at a rate of 0.08 kPa/s and the temperature is increasing at a rate of 0.17 K/s. Use the
equation P V = 8.31T to find the rate of change of the volume when the pressure is 10 kPa and the temperature is 340 K.
Please round your answer to the nearest hundredth.
dV =
dt
Problem
code:
stet.
14.05.41
PAGE 4
________
L/s
ANSWER KEY
Homework 14.5 Math 22 Spring 2007, Bauerle
1. e
2. c
3. e
5. b
6. a
7. 4
ANSWER KEY Page 1
53
157
8. 2.12
4.