Math 2263 Multivariable Calculus
Homework 8: 14.5 #42, 14.6 #16
June 24, 2011
14.5#42
Car A is traveling north on Highway 16 and car B is traveling west on Highway 83. Each car
is approaching the intersection of these highways. At a certain moment, car A is .3 km from
the intersection and traveling 90 km/h while car B is .4 km from the intersection and traveling
at 80 km/h. How fast is the distance between the cars changing at that moment?
If A is the distance from car A to the intersection and B is the distance from car B to
the intersection, the Pythagorean Theorem tells us that the distance x between the
cars is
√
x = A2 + B 2 .
Viewing A and B as functions of time t, we have A0 (t) = −90 and B 0 (t) = −80. We
.
need to find dx
dt
xA = (1/2)(A2 + B 2 )−1/2 (2A) = √
A
.3
.3
=
= .6
=√
2
.5
.16 + .09
+B
A2
xB = (1/2)(A2 + B 2 )−1/2 (2B) = √
B
.4
= .8
=
.5
A2 + B 2
dx
= xA At + xB Bt = (.6)(−90) + (.8)(−80) = −118
dt
14.6#16
Find the directional derivative of the function at the given point in the direction of the vector v.
√
f (x, y, z) = xyz, (3, 2, 6), v = h−1, −2, 2i
yz
xz
xy
∇f (3, 2, 6) = h √
, √
, √
i|(3,2,6)
2 xyz 2 xyz 2 xyz
= h1, 3/2, 1/2i
To find the directional derivative, we need to use the unit vector in the given
direction.
v
h−1, −2, 2i
=
= h−1/3, −2/3, 2/3i
u=
|v|
3
Du f (3, 2, 6) = h1, 3/2, 1/2i · h−1/3, −2/3, 2/3i = −1/3 − 1 + 1/3 = −1