Ch. 14
1.
2.
Let
Handout (14.3 – 14.6)
f ( x, y)  x 2  3 y 2  4 x. Find the maximum rate of change of f at (2, 1).
Given that f(2, 4) = 1.5 and f(2.1, 4.4) = 2.1, estimate the value of f u (2, 4) , where
u is the unit vector in the direction of 1i  4 j. Give your answer to four decimal
places.
3.
Suppose that as you move away from the point (2, 0, 2), the function increases
most rapidly in the direction 0.6i + 0.8 j and the rate of increase of f in this
direction is 7. At what rate is f increasing as you move away from (2, 0, 2) in the
direction of i + j  k ? Give your answer to 4 decimal places.
4.
. Suppose that as you move away from the point (–1, –1, –1), the function
f ( x, y, z) increases most rapidly in the direction 0.6i + 0.8 j and the rate of
increase of f in this direction is 4. At what rate is f increasing as you move away
from (–1, –1, –1) in the direction of i + j  k ? Give your answer to 4 decimal
places.
5.
6.
Find an equation of the tangent plane to the surface 3xyz  x  y  z
the point P(1, –3, 2).
–15 x + 33 y + 3 z  –108
A)
C) 15 x + 33 y + 3z  –108
–15 x + 33 y + 3z  108
–15 x – 33 y + 3 z  –108
B)
D)
3
3
3
 15 at
Find the directional derivative of f ( x, y, z )  3xyz  4 xz at the point (3, 3, 2), in
2
the direction of the vector v  2i  3 j  k .
7.
Consider the function g ( x, y, z )  x 2  y 2  z 2 .
(a) Describe the level set g = 16.
(b) Find a vector perpendicular to the tangent plane to the level set g = 16 at the
point (–1, 2, 2).
8. Find an equation of the tangent plane to x 2  2 xy  4 y  6  z 2 at the point (–4, 1, 3).
6x  4 y  6z  2
6 x  4 y  6 z  –4
A)
D)
–6 x  4 y  6 z  –2
6 x  4 y  6 z  –2
B)
E)
6 x  4 y  6 z  –2
C)
9.
An ant is walking along the surface which is the graph of the function
f ( x, y)  x 2  sin( y).
(a) When the ant is at the point (1, 0, 1), what direction should it move in order to
be moving on the surface in the direction of greatest ascent?
(b) If the ant moves in this direction at a speed of 6 units per second, what is the
rate of change of height of the ant?
10.
The quantity z can be expressed as a function of x and y as follows: z = f(x, y).
Now x and y are themselves functions of r and , as follows: x  g (r, ) and
y  h(r, ).
Suppose you know that 𝑔 (1, 𝜋⁄ ) = −1, and ℎ(1, 𝜋⁄2) = 1. In addition, you are
2
told that
f
(–1,1)  1,
x
g π
(1, )  7,
 2
Find
z
(1, π / 2).
r
f
g π
(–1,1)  6,
(1, )  7,
y
r 2
h π
h π
(1, )  6,
(1, )  4.
r 2
 2