Math 101C Ch. 12 Test
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1. (10 points) Find the equation of the line (in vector form) that passes through (2, 5, −8) and (4, 3, 1).
Solution:
r(t) = h2 + 2t, 5 − 2t, −8 + 9ti
2. (10 points) Find the equation of the plane that passes through the points (3, 1, 0), (2, 5, −1) and (4, 3, 1).
Solution:
z = x−3
3. (10 points) The electric potential in the xy-plane associated with two positive charges, one at (0, 1) with
twice the magnitude as the charge at (0, −1), is
φ( x, y) = p
2
x 2 + ( y − 1)2
+p
1
x 2 + ( y + 1)2
(a) Compute the value of φx (0, 2).
Solution: Computing the partial derivative gives φx (0, 2) = 0
(b) What is the meaning of your answer in part (a)? Note that φ is measured in volts and x is measured
in meters.
Solution: At the point (0, 2), a small change in the position x will result in no change in the
voltage φ.
Points on page:
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Math 101C
Ch. 12 Test
Page 2 of 4
4. (10 points) Assuming that z is an implicitly defined function of x and y, find z x if xz + yz3 − x2 = 0.
Solution:
zx = −
Fx
2x − z
=
Fz
x + 3yz2
√
5. (10 points) Find the equation of the tangent plant to x2 + y2 + z2 = 100 at the point (4, 3, 5 3).
Solution:
√
4x + 3y + 5 3z = 100
6. (10 points) Mark each question as “True” or “False”. Assume z = f ( x, y) is a three-dimensional function and w = F ( x, y, z) is a four-dimensional function. Also assume that if w = 0, the resulting level
surface is z = f ( x, y).
(a) The directional derivative Du f is a vector. True
False
(b) The gradient ∇ f is a vector that is orthogonal to the surface z = f ( x, y). True
False
(c) The gradient ∇ F is a vector that is orthogonal to the surface z = f ( x, y). True
False
(d) The directional derivative Du f is orthogonal to the level curves of z = f ( x, y). True
(e) The maximum value of the directional derivative Du f is given by k∇ f k. True
False
False
7. (10 points) Find the direction(s) in the xy-plane in which the function f ( x, y) = 12 − 4x2 − y2 at the
point (1, 2, 4) has a zero change. Express the direction in terms of a unit vector.
Solution: At the point ( x, y) = (1, 2), ∇ f = h−8, −4i. Since we want the function to have no change,
h−1, 2i
h1, −2i
or √ .
Du f = ∇ f · u = 0. Therefore, the unit vector u = √
5
5
Points on page:
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Math 101C
Ch. 12 Test
Page 3 of 4
kw
, where w is the weight measured in pounds
h2
and h is the height measured in inches. For a given point (w0 , h0 ), what ratio of the change in the weight
to the change in height will result in the maximum change in the BMI? Hint: think u = h∆w, ∆hi.
8. (10 points) The body mass index (BMI) is given by B =
Solution: To get the maximum change in the BMI means that we want t find the maximum change in
the directional
derivative
of the BMI. This means the the unit vector u must have the same direction
∆w
−h
k −2k
k/h2
,
.
Thus
the
ratio
=
as ∇ B =
=
.
2
3
3
∆h
2w
h
h
−2kw/h
9. Sketch the graphs of each of the following. Note
• All the graphs are in three dimensions.
• Be sure to pay attention to the given domain.
• Be sure to show the orientation (direction) of the curve.
(a) (5 points) r(t) = ht cos t, t sin t, 4π − ti , 0 ≤ t ≤ 4π
Solution: Two revolutions of a spiral, starting at (0, 0, 4π ) and ending on the xy plane at (4π, 0, 0)
(b) (5 points) z2 = x2 + y2
Solution: A double cone with opening both upward and downward from the origin.
p
(c) (5 points) z = 5 + 4 − x2 − y2
Solution: The top half of a sphere of radius 2, with center at (0, 0, 5).
10. (10 points) The level curves of two functions
f ( x, y) =
1
p
1 + x 2 + y2
g( x, y) = p
and
1
x2
+ y2
are shown in the picture. Identify which graph goes with which function. Be sure to explain how you
made the choice.
1.0
1.0
0.5
0.5
0.0
0.0
-0.5
-0.5
-1.0
-1.0
-0.5
0.0
0.5
1.0
-1.0
-1.0
-0.5
0.0
0.5
1.0
Solution: The graph of g( x, y) is shown on the right. The reason is as follows. Since g( x, y) becomes
undefined as ( x, y) −→ (0, 0), the graph becomes increasingly steep. Therefore its level curves must
be getting closer together, as shown in the drawing on the right.
11. (10 points) For each of the given graphs, match the appropriate equations from the list. Note the following
• Some of the equations may not be used.
• All graphs are drawn in standard position with the z axis vertical.
Points on page:
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Math 101C
Ch. 12 Test
z = sin( x )
Page 4 of 4
z = ( y − 3)2
1. z = sin( x + y)
2. z = sin( xy)
3. z = sin( x )
4. z = sin(y)
5. z = x2
6. z = ( x − 3)2
7. z = y2
8. z = (y − 3)2
z = sin( x + y)
9. z = x2 + y2
10. z = ( x − 3)2 + (y − 3)2
Points on page:
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