## Transcribed Text

1. Find the equations of planes that just touch the sphere (x −
3)
2 + (y − 1)
2 + (z − 2)
2 = 25 and are parallel to
(a) The xy-plane: and
(b) The yz-plane: and
(c) The xz-plane: and
2. Without a calculator or computer, match the func-tions
with their graphs in the figures below. Note that two of the
functions do not have a matching graph.
(a) z = 2+x
2 +y
2
?
(b) z = 2−x
2 −y
2
?
(c) z = 2(x
2 +y
2
) ?
(d) z = 2+x−2y ?
(e) z = −2 ?
(f) z = 2−x ?
1. 2.
3. 4.
3. On a piece of paper, sketch each of the following
surfaces:
(i) z = x
2 +y
2 +4
(ii) 4 = x+y
Use your graphs to fill in the following descriptions of crosssections of the surfaces.
(a) For (i) (z = x
2 +y
2 +4):
Cross sections with x fixed give ?
Cross sections with y fixed give ?
Cross sections with z fixed give ?
(b) For (ii) (4 = x+y):
Cross sections with x fixed give ?
Cross sections with y fixed give ?
Cross sections with z fixed give ?
4. By setting one variable constant, find a plane that
intersects the graph of z = 2y
2 − 6x
2 + 3 in a:
(a) Parabola opening upward: the plane =
(Give your answer by specifying the variable in the first answer
blank and a value for it in the second.)
(b) Parabola opening downward: the plane =
(Give your answer by specifying the variable in the first answer
blank and a value for it in the second.)
(c) Pair of intersecting straight lines: the plane =
(Give your answer by specifying the variable in the first answer
blank and a value for it in the second.)
5. Find an equation for the contour of f(x,y) = 2x
2
y+
6x+10 that goes through the point (4,3).
Equation:
6. Find an equation for the plane containing the line in the
xy-plane where x = 3, and the line in the yz-plane where z =
4.5.
equation:
7. Find a possible equation for the linear function g(x,y)
shown in the graph below:
g(x,y) =
8. Find a formula for a function g(x,y,z) whose level
surfaces are planes parallel to the plane z = 8x+4y−3.
g(x,y,z) =
9. Find the limit of the function
f(x,y) = sin(5
p
x
2 +y
2)
5
p
x
2 +y
2
as (x,y) → (0,0). Assume that polynomials, exponentials, logarithmic, and trigonometric functions are continuous. [Hint:
limt→0
sint
t = 1.]
lim
(x,y)→(0,0)
sin(5
√
x
2+y
2)
5
√
x
2+y
2
=
(Enter DNE if the limit does not exist.)
10. In this problem we show that the function
f(x,y) = x
2 −y
2
x
2 +y
2
does not have a limit as (x,y) → (0,0).
(a) Suppose that we consider (x,y) → (0,0) along the curve
y = 4x. Find the limit in this case:
lim
(x,4x)→(0,0)
x
2−y
2
x
2+y
2 =
1
(b) Now consider (x,y) → (0,0) along the curve y = 5x. Find
the limit in this case:
lim
(x,5x)→(0,0)
x
2−y
2
x
2+y
2 =
(c) Note that the results from (a) and (b) indicate that f has
no limit as (x,y) → (0,0) (be sure you can explain why!) .
To show this more generally, consider (x,y) → (0,0) along the
curve y = mx, for arbitrary m. Find the limit in this case:
lim
(x,mx)→(0,0)
x
2−y
2
x
2+y
2 =
(Be sure that you can explain how this result also indicates that
f has no limit as (x,y) → (0,0).
11. Show that the function
f(x,y) = x
2
y
x
4 +y
2
.
does not have a limit at (0,0) by examining the following limits.
(a) Find the limit of f as (x,y) → (0,0) along the line y = x.
lim
(x,y)→(0,0)
y=x
f(x,y) =
(b) Find the limit of f as (x,y) → (0,0) along the line y = x
2
.
lim
(x,y)→(0,0)
y=x
2
f(x,y) =
(Be sure that you are able to explain why the results in (a)
and (b) indicate that f does not have a limit at (0,0)!
12. What value of c makes the following function continuous at (0,0)?
f(x,y) = x
2 +y
4 +4,(x,y) =6 (0,0)c,(x,y) = (0,0)
c =
13. Find the value(s) of a making~v = 7a~i −3~j parallel to ~w
= a
2~i + 6~j.
a =
(If there is more than one value of a, enter the values as a
comma-separated list.)
14. (a) Find a unit vector from the point P = (2,2) and
toward the point Q = (6,5).
~u =
(b) Find a vector of length 15 pointing in the same direction.
~v =
15. Find all vectors ~v in 2 dimensions having ||~v|| = 5
where the ˜i-component of~v is 3˜i.
vectors:
(If you find more than one vector, enter them in a commaseparated list.)
16. A plane is heading due west and climbing at the rate of
100 km/hr. If its airspeed is 500 km/hr and there is a wind
blowing 90 km/hr to the northwest, what is the ground speed of
the plane?
ground speed =
17. Let ~a,~b,~c and~y be the three dimensional vectors
~a = 4˜j+4 ˜k, ~b = 5˜i+6˜j+5 ˜k, ~c = 4˜i+5˜j, ~y = −5˜i−3˜j
Perform the following operations on these vectors:
(a)~c ·~a+~a ·~y =
(b) (~a ·~b)~a =
(c) ((~c ·~c)~a)·~a =
18. The force on an object is ~F = −25˜j. For the vector ~v = ˜i
+ 5˜j, find:
(a) The component of ~F parallel to~v:
(b) The component of ~F perpendicular to~v:
The work, W, done by force ~F through displacement ~v:
˜
19. Compute the angle between the vectors ˜i + ˜j + and −˜i k
− ˜j − ˜k.
angle = radians
(Give your answer in radians, not degrees.)
20. Find the equation of a plane that is perpendicular to the vector −2˜i + ˜j − 2 ˜k and passing through the point
(−1,5,−1).
The plane is given by:
21. Use the algebraic definition to find ~v × ~w if
~v =˜i−˜j+3˜k and ~w = −3˜i−3˜j−˜k.
~v×~w =
22. Find an equation for the plane through the points
(5,3,5),(−1,2,0),(−1,0,−2).
The plane is
23. Consider the planes given by the equations
x+y−4z = 3,
4x+y+z = 8.
(a) Find a vector ~v parallel to the line of intersection of the
planes.
~v =
(b) Find the equation of a plane through the origin which is
perpendicular to the line of intersection of these two planes.
This plane is
24. Find a vector ~v parallel to the intersection of the planes
4z − (3x + 5y) = 4 and 3x + 2y − z = 7.
~v =
25. Let P = (0,0,1),Q = (1,−1,2),R = (−2,1,1). Find
(a) The area of the triangle PQR.
area =
(b) The equation for a plane that contains P, Q, and R. This
plane is
26. Suppose ~v · ~w = 7 and ||~v×~w|| = 4, and the angle
between~v and ~w is θ. Find
(a) tanθ =
(b) θ =
2
27. Find the partial derivative indicated. Assume the
variables are restricted to a domain on which the function is defined.
z =
2x
2
y
6 −y
6
12xy−5
.
zy =
28. Find the partial derivative indicated. Assume the
variables are restricted to a domain on which the function is defined.
∂
∂x
(x
3
e
√
5xy) =
z = sin
4x
5
y−4xy3
29. Find the partial derivative indicated. Assume the
variables are restricted to a domain on which the function is defined.
.
zx =
30. Find the partial derivatives indicated Assume the
variables are restricted to a domain on which the function is defined.
z = x
5 +3
y +x
y
.
zx =
zy =
31. Find the equation of the tangent plane to the surface
determined by
x
3
y
4 +z−20 = 0
at x = 2, y = 3.
z =
32.
Find the equation of the sphere centered at (10,10,8) with
radius 10.
= 0.
Give an equation which describes the intersection of this
sphere with the plane z = 9.
= 0.
33.
Find the area of the parallelogram with vertices:
P(0,0,0), Q(2,-1,4), R(2,-2,5), S(4,-3,9).
34.
Find the distance the point P(3, -2, -4), is to the plane through
the three points
Q(-1, 0, -2), R(1, -1, -5), and S(1, -4, -5).
35.
Find the distance from the point (-1, 5, -4) to the plane −4x +
2y+2z = 6.
36. Find the limits, if they exist, or type DNE for any which
do not exist.
lim
(x,y)→(0,0)
2x
2
3x
2 +4y
2
1) Along the x-axis:
2) Along the y-axis:
3) Along the line y = mx :
4) The limit is:
37.
Find the first partial derivatives o f f(x,y,z) = z arctan(
y
x
) at
the point (2, 2, -2).
A. ∂ f
∂x
(2,2,−2) =
B. ∂ f
∂y
(2,2,−2) =
C. ∂ f
∂z
(2,2,−2) =
38.
Find the partial derivatives of the function
f(x,y) = −4x+2y
2x+1y
fx(x,y) =
fy(x,y) =
39.
Find the partial derivatives of the function
w =
p
1r
2 +1s
2 +2t
2
∂w
∂r =
∂w
∂s =
∂w
∂t =
40.
p
Find the linearization of the function f(x,y) =
54−1x
2 −4y
2 at the point (3, 3).
L(x,y) =
Use the linear approximation to estimate the value of f(2.9,3.1)
=
3