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Duplicate of Multivariable Calculus Partial Derivatives Test Review
Duplicate of Multivariable Calculus Partial Derivatives Test Review
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Answers
1
If
f
is a function, then lim
(x,y)→(2,5)
f(x,y)=f(2,5).
TRUE
FALSE
2
If f
xx
f
yy
- (f
xy
)
2
< 0, this means . . .
No solution
saddle point
its time to move on.
there is a rel. extremum
3
The directional derivative of z = (x^2)(e^-y) at the point (-1,0) in the direction of (4,-3) is
-11/5
-5
-1
-3/5
4
f(x,y)=x
3
+ x
2
y
3
- 2y
2
. Find f
x
(2,1) and f
y
(2,1).
12 and -8
12 and 8
16 and 8
8 and 16
5
A gradient is a(n) _____
integral
vector
number
scalar
6
Find the equation of the tangent plane to f(x,y) = e
x .
cos y at the point (0,0,1).
z = 2x + y
z = y + 1
z = x + y + 1
z = x + 1
7
D
k
f(x,y,z) = f
z
(x,y,z).
TRUE
FALSE
8
Find the loc maxes and min values and saddle points of the function f(x,y)=3xy - x
2
y - xy
2
.
loc max of 1 and saddle pts. at (0,0), (0,3), and (3,0).
loc min of 1 and saddle points at (0,0), (0,3), and (3,0).
loc. max of 1 and loc mins of 0.
loc. max of 0 and loc. min of 1.
9
If
f
(x,y) = sin x + sin y, then −√2 ≤ D
u
f(x,y) ≤ √2.
TRUE
FALSE
10
A Pringle
TM
is an example of a(n) _____
all of the above
hyperbolic paraboloid
a surface with a saddlepoint
potato chip
11
The "Lagrange multiplier" method is
always equal to π
a greek letter
a way to solve constraint problems
a way to integrate with 3 variables
12
Given f(x, y, z) = x
30
y
26
z, what is f
xyzyzzyyxx
?
85700x27y22
4320x27y22
0
2360x27y22
13
If
f
has a local minimum at (a, b) and
f
is differentiable at (a, b), then Grad(f)(a, b) =
0
.
FALSE
TRUE
14
A gradient at a point will behave
how
relative to a level curve?
perpendicular
parallel
any direction
tangent
15
A gradient points in what direction?
up
tangent to the curve
maximum rate of change
toward the center
16
∇f(x, y), where f(x, y) = 10x
5
y is
<5x6/3, y2/2>
<50x4y, 10x5>
<10x5, y>
<50x4, 10x5>
17
Clairaut's Theorem states that _____
fxy = fyx
fx = fy
fxy = fxx
fxx = fyy
18
Given f(x,y) below and x = sin(2t) and y = cos t, find dz/dt when t = 0.
-6
6
3
0
19
Evaluate the limit:
2
Does not exist
infinity
0/0
20
If f(x,y) has two local maxima, then f must have a local minimum.
FALSE
TRUE
21
If
f
(x,y) = ln y, then Grad(f)(x,y) = 1/y.
TRUE
FALSE
22
Which equation below gives the graph of this quadric surface?
4x^2 - y^2 + 2z^2 + 4 = 0
I don't know!
4x^2 - y^2 - z + 8x + 8z + 24 = 0
x^2 + 2z^2 -6x - y + 10 = 0
23
Is fxy at P positive or negative?
positive
negative
0
24
The "D" in the multivariable second derivative test is:
fxxfyy + (fxy)2
fxxfyy - (fxy)2
fxxfyy - fxy
fxxfyy(fxy)2
