Practice Problems #8 (7092662)
Current Score:
Question
Points
1.
0/59
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
0/1 0/1 0/1 0/1 0/2 0/2 0/3 0/3 0/6 0/2 0/13 0/1 0/5 0/6 0/2 0/7 0/3
0/1 points
Total
0/59
SCalcET7 14.2.037. [2004965]
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SCalcET7 14.2.501.XP. [1889082]
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SCalcET7 14.2.503.XP. [1889071]
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SCalcET7 14.2.504.XP. [1898639]
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Determine the set of points at which the function is continuous.
{(x, y) | x
}
and y
{(x, y) | (x, y) ≠ (0, 0)}
{(x, y) | x
and y ≠ 0}
{(x, y) | x · y ≠ 0}
{(x, y) | x > 0 and y > 0}
2.
0/1 points
Find the limit, if it exists. (If an answer does not exist, enter DNE.)
lim
y4
(x, y) → (0, 0) x4 + 2y4
3.
0/1 points
Find the limit, if it exists. (If an answer does not exist, enter DNE.)
lim
xy cos y
(x, y) → (0, 0) 5x2 + y2
4.
0/1 points
Find the limit, if it exists. (If an answer does not exist, enter DNE.)
lim
18x3y
(x, y) → (0, 0) 5x4 + y4
5.
0/2 points
SCalcET7 14.3.005. [1888968]
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SCalcET7 14.3.017. [1898016]
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Determine the signs of the partial derivatives for the function f whose graph is shown below.
(a)
fx(x0, y0)
positive
negative
(b)
fy(x0, y0)
positive
negative
6.
0/2 points
Find the first partial derivatives of the function.
f(x, t) = e−8t cos πx
fx(x, t) =
ft(x, t) =
7.
0/3 points
SCalcET7 14.3.032. [1898020]
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SCalcET7 14.3.033. [1898031]
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SCalcET7 14.3.041.MI.SA. [1724196]
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Find the first partial derivatives of the function.
f(x, y, z) = 4x sin(y − z)
fx(x, y, z) =
fy(x, y, z) =
fz(x, y, z) =
8.
0/3 points
Find the first partial derivatives of the function.
w = ln(x + 8y + 4z)
∂w =
∂x
∂w =
∂y
∂w =
∂z
9.
0/6 points
This question has several parts that must be completed sequentially. If you skip a part of the question, you will not receive
any points for the skipped part, and you will not be able to come back to the skipped part.
Tutorial Exercise
Find the indicated partial derivatives.
10.
0/2 points
SCalcET7 14.3.049. [1857237]
Use implicit differentiation to find ∂z/∂x and ∂z/∂y.
e3z = xyz
∂z
=
∂x
∂z
=
∂y
-
11.
0/13 points
SCalcET7 14.3.056.MI.SA. [1724053]
-
This question has several parts that must be completed sequentially. If you skip a part of the question, you will not receive
any points for the skipped part, and you will not be able to come back to the skipped part.
Tutorial Exercise
Find all the second partial derivatives.
12.
0/1 points
SCalcET7 14.4.003. [1905048]
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SCalcET7 14.4.015. [1853571]
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Find an equation of the tangent plane to the given surface at the specified point.
z=
13.
xy ,
(2, 2, 2)
0/5 points
Explain why the function is differentiable at the given point.
f(x, y) = 9 + e−xycos y,
(π, 0)
The partial derivatives are fx(x, y) =
fx(π, 0) =
and fy(x, y) =
and fy(π, 0) =
, so
. Both fx and fy are continuous functions, so f is differentiable at
(π, 0).
Find the linearization L(x, y) of f(x, y) at (π, 0).
L(x, y) =
14.
0/6 points
SCalcET7 14.4.018. [1905011]
Verify the linear approximation at (2π, 0).
f(x, y) =
Left f(x, y) =
y + cos2x ≈ 1 +
1
y
2
y + cos2x . Then fx(x, y) =
continuous functions for y >
fx(2π, 0) =
and fy(2π, 0) =
and fy(x, y) =
. Both fx and fy are
, so f is differentiable at (2π, 0) by this theorem. We have
, so the linear approximation of f at (2π, 0) is
f(x, y) ≈ f(2π, 0) + fx(2π, 0)(x − 2π) + fy(2π, 0)(y − 0) =
.
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15.
0/2 points
SCalcET7 14.5.016. [1905019]
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Suppose f is a differentiable function of x and y, and g(r, s) = f(8r − s, s2 − 2r). Use the table of values below to calculate
gr(1, 8) and gs(1, 8).
f
g
fx
fy
(0, 62)
1
4
5
2
(1, 8)
4
1
8
7
gr(1, 8) =
gs(1, 8) =
16.
0/7 points
SCalcET7 14.5.035.MI.SA. [1724042]
-
This question has several parts that must be completed sequentially. If you skip a part of the question, you will not receive
any points for the skipped part, and you will not be able to come back to the skipped part.
Tutorial Exercise
The temperature at a point (x, y) is T(x, y), measured in degrees Celsius. A bug crawls so that its position after t
seconds is given by the following function, where x and y are measured in centimeters.
The temperature function satisfies Tx(5, 10) = 9 and Ty(5, 10) = 8. How fast is the temperature rising on the
bug's path after 2 seconds?
17.
0/3 points
SCalcET7 14.5.501.XP. [1905050]
Use the Chain Rule to find the indicated partial derivatives.
z = x4 + xy3,
∂z , ∂z , ∂z
∂u ∂v ∂w
∂z
=
∂u
∂z
=
∂v
∂z
=
∂w
Assignment Details
x = uv4 + w3,
y = u + vew
when u = 1, v = 1, w = 0
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