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Practice Problems #9 (7143494)
Current Score:
0/71
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0/3 0/11 0/2 0/16 0/3 0/3 0/1 0/3 0/5 0/1 0/1 0/2 0/2 0/15 0/1 0/2
Question
Points
1.
Total
0/71
0/3 points
SCalcET7 14.6.008. [1921400]
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SCalcET7 11.11.003.MI.SA. [1724072]
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Consider the following equation.
f(x, y) = y3/x,
P(1, 3),
u=
1
2i +
3
5j
(a) Find the gradient of f.
∇f(x, y) =
(b) Evaluate the gradient at the point P.
∇f(1, 3) =
(c) Find the rate of change of f at P in the direction of the vector u.
Duf(1, 3) =
2.
0/11 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 Taylor polynomial Tn(x) for the function f at the number a. Graph f and T3 on the same paper.
3.
0/2 points
SCalcET7 11.11.005. [1760721]
Find the Taylor polynomial T3(x) for the function f centered at the number a.
f(x) = cos x,
a = π/2
T3(x) =
Graph f and T3 on the same screen.
-
4.
0/16 points
SCalcET7 11.11.019.MI.SA. [1724113]
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.
Consider the following function.
Exercise (a)
Approximate f by a Taylor polynomial with degree n at the number a.
Exercise (b)
Use Taylor's Inequality to estimate the accuracy of the approximation f ≈ Tn(x) when x lies in the given interval.
Exercise (c)
Check your result in part (b) by graphing |Rn(x)|.
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5.
0/3 points
SCalcET7 11.11.017. [1655286]
Consider the following function.
f(x) = sec x,
a = 0,
n = 2,
−0.3 ≤ x ≤ 0.3
(a) Approximate f by a Taylor polynomial with degree n at the number a.
T2(x) =
(b) Use Taylor's Inequality to estimate the accuracy of the approximation f(x) ≈ Tn(x) when x lies in the given
interval. (Round your answer to six decimal places.)
|R2(x)| ≤
(c) Check your result in part (b) by graphing |Rn(x)|.
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6.
0/3 points
SCalcET7 11.11.021. [1654423]
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Consider the following function.
f(x) = x sin x,
a = 0,
n = 4,
−0.5 ≤ x ≤ 0.5
(a) Approximate f by a Taylor polynomial with degree n at the number a.
T4(x) =
(b) Use Taylor's Inequality to estimate the accuracy of the approximation f(x) ≈ Tn(x) when x lies in the given
interval. (Round your answer to four decimal places.)
|R4(x)| ≤
(c) Check your result in part (b) by graphing |Rn(x)|.
7.
0/1 points
SCalcET7 11.11.025. [1655122]
Use Taylor's Inequality to determine the number of terms of the Maclaurin series for ex that should be used to estimate e0.1 to
within 0.000001.
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8.
0/3 points
What can you say about the series
(a)
an+1
an
n→∞
lim
SCalcET7 11.6.001. [1853438]
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SCalcET7 11.6.008.MI.SA. [1724107]
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an in each of the following cases?
=8
absolutely convergent
conditionally convergent
divergent
cannot be determined
(b)
an+1
an
n→∞
lim
= 0.9
absolutely convergent
conditionally convergent
divergent
cannot be determined
(c)
an+1
an
n→∞
lim
=1
absolutely convergent
conditionally convergent
divergent
cannot be determined
9.
0/5 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
Determine whether the series is absolutely convergent, conditionally convergent, or divergent.
∞
n=1
n!
101n
10.
0/1 points
SCalcET7 11.6.012. [1759119]
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SCalcET7 11.6.013. [1759176]
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SCalcET7 11.10.004. [1655006]
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Determine whether the series is absolutely convergent, conditionally convergent, or divergent.
∞
sin 6n
6n
n=1
absolutely convergent
conditionally convergent
divergent
11.
0/1 points
Determine whether the series is absolutely convergent, conditionally convergent, or divergent.
∞
14n
n=1
(n + 1)52n + 1
absolutely convergent
conditionally convergent
divergent
12.
0/2 points
Find the Taylor series for f centered at 9 if
f(n) (9) =
∞
n=0
∞
n=0
∞
n=0
∞
n=0
∞
n=0
(−1)nn!
8n(n + 6)
.
(x − 9)n
8n(n + 6)
(−1)n(x − 9)n
8n(n + 6)n!
(−1)n(x − 9)n
8n(n + 6)
(−1)n(n + 6)(x − 9)n
8nn!
(−1)nxn
8n(n + 6)
What is the radius of convergence R of the Taylor series?
R=
13.
0/2 points
SCalcET7 11.10.007. [1654266]
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Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do
not show that Rn(x) → 0.]
f(x) = sin
π x
3
∞
f(x) =
n=0
Find the associated radius of convergence R.
R=
14.
0/15 points
SCalcET7 11.10.019.MI.SA. [1724006]
-
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 Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Do
not show that R(x) → 0.]
f(x) = 5 cos x, a = 13π
15.
0/1 points
SCalcET7 14.6.015. [1905025]
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SCalcET7 14.6.021. [1853539]
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Find the directional derivative of the function at the given point in the direction of the vector v.
f(x, y, z) = xey + yez + zex,
(0, 0, 0),
v = 6, 1, −3
Duf(0, 0, 0) =
16.
0/2 points
Find the maximum rate of change of f at the given point and the direction in which it occurs.
f(x, y) = 8y
x,
maximum rate of change
direction vector
Assignment Details
(16, 7)