Find the Taylor series for the given function.
137)
f(x) = ln(1 +8x)
137)
A)
8x +82
2x2–83
3x3+84
4x4+ . . . +(–1)n8n+1
n + 1 xn+1+ . . .
B)
–8x –82
2! x2+83
3! x3–84
4! x4+ . . . +(–1)n8n
(n + 1)! xn+ . . .
C)
8x –82
2x2+83
3x3–84
4x4+ . . . +(–1)n8n+1
n + 1 xn+1+ . . .
D)
8x –82
2! x2+83
3! x3–84
4! x4+ . . . +(–1)n8n+1
(n + 1)! xn+1+ . . .
Determine whether the given series converges or diverges, and find the sum if it converges.
138)
6+3.6 +2.16 +1.296 + …
138)
A)
Converges; sum =22.5
B)
Converges; sum =15
C)
Converges; sum =9
D)
Diverges
Solve the problem.
139)
A ball is dropped from a height of 6 meters and returns to about 7/8 of its previous height on each
bounce. About how far will the ball travel before it comes to rest?
139)
A)
90 m
B)
174 m
C)
132 m
D)
48 m
140)
Determine the sum of the series
n = 1
3n
4n – 1 .
140)
A)
12
B)
1
4
C)
3
4
D)
3
141)
Let f(x) = ln(1 + x). Determine the third Taylor polynomial of f(x) at x = 0.
141)
A)
x +1
2x2+x3
B)
x –1
2x2+1
3x3
C)
x +1
2x2+1
3x3
D)
x +x2+x3
Find the Taylor series for the given function.
142)
f(x) =x5e–x
142)
A)
x5–x6+1
2x7–1
6x8+ . . . +(–1)n
n! x5+n+ . . .
B)
1 –x5+x6–1
2x7+1
6x8– . . . +(–1)n
n! x5+n+ . . .
C)
1 +x5–x6+1
2x7–1
6x8+ . . . +(–1)n
n! x5+n+ . . .
D)
x5+x6–1
2x7+1
6x8+ . . . +(–1)2n
n! x5+n+ . . .
Determine whether the given series converges or diverges, and find the sum if it converges.
143)
1
15 +4
15 +16
15 +64
15 + …
143)
A)
Converges; sum =4.27
B)
Converges; sum =5.67
C)
Converges; sum =6.10
D)
Diverges
Determine the third Taylor polynomial of the function at x = 0.
144)
f(x) =e4x
144)
A)
1 + 4x + 8x2+32
9x3
B)
1 +16x + 128x2+1024
3x3
C)
1 + 4x + 8x2+32
3x3
D)
1 + 4x + 8x2+64
3x3
145)
Find the Taylor series at x = 0 of the function f(x) =1
1 – 3x by computing three or four derivatives
and using the definition of the Taylor series.
145)
A)
1 +3x
1x +9x
2x2+ …
B)
1 – 3x + 9x2– 27x3+ …
C)
1 + 3x + 9x2+ 27x3+ …
D)
1 +(3x)2+(3x)4+(3x)6+ …
E)
none of these
Solve the problem.
146)
Suppose that an investment of $1000 yields returns of $258.48, $348.48, and $500.00 at the end of
the first, second, and third months, respectively. Determine the internal rate of return on this
investment.
146)
A)
4.95%
B)
4.41%
C)
4.72%
D)
5.72%
Determine the third Taylor polynomial of the function at x = 0.
147)
f(x) =1
x +7
147)
A)
1
7x –1
49 x2+1
343x3–1
2401 x4
B)
1
7x +1
49 x2+1
343x3+1
2401 x4
C)
1
7+1
49 x +1
343x2+1
2401 x3
D)
1
7–1
49 x +1
343x2–1
2401 x3
148)
Find the Taylor series at x = 0 of the function f(x) = x ln(1 + 2x) by computing three or four
derivatives and using the definition of the Taylor series.
148)
A)
2x2–4x3
2
B)
2x2+4x3
2!
C)
2x –4x2
2! +8x3
3!
D)
2x2–4x3
3
E)
none of these
149)
Suppose f(x) =x4– 7x3+ 2. The third Taylor polynomial of f(x) at x = 0 is p3(x) = 2 – 7x3.
149)
A)
True
B)
False
Determine the third Taylor polynomial of the function at x = a.
150)
f(x) =x2+ x + 1, a =5
150)
A)
6+11(x –5) +16(x –5)2
B)
31 +11(x –5) +11(x –5)2+31(x –5)3
C)
31 +11(x –5) +(x –5)2
D)
1 + 3(x –5) + 3(x –5)2+(x –5)3
37
151)
The geometric series 1 +(0.2)3+(0.2)6+(0.2)9+ …
(I) converges
(II) is equal to
k = 0
1
125
k
(III) is equal to
k = 0
(0.2)3k
151)
A)
III only
B)
I, II, and III
C)
I and II
D)
II and III
E)
I and III
152)
Let f(x) = ln x. Find the second Taylor polynomial of f(x) at x = 2.
152)
A)
1 +1
2(x – 2) +1
8(x – 2)2
B)
ln 2 +1
2(x – 2) –1
8(x – 2)2
C)
ln 2 –1
2(x – 2) –1
8(x – 2)2
D)
1 +1
2(x – 2) –1
8(x – 2)2
B
153)
Determine the sum of the series
n = 1
e
n.
153)
A)
e
B)
e
e –
C)
e –
D)
none of these
C
B
Determine the third Taylor polynomial of the function at x = 0.
154)
f(x) =x +25
154)
A)
5–1
10 x +1
1000 x2–1
50,000 x3
B)
5+1
10 x –1
1000 x2+1
25,000 x3
C)
5+1
10 x –1
1000 x2+1
50,000 x3
D)
5–1
10 x +1
1000 x2–1
25,000 x3
155)
The Bessel function of order 1 has the Taylor Series at x = 0 given by f(x) =x
2–x3
16 +x5
384 – …, find
f(5)(0).
155)
A)
5
16
B)
5
384
C)
1
384
D)
none of these
Solve the problem.
156)
After being struck with a hammer, a gong vibrates 50 vibrations in the first second and in each
second thereafter makes 5
6 as many vibrations as in the previous second. Find how many
vibrations the gong makes before it stops vibrating.
156)
A)
60 vibrations
B)
310 vibrations
C)
55 vibrations
D)
300 vibrations
Use the Newton–Raphson algorithm to approximate the given root to the nearest thousandth.
157)
312
157)
A)
2.295
B)
2.301
C)
2.289
D)
2.292
158)
Suppose f(x) =x4– 7x3+ 2. The fifth Taylor polynomial of f(x) at x = 1 is p5(x) =x5+x4– 7x3+ 2.
158)
A)
True
B)
False
39
Use the Integral Test to determine whether the series converges.
159)
n=1
8n–3/2
159)
A)
Diverges
B)
Converges
Solve the problem.
160)
England decides to decrease its taxes by £7 billion. It is estimated that of each pound received, a
typical citizen will spend 90%. The level of economic activity generated by the tax cut is therefore
estimated to be: 7·(0.90) + 7·(0.90)2+ 7·(0.90)3+ … billion pounds. This amount is equal to what?
160)
A)
£70 billion
B)
£63 billion
C)
£3.6 billion
D)
£45 billion
E)
none of these
B
161)
Determine the sum of the series
n = 0
1 –2n
3n.
161)
A)
2
3
B)
3
C)
–3
2
D)
none of these
C
Use the Integral Test to determine whether the series converges.
162)
n=1
12
n
162)
A)
Diverges
B)
Converges
A
B
Find the nth Taylor polynomial for the function at x = 0, and use it to estimate the value of the function at the given value
of x. Round to seven decimal places.
163)
f(x) =ex, n = 4, estimate e0.42
163)
A)
0.9877605
B)
0.6805480
C)
1.5205480
D)
0.9130933
Determine the third Taylor polynomial of the function at x = 0.
164)
f(x) =e–2x
164)
A)
1 – 4x + 8x2–16
3x3
B)
1 + 2x + 2x2+4
3x3
C)
1 – 2x + 2x2–8
3x3
D)
1 – 2x + 2x2–4
3x3
Solve the problem.
165)
The infinite series a1+a2+a3+ … has partial sums given by Sn=10 –1
n. Find
10
k–1
ak.
165)
A)
10.1
B)
99
C)
9.99
D)
9.9
Determine the third Taylor polynomial of the function at x = 0.
166)
f(x) =1
9– x
166)
A)
1
9–1
81 x +1
729x2–1
6561 x3
B)
1
9+1
81 x +1
729x2+1
6561 x3
C)
1
9x +1
81 x2+1
729x3+1
6561 x4
D)
1
9x –1
81 x2+1
729x3–1
6561 x4
167)
The Bessel function f(x) of order zero has the Taylor series at x = 0 given by
f(x) = 1 –x2
4+x4
64 –x6
2304 + … . What is f(4)(0) ?
167)
A)
1
4
B)
3
8
C)
1
64
D)
none of these
Determine whether the given series converges or diverges, and find the sum if it converges.
168)
4+4
7+4
49 +4
343 + …
168)
A)
Converges; sum =5.25
B)
Converges; sum =4.6667
C)
Converges; sum =7
D)
Diverges
42
Answer Key
Testname: C11
43
Answer Key
Testname: C11
Answer Key
Testname: C11
Answer Key
Testname: C11