Find the Taylor series for the given function. Give the interval of convergence.
83)
f(x) =4x2
3– x
83)
A)
4
3x +4
9x2+4
27 x3+ . . . +4
3nxn+ . . . ; (–3, 3)
B)
4
3x +4
9x2+4
27 x3+ . . . +4
3nxn+ . . . ; (–4, 4)
C)
4
3x2+4
9x3+4
27 x4+ . . . +4
3n+1xn+2+ . . . ; (–3, 3)
D)
4
3x2+4
9x3+4
27 x4+ . . . +4
3nxn+1+ . . . ; (–4, 4)
The nth term of a sequence is given. Calculate the fifth partial sum.
84)
an=1
n +9
84)
A)
8633
21,840
B)
25,381
60,060
C)
81,841
240,240
D)
44,201
120,120
Find the indicated term of the geometric sequence.
85)
a2= – 9, r = – 3; Find a5.
85)
A)
27
B)
–81
C)
–243
D)
243
Find an for the given geometric sequence.
86)
4, –4
5, 4
25 , –4
125, 4
625, . . .
86)
A)
an=4–1
25
n – 1
B)
an=4–1
5
n
C)
an=41
5
n – 1
D)
an=4–1
5
n – 1
The nth term of a sequence is given. Calculate the fifth partial sum.
87)
an=(–1)n(n+1)/2
n +3
87)
A)
–2027
6720
B)
–139
480
C)
–373
1344
D)
–223
840
Use the appropriate Taylor polynomial of degree 3 at x = 0 to approximate the quantity. Round the answer to four
decimal places.
88)
38.5
88)
A)
2.0408
B)
2.0816
C)
1.0408
D)
2.0400
Solve the problem.
89)
Assuming two parents, four grandparents, etc., what is the total number of ancestors a person has
going back 10 generations?
89)
A)
2046 ancestors
B)
512 ancestors
C)
1024 ancestors
D)
1022 ancestors
Find the common ratio for the geometric sequence.
90)
3, –9, 27, –81, 243, . . .
90)
A)
r = – 9
B)
r = – 3
C)
r =(–3)n–1
D)
r =3
Solve the problem.
91)
If P dollars are invested at an annual interest rate of r compounded n times a year, then the
accumulated amount after t years is given by
A = P 1 +r
n
nt.
Find the Taylor polynomial of degree 2 at r = 0 for
f(r) = P 1 +r
n
nt.
91)
A)
P + Ptr +nt – 1
2n Ptr2
B)
1 + Ptr –nt – 1
2n Ptr2
C)
P +1
2Ptr +nt – 1
6n Ptr2
D)
P +1
2Ptr +nt – 1
3n Ptr2
Find the Taylor series for the given function. Give the interval of convergence.
92)
f(x) =x5e–x
92)
A)
x5–x6+1
2x7–1
6x8+ . . . +(–1)n
n! x5+n+ . . . ; –1
5, 1
5
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+ . . . ; –1
5, 1
5
D)
x5–x6+1
2x7–1
6x8+ . . . +(–1)n
n! x5+n+ . . . ; (–, )
Find the indicated term of the geometric sequence.
93)
a1=960, r =1
4; Find a4.
93)
A)
a4= – 1
15
B)
a4=15
C)
a4= – 15
D)
a4=1
15
Use Newton’s method to find a solution for the equation in the given interval. Round your answer to the nearest
hundredth.
94)
2x4– 3x2– 7x + 1 = 0; [0, 1]
94)
A)
0.13
B)
0.15
C)
0.14
D)
0.12
95)
2x3–x2+ 5x + 6 = 0; [–1, 0]
95)
A)
–0.84
B)
–0.82
C)
–0.81
D)
–0.83
Identify which geometric series converge. Give the sum of a convergent series.
96)
52 +26 +13 +13
2+ . . .
96)
A)
Converges to 26
B)
Converges to 130
C)
Converges to 104
D)
Diverges
24
Use the formula for the sum of the first n terms of a geometric sequence to evaluate the sum.
97)
4
i=0
21
4
i
97)
A)
341
128
B)
1
512
C)
341
256
D)
162
Find the Taylor polynomial of degree 3 at 0.
98)
f(x) = ln(1 – 4x)
98)
A)
–4x – 8x2–32
3x3
B)
–4x – 16x2– 64x3
C)
–4x + 8x2–64
3x3
D)
–4x – 8x2–64
3x3
Find the common ratio for the geometric sequence.
99)
–7, –14, –28, –56, –112, . . .
99)
A)
r =2n
B)
r = – 2
C)
r =2
D)
r = – 14
Use l’Hospital’s rule, if applicable, to find the limit.
100)
lim
x
0
1 +x
8–(1 + x)1/8
x2
100)
A)
7
128
B)
7
256
C)
7
192
D)
Does not exist
Solve the problem.
101)
A ball is dropped from a height of 9 meters and returns to about 6/7 of its previous height on each
bounce. About how far will the ball travel before it comes to rest?
101)
A)
171 m
B)
225 m
C)
117 m
D)
63 m
Find the common ratio for the geometric sequence.
102)
4
3, 16
3, 64
3, 256
3, 1024
3, …
102)
A)
r =4
B)
r =4
3
C)
r =1
3
D)
r =1
4
Use l’Hospital’s rule, if applicable, to find the limit.
103)
lim
x
100
x–10
x –100
103)
A)
1
10
B)
1
20
C)
0
D)
Does not exist
Find the present value of the ordinary annuity. Round to the nearest cent.
104)
Payments of $900 are made semiannually for 16 years at 4% compounded semiannually.
104)
A)
$16,086.20
B)
$32,329.27
C)
$21,121.50
D)
$12,219.94
Find the Taylor polynomial of degree 3 at 0.
105)
f(x) =(4 – x)3/2
105)
A)
8+3x –3
16 x2+1
64 x3
B)
8–3x +3
16 x2+1
64 x3
C)
8–3x +3
16 x2+1
128x3
D)
8+3x –3
16 x2+1
128x3
Use Newton’s method to find a solution for the equation in the given interval. Round your answer to the nearest
hundredth.
106)
x1/3 – x + 8; [10, 11]
106)
A)
10.19
B)
10.21
C)
10.18
D)
10.17
Find the periodic payment that will amount to the given sum under the given conditions. Round to the nearest cent.
107)
S =$40,000; interest is 12% compounded quarterly; payments are made at the end of each quarter
for 5 years.
107)
A)
$1997.64
B)
$1488.63
C)
$1343.27
D)
$2150.66
Use Newton’s method to find the given root to the nearest thousandth.
108)
95
108)
A)
95.000
B)
9.744
C)
9.752
D)
9.747
Find the Taylor series for the given function. Give the interval of convergence.
109)
f(x) =7
1 +8x
109)
A)
7x –56x2+ . . . +(–1)n7·8nxn+1+ . . . ; –1
7,
1
7
B)
7x +56x2+ . . . +(–1)n7·8nxn+1+ . . . ; –1
7,
1
7
C)
7+56x + . . . +(–1)n7·8nxn+ . . . ; –1
8,
1
8
D)
7–56x + . . . +(–1)n7·8nxn+ . . . ; –1
8,
1
8
Find the present value of the ordinary annuity. Round to the nearest cent.
110)
Payments of $9000 are made quarterly for 10 years at 8% compounded quarterly.
110)
A)
$282,812.45
B)
$404,595.35
C)
$178,134.97
D)
$147,162.90
Use l’Hospital’s rule, if applicable, to find the limit.
111)
lim
x
0
ex
16x3–14x2+ 3x
111)
A)
1
B)
1
16
C)
0
D)
Does not exist
The nth term of a sequence is given. Calculate the fifth partial sum.
112)
an=6n + 1
6n + 2
112)
A)
6271
2080
B)
56,047
14,560
C)
80,347
14,560
D)
68,197
14,560
28
Use Newton’s method to find a solution for the equation in the given interval. Round your answer to the nearest
hundredth.
113)
8x2+5x –18 = 0; [1, 2]
113)
A)
1.25
B)
1.22
C)
1.23
D)
1.24
Use l’Hospital’s rule, if applicable, to find the limit.
114)
lim
x
0
e5x +e–5x – 2
x
114)
A)
10
B)
0
C)
5
D)
Does not exist
Find the common ratio for the geometric sequence.
115)
4, 2, 1, 1
2, 1
4, . . .
115)
A)
r =1
2
B)
r =2
C)
r =4
D)
r =1
4
Use Newton’s method to find a solution for the equation in the given interval. Round your answer to the nearest
hundredth.
116)
4x1/3 –3x2+10 = 0; [–2, –1]
116)
A)
–1.37
B)
–1.38
C)
–1.39
D)
–1.36
29
Solve the problem.
117)
Eloise contracts to work for 20 days, receiving $0.03 the first day, $0.09 the second day, $0.27 the
third day, and so on, with each day’s pay triple that of the previous day. How much will she earn
on the last day of the contract?
117)
A)
$15,728.64
B)
$104,603,532
C)
$34,867,844.01
D)
$11,622,614.67
Find the payment necessary to amortize the loan. Round to the nearest cent.
118)
$49,000, 12% compounded semiannually, 9 semiannual payments
118)
A)
$7890.76
B)
$1846.85
C)
$7204.09
D)
$6657.53
Solve the problem.
119)
At a certain university, 1 in 6 students is an engineering major. Suppose we randomly select
students until we find one who is an engineering major. What is the probability that we will find
an engineering major within the first four students we select? (Assume a geometric distribution.)
119)
A)
0.800154
B)
1.082562
C)
0.517747
D)
0.235340
Use l’Hospital’s rule, if applicable, to find the limit.
120)
lim
x
5
x2– 7x + 10
x – 5
120)
A)
–2
B)
3
C)
10
D)
17
121)
lim
x
1
x3– 6x2+ 5
x – 1
121)
A)
12
B)
–9
C)
15
D)
9
Solve the problem.
122)
The strain S on a small experimental I–beam under a central point load of x pounds is given by
S(x) = ln x2
11 + 1 (see figure below).
Use a Taylor polynomial of degree 2 at x = 0 to estimate the strain on the beam when loaded with a
5 lb point load. Round to the nearest hundredth.
122)
A)
–0.31
B)
1.53
C)
2.27
D)
1.19
Find the amount of the ordinary annuity. Round to the nearest cent.
123)
R =$10,300, 8% interest compounded semiannually for 7 years
123)
A)
$445,906.69
B)
$188,406.69
C)
$171,256.43
D)
$204,310.07
Use l’Hospital’s rule, if applicable, to find the limit.
124)
lim
x
3
x2–6x +9
x3–13x2+51x –63
124)
A)
0
B)
–1
8
C)
–1
4
D)
1
10
Find the common ratio for the geometric sequence.
125)
4, – 1, 1
4, –1
16 , 1
64 , . . .
125)
A)
r =1
4
B)
r = – 1
4
C)
r = – 1
16
D)
r =4
Use l’Hospital’s rule, if applicable, to find the limit.
126)
lim
x
0+
x ln(ex– 1)
126)
A)
0
B)
1
2
C)
–1
D)
Solve the problem.
127)
Use Newton’s method to find the critical point of the function f(x) =x3– 6x2+ 11x – 5 that
corresponds to a relative minimum. Round your answer to the nearest hundredth.
127)
A)
2.61
B)
2.58
C)
2.57
D)
2.59
Use l’Hospital’s rule, if applicable, to find the limit.
128)
lim
x
2
x2+6x +4
x –2
128)
A)
–1
2
B)
1
C)
0
D)
Does not exist
Find the sum of the first five terms of the indicated geometric series.
129)
3, 3
2, 3
4, 3
8, . . .
129)
A)
9
16
B)
3
4
C)
93
16
D)
93
The nth term of a sequence is given. Calculate the fifth partial sum.
130)
an=1
n –6
130)
A)
1159
60
B)
511
60
C)
–137
60
D)
–157
12
Find the Taylor series for the given function. Give the interval of convergence.
131)
f(x) =2x
1 +3x
131)
A)
2x –2·3x2+ . . . +(–1)n2·3nxn+1+ . . . ; (–3, 3)
B)
2x +2·3x2+ . . . +2·3nxn+1+ . . . ; –1
3, 1
3
C)
2x +2·3x2+ . . . +2·3nxn+1+ . . . ; (–3, 3)
D)
2x –2·3x2+ . . . +(–1)n2·3nxn+1+ . . . ; –1
3, 1
3
132)
f(x) =2
1 +x2
132)
A)
2–2x +2x2–2x3+ . . . +(–1)n2xn+ . . . ; (–1, 1)
B)
2–2x2+2x4–2x6+ . . . +(–1)n2x2n + . . . ; ( , )
C)
2–2
2! x2+2
4! x4–2
6! x6+ . . . +(–1)n2
(2n)! x2n + . . . ; (–1, 1)
D)
2–2x2+2x4–2x6+ . . . +(–1)n2x2n + . . . ; (–1, 1)
Solve the problem.
133)
A company makes a very durable product. The company sells 20,000 products in the first year, but
will have diminishing sales due to the product’s durability, so that each year it can expect to sell
only seventy–five percent of the quantity it will have sold the year before. How many of the
product can the company expect to eventually sell?
133)
A)
40,000 products
B)
26,667 products
C)
35,000 products
D)
80,000 products
Find an for the given geometric sequence.
134)
a1=2, r = – 2
134)
A)
an= – 2(2)n–1
B)
an=2(–2)n–1
C)
an=2(–2)n
D)
an= – 2(2)n
Find the periodic payment that will amount to the given sum under the given conditions. Round to the nearest cent.
135)
S =$89,000; interest is 6% compounded semiannually; payments are made at the end of each
semiannual period for 8 years.
135)
A)
$4415.37
B)
$10,008.62
C)
$8992.20
D)
$3466.74