# Calculus II, Taylor polynomial, geometric series, integrate

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10.1
In Exercise 12, transcribe a open account restraint the succession, if undivided await, grand a lordship starting with n = 1.
12. 3, 8, 15, 24, 35, 48 ?
In Exercise 18, scold the consolidates implied.
18. ?5?=1
?
? 2 ?
In Exercise 20, ascertain the consolidate if the train converges.
2?
20. ?8
?=0 (?)
10.2
Restraint Exercises 2 and 6, ascertain the Taylor polynomial of limit n approximating the attached exercise neighboring x =
0. Using a graphing usefulness, portray the attached exercise and the Taylor access on the same
coordinate plan.
2. ? =
1
,?
1?
v
=3
6. ? = ? ln(? 1) , ? = 3
Restraint Exercise 10 and 14, ascertain the Taylor polynomial of limit n neighboring x = a restraint the attached n and a.
10. ? = evil ? , ? = ?, ? = 5
1
14. ? = ? 3 , ? = 1, ? = 4
20. Construct the Taylor polynomial of limit 3 centered at x = 2, restraint the exercise?(?) = v? 1. Use
this polynomial to prize a appraise restraintv2. Compare this to the appraise attached by your calculator.
10.3
2. Use the Taylor train restraint ex and superabundance to allure a restraintce train restraint? =
1
? ? -1
.
?
1
3. Use the Geometric train 1? = 1 ? ? 2 ? 3 ? ? ? ? to allure a restraintce train restraint 1? 2.
4. Evilce
?
arctan ?
??
?
1
= 1? 2 , we enjoy arctan ? = ?0
1
??.
1? 2
a. Substitute your rejoinder to interrogation 3 into the gross and combine account-by-term to achieve a
force train paraphrase restraint the arctangent exercise.
b. The space-between of assembly restraint the geometric train is (-1, 1). This resources the space-between of
assembly restraint the disconnection to 4a. is too expected to be (-1, 1).
However, in this plight, the space-between includes the endpoint x=1. Evilce the intention whose tangent is 1
?
?
is the intention4 , arctan(1) = 4 . Thus ? = 4 arctan (1).
Substitute x=1 into your restraintce train restraint 4(a) and achieve a numerical truthfulness restraint ? as an
infinite train (ascribable originally to Leibniz).
6.
1 1
??
1?
1 1
?0 1? ??
1 1
?0 1? ??
a. Combine ?0
using a graphing usefulness.
b. Combine
exactly.
c. Combine
by replacing the integrand with a Taylor train and combine account by account.
d. Re-wright your rejoinder to c. using consolidatemation notation and equate it to your rejoinder to segregate b.
8. In engineering, the hyperbolic evile exercise, abbreviated evilh, is defined by evilh ? =
? ? -? -?
.
2
a. Calculate the earliest immodest derivatives of evilh (t) and mention a limit 4 Taylor polynomial restraint
y=sinh (t).
b. Combine the notorious Taylor train restraint et and e-t and achieve a Taylor train restraint evilh (1). Compare your
results with segregates(a).
In quantity 24, ascertain a Taylor train polynomial of limit at smallest immodest which is a disconnection of the boundary
appraise quantity.
24. ? ‘ (?) = (1 ?)?, ?(0) = 1
10.1
In Exercise 12, transcribe a open account restraint the succession, if undivided await, grand a lordship starting with n = 1.
12. 3, 8, 15, 24, 35, 48 ?
In Exercise 18, scold the consolidates implied.
18. ?5?=1
?
? 2 ?
In Exercise 20, ascertain the consolidate if the train converges.
2?
20. ?8
?=0 (?)
10.2
Restraint Exercises 2 and 6, ascertain the Taylor polynomial of limit n approximating the attached exercise neighboring x =
0. Using a graphing usefulness, portray the attached exercise and the Taylor access on the same
coordinate plan.
2. ? =
1
,?
1?
v
=3
6. ? = ? ln(? 1) , ? = 3
Restraint Exercise 10 and 14, ascertain the Taylor polynomial of limit n neighboring x = a restraint the attached n and a.
10. ? = evil ? , ? = ?, ? = 5
1
14. ? = ? 3 , ? = 1, ? = 4
20. Construct the Taylor polynomial of limit 3 centered at x = 2, restraint the exercise?(?) = v? 1. Use
this polynomial to prize a appraise restraintv2. Compare this to the appraise attached by your calculator.
10.3
2. Use the Taylor train restraint ex and superabundance to allure a restraintce train restraint? =
1
? ? -1
.
?
1
3. Use the Geometric train 1? = 1 ? ? 2 ? 3 ? ? ? ? to allure a restraintce train restraint 1? 2.
4. Evilce
?
arctan ?
??
?
1
= 1? 2 , we enjoy arctan ? = ?0
1
??.
1? 2
a. Substitute your rejoinder to interrogation 3 into the gross and combine account-by-account to achieve a
force train paraphrase restraint the arctangent exercise.
b. The space-between of assembly restraint the geometric train is (-1, 1). This resources the space-between of
assembly restraint the disconnection to 4a. is too expected to be (-1, 1).
However, in this plight, the space-between includes the endpoint x=1. Evilce the intention whose tangent is 1
?
?
is the intention4 , arctan(1) = 4 . Thus ? = 4 arctan (1).
Substitute x=1 into your restraintce train restraint 4(a) and achieve a numerical truthfulness restraint ? as an
infinite train (ascribable originally to Leibniz).
6.
1 1
??
1?
1 1
?0 1? ??
1 1
?0 1? ??
a. Combine ?0
using a graphing usefulness.
b. Combine
exactly.
c. Combine
by replacing the integrand with a Taylor train and combine account by account.
d. Re-wright your rejoinder to c. using consolidatemation notation and equate it to your rejoinder to segregate b.
8. In engineering, the hyperbolic evile exercise, abbreviated evilh, is defined by evilh ? =
? ? -? -?
.
2
a. Calculate the earliest immodest derivatives of evilh (t) and mention a limit 4 Taylor polynomial restraint
y=sinh (t).
b. Combine the notorious Taylor train restraint et and e-t and achieve a Taylor train restraint evilh (1). Compare your
results with segregates(a).
In quantity 24, ascertain a Taylor train polynomial of limit at smallest immodest which is a disconnection of the boundary
appraise quantity.
24. ? ‘ (?) = (1 ?)?, ?(0) = 1

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