Math 116 CALCULUS II
SOLUTION FOR MIDTERM EXAM II (In-class; 11/14)
November 14 (Thu), 2013
Instructor:
Yasuyuki Kachi
Line #: 23590.
[I] (8pts)
(1)
f x is some unknown function, but suppose
the Taylor polynomial of order 2 for f x
1
We may nd the Taylor
Exam Seat # :
Regular Seat # :
Math 116 CALCULUS II
MIDTERM EXAM II (In-class)
November 14 (Thu), 2013
Instructor:
Yasuyuki Kachi
Line #: 23590.
ID # :
Name :
This in-class portion of the exam is worth 80 points, plus extra
20 points. Note that there is a
Math 116 CALCULUS II
SOLUTION FOR MIDTERM EXAM II (Take-home; 11/14)
November 19 (Tue), 2013
Instructor:
Yasuyuki Kachi
Line #: 23590.
[I] (Take-home; 10pts)
m
(a)
fx
=
n=0
f (n) 0
n!
If
lim
m
x
fx
tm dt
m!
t=0
=
n=0
f (n) 0
n!
tm dt.
m!
t=0
then
+
f (m+
Seat # :
Math 116 CALCULUS II
MIDTERM EXAM II (Take-home)
November 14 (Thu), 2013
Due date: November 19 (Tue), 2013
Instructor:
Yasuyuki Kachi
Line #: 23590.
ID # :
Name :
This take-home part of Midterm Exam II is worth 40 points, plus
extra 30 points, an
Math 116 CALCULUS II
SOLUTION FOR QUIZ XXVIII (11/14)
November 26 (Tue), 2013
Instructor:
Yasuyuki Kachi
Line #: 23590.
[I] (5pts)
1
The radius of convergence for
2n + 1 !
n=0
Work:
Set
1
an =
.
xn is R = +.
Then
2n + 1 !
=
R=
lim
n
an
1
=
,
2n + 1 !
an+
Math 116 CALCULUS II
SOLUTION FOR QUIZ XXXI (12/05)
December 5 (Thu), 2013
Instructor:
Yasuyuki Kachi
Line #: 23590.
[I] (2pts)
The denition of tan x is
sin x
.
cos x
tan x =
[II] (10pts)
(1)
Find the following special values:
(6)
tan 0 = 0,
tan
2
3
= 3,
Math 116 CALCULUS II
QUIZ XXXI (In-Class) STUDY GUIDE
November 26 (Tue), 2013
Instructor:
Yasuyuki Kachi
Line #: 23590.
Quiz XXXI (In-class) is scheduled on Thursday, December 5th.
Denition of tan x.
sin x
cos x
tan x =
.
Basic special values for tan x
Math 116 CALCULUS II
SOLUTION FOR QUIZ XXIX (11/19)
November 19 (Tue), 2013
Instructor:
Yasuyuki Kachi
Line #: 23590.
[I] (10pts)
J0
=1
Let
Jx
be dened over the interval
, + . Assume
and moreover that for an arbitrary positive integer
2 1
J (21) 0
J (2)
Math 116 CALCULUS II
SOLUTION FOR PRACTICE EXAM FINAL A (12/05)
December 5 (Thu), 2013
Instructor:
Yasuyuki Kachi
Line #: 23590.
[I] (30pts)
(1)
Use your calculator to give a ballpark estimate of the following
106 ! = 8.26393. 105565708 .
denite integral,
Math 116 CALCULUS II
SOLUTION FOR PRACTICE EXAM FINAL B (12/05)
December 5 (Thu), 2013
Instructor:
Yasuyuki Kachi
Line #: 23590.
[I] (30pts)
J0
=1
Let
be dened over the interval
Jx
and moreover that for an arbitrary positive integer
J (21) 0
=
1
= 0,
J (
Math 116 CALCULUS II
SOLUTION FOR PRACTICE EXAM FINAL C (12/05)
December 5 (Thu), 2013
Instructor:
Yasuyuki Kachi
Line #: 23590.
[I] (40pts)
arctan x = 1
1
=
The radius of convergence
d
dx
+
(2b)
x=0
17
x+
7
19
x
9
1
x11 +
11
n
x
2n + 1
n= 0
(2a)
15
x
Math 116 CALCULUS II
SOLUTION FOR QUIZ XXX (12/03)
December 3 (Tue), 2013
Instructor:
Yasuyuki Kachi
Line #: 23590.
[I] (8pts)
Find the following special values:
(1)
sin 0
(2)
sin
6
4
(3)
cos
(4)
sin
3
=
0,
=
1
.
2
(6)
=
1
.
2
(7)
=
(5)
3
.
2
(8)
2
[II] (
Math 116 CALCULUS II
FINAL EXAM (Take-home) SOME HINTS
December 12 (Thu), 2013
Instructor:
Yasuyuki Kachi
Line #: 23590.
Reminder
:
The take-home part of Final Exam is due in class
Tuesday, December 17th, 2013. Submission after 1:30 pm, December
17th will
Integration by Parts Supplement
Integration by parts is a technique for evaluating integrals whose integrand is the product of two
functions.
For example, x 2 sin x dx or x e x dx . The rule is:
u( x ) v ' ( x ) dx
u( x ) v ( x )
v ( x ) u ' ( x )dx
(1)
Seat # :
Math 116 CALCULUS II
PRACTICE EXAM FINAL A (For In-Class)
December 5 (Thu), 2013
Instructor:
Yasuyuki Kachi
Line #: 23590.
Name :
ID # :
This is Version A of the practice exam. (There are Versions B,C.)
The actual exam may not be very similar to
Seat # :
Math 116 CALCULUS II
PRACTICE EXAM FINAL C (For In-Class)
December 5 (Thu), 2013
Instructor:
Yasuyuki Kachi
Line #: 23590.
Name :
ID # :
This is Version C of the practice exam. (There are Versions A,B.)
The actual exam may not be very similar to
Seat # :
Math 116 CALCULUS II
PRACTICE EXAM FINAL B (For In-Class)
December 5 (Thu), 2013
Instructor:
Yasuyuki Kachi
Line #: 23590.
ID # :
Name :
This is Version B of the practice exam. (There are Versions A,C.)
The actual exam may not be very similar to
Math 116 CALCULUS II
SOLUTION FOR EXTRA CREDIT HOMEWORK VII (10/31)
November 12 (Tue), 2013
Instructor:
Yasuyuki Kachi
Line #: 23590.
Problem 1.
For a positive integer n, dene
n
Hn
=
m=1
1
m
the n-th harmonic number
.
Use below Formula 1, on page 1 in the
Seat # :
Math 116 CALCULUS II
EXTRA CREDIT HOMEWORK VIII
November 21 (Thu), 2013
Due date: December 5 (Thu), 2013
Instructor:
Yasuyuki Kachi
Line #: 23590.
ID # :
Name :
Problem 1.
(0)
Write out the following power series in an honest form:
2n !
n
Ix
=
1
Name: _
AP Calculus AB
Integration By Parts
udv uv vdu
1) xe x dx
2) xe 3x dx
4) x 2 e 2 x dx
5) x sin 2xdx
3) x 2 e x dx
6) x cos 3xdx
7) x 2 cos xdx
10) x ln xdx
13) ln( 2 x 3)dx
16) cos 1 (2 x)dx
8) x 2 sin xdx
11) (ln x) 2 dx
14) ln( x 2 4)dx
9) x ln
Math 116 CALCULUS II
SOLUTION FOR QUIZ XV (10/01)
October 1 (Tue), 2013
Instructor:
Yasuyuki Kachi
Line #: 23590.
[I] (20pts)
Observe:
Formula A.
1
1
12
13
14
15
16
+
x+
x+
x+
x+
x+
x +
0!
1!
2!
3!
4!
5!
6!
ex =
(1)
( )
(2)
An alternative way to write Fo
Math 116 CALCULUS II
SOLUTION FOR QUIZ XIV (09/26)
September 26 (Thu), 2013
Instructor:
Yasuyuki Kachi
Line #: 23590.
[I] (6pts)
We may use the fact that an antiderivative of
1
x
2
x3
+
ex
is
1
x
x=2
x=2
1
x
=
2
x3
+
1
x
1
+
=
1
+
=
00
=
=
1
4
1
x2
ex ,
1
Math 116 CALCULUS II
SOLUTION FOR QUIZ I (08/27)
August 27 (Tue), 2013
Instructor:
Yasuyuki Kachi
Line #: 23590.
[I] (8pts)
(1)
is dened as the unique positive real number such that
e
eh 1
h
lim
h 0
=1.
Review of Lectures I; page 34, Denition 1.
Thus, the
Math 116 Exam 1 Review
Directions: Do the following problems for extra credit by Friday, February 12. They are worth 5 points each, 30 points total. 1. (x + 1)2 1 1 x dx = (x2 + 2x + 1) 1 1 x dx = x2 + x 1 1 x dx
1 1 = x3 + x2 x ln |x| + C 3 2 2. xex +
2
Math 116 Exam 2 Review
Directions: Do the following problems for extra credit by Monday, March 22. They are worth 5 points each, 30 points total.
e
1.
1
x2 ln x dx e2x dx 1 + 3ex (Hint: e2x = ex ex and then use form 5 on p. 500 with u = ex , du = ex dx.)
Math 116 Exam 1 Review
Directions: Do the following problems for extra credit by Friday, February 12. They are worth 5 points each, 30 points total. 1. 2.
4
(x + 1)2 1 xex + ex x
2
2
1 x
dx dx
ex ex + 3
3.
0
2
0
2
4. Given that
1
f (x) dx = 2 and
0
f (x)
Math 116 Homework #2 1. Find the derivative of f (x) = (x 1)(x2 9)(x3 + 2) 2. Find the indenite integral xex dx
2
3. In calm waters, the oil spilling from the ruptured hull of a grounded tanker forms an oil slick that is circular in shape. If the radius r
Math 116 Exam 2
Directions: Do the following problems by Friday, March 26. They are worth 5 points each, 30 points total.
e
1.
1
x2 ex dx e2x dx 2 + ex (Hint: e2x = ex ex and then use form 5 on p. 500 with u = ex , du = ex dx.)
e
2.
3.
0
x2 ln x dx
2
4.