MATH 222 Midterm: MATH 222 UW Madison Summer02 AFTRNOON
Mathematis 222 Summer 2002 Wilson
Your Name:
Please irle the name of your disussion instrutor:
Adrian Jenkins Bob Wilson
Final Exam August 8, 2002
Afternoon Portion
Write your answers to the four problems in the spaes provided. If you must ontinue
an answer somewhere other than immediately after the problem statement, be sure (a)
to tell where to look for the answer, and (b) to label the answer wherever it winds up.
In any ase, be sure to irle your nal answer to eah problem.
On the other side of this sheet there is a olletion of fats and formulas.
Wherever appliable, leaveyour answers in exat forms (using ,e,
p
3,ln(2), and similar
numbers) rather than using deimal approximations.
You may refer to notes you have brought in on one or two sheets of paper as announed
in lass.
BE SURE TO SHOW YOUR WORK, AND EXPLAIN WHAT YOU DID. YOU MAY RE-
CEIVE REDUCED OR ZERO CREDIT FOR UNSUBSTANTIATED ANSWERS. (\I did it
on my alulator" and \I used a formula from the book" are not suÆient substantiation...)
Problem Points Sore
7 20
8 20
9 20
10 20
TOTAL 80
page 2
Some formulas, identities, limits, and numeri values you might nd useful:
Values of trig funtions:
sinostan
0 0 1 0
6
1
2
p
3
2
p
3
3
4
p
2
2
p
2
2
1
3
p
3
2
1
2
p
3
2
1 0 |
Trig fats:
1. tan=
sin
os
2. se=
1
os
3. sin
2
+ os
2
=1
4. se
2
= tan
2
+1
5. sin(x+y) = sin(x)os(y)+os(x)sin(y)
6. os(x+y) = os(x)os(y)sin(x)sin(y)
7. tan(x+y)=
tan(x)+tan(y)
1tan(x) tan(y)
8. sin
2
x=
1
2
(1 os2x)
9. os
2
x=
1
2
(1 + os 2x)
Derivative formulas:
1.
d
dx
tanx= se
2
x
2.
d
dx
sex= sextan x
3.
d
dx
sin
1
x=
1
p
1x
2
4.
d
dx
tan
1
x=
1
1+x
2
5.
d
dx
se
1
x=
1
jxj
p
x
2
1
6.
d
dx
lnx=
1
x
7.
d
dx
e
x
=e
x
Integral formulas: (We assume you know, and
you are ertainly allowed to use, basi formu-
las for integrals of funtions suh as x
n
,e
x
,
sinx, osx, et., and how to use substitution
to extend these.)
1.
R
1
u
du =lnjuj+C
2.
R
du
p
1u
2
= sin
1
u+C
3.
R
du
1+u
2
= tan
1
u+C
4.
R
se(u)du =lnjse(u) + tan(u)j+C
5.
R
udv=uv
R
vdu
Algebra formulas:
1. a
x+y
=a
x
a
y
2. a
b
=e
bln a
3. ln(xy)=ln(x)+ln(y)
Some ommonly enountered limits:
1. lim
n!1
lnn
n
=0 2. lim
n!1
n
p
n=1
3. lim
n!1
x
1
n
=1 (x>0) 4. lim
n!1
x
n
=0 (jxj<1)
5. lim
n!1
1+
x
n
n
=e
x
(any x) 6. lim
n!1
x
n
n!
=0 (any x)
