1220-90 Exam 1
Spring 2013
KEY
Name
Instructions.
Show all work and include appropriate explanations when space is provided. Correct
answers unaccompanied by work may not receive full credit. Please circle your final answer. The last page
contains some useful identities.
1. (2Opts) Find the indicated derivatives
(a) (5ptsSx
(b)
9 sm
(c) (5pts) D(log
1
x))
D(x
x
)
(d) (5pts) 7
1)C
x
x7
----
()
(
2. (5pts) Suppose the function
f
is one-to-one. Listed below are a few values of
xf(x)f’(x)
-2
I 0
3
5
6
f
and its derivative:
/
j lx)
Fill in the blank: (f)’(3)
(o.
1
2-
3. (llpts) Consider the function
x>1.
Note that the domain of
(a) (4pts) Show that
If ‘f)
f
is only numbers greater than 1.
f(x) has an inverse. Hint: Conclude something from the derivative.
>O
wt,-’X>I.
—
(b) (-lpts) Find a formula for f(i).
‘C
-yy
(c) (3pts) What is the domain of
_I)
-)
f’(x)?
e
3
4. (lOpts) The half-life of Sodium-22 is 2.6 years (in other words. after 2.6 years, a given sample will
contain half as much Sodium-22 as it did initially). Suppose a test tube contains 10 grams of Sodium22. Let S(t) denote the number of grams of Sodium-22 in the test tube t years in the future.
(a) (6pts) Find constants C and k such that S(t)
No need to simplify.
=
Cet. Write your aiiswers in the provided blanks.
—
c
q)
7-)
.
—
—
=________
k=_____
c/o
(b) (pts) How long will ft take before there is only 1
to simplify.
gram of
Sodium-22 in the test tube? No need
/Oe
I
/
C)
(G
2
2.
(o
(ok)
C’
—
0
C.)
0
0
0
C,)
bO
C,)
C,)
bO
0
CO
)l)
-TO
TO
0
C)
CTC
TO
TO
C,)
,—1
1
-TO
ç4
I
IL
x
e..,., —-
11
+
/I
-TO
+
C”
Cl)
C)
CO
4
k
x
cf
(
V
4
—
U
u
-•
U’
k
4
-it
I’
‘C
_:D__)
cD
6. (l8pts) Evaluate the following indefinite integrals. Remember +C!
(a) (9pts) fsec4xtan3xdx
J
j
,
ce-c- ?c
z.
/
—I
(H
oLLL
-L)c
)()
4t2)(
4X
5 ,LL4
Lt3# LL
-
4t
(b) (9pts)
f
9
—
2
Hint: make the rationalizing substitution x
dx
=
3 sin t.
x
3fr
tX3t’ oL
-
1CDS -t-)
-
6c;.StL4
.
;t(2-)+C-
-fC
‘‘I $..
L
)
LL”4-t
7. (l8pts) Use integration by parts to find the following indefinite integrals. Remember -i-C!
(a) (9pts)
f
2
x
sin (3x)
dx
&f-4t-2
Av’5”t(3x)
V
‘
c’. (‘3c) x
(3x)
gAV CoS(3c)
V5ut3xJ
3)c)DL?c)
—
2.
2c
I1q,3)c)
—
(b)
c__
)
Wos(x)
(9Pts)Jtan1
(x) dx
f$(3X
_(
—
Hint: Set u
=
tan (x) and dv
=
1 dx.
I
A(- j
2
(
J-,
(x
—1
XfrA-X
OL
K
5
‘tx
Trigonometric Formulas:
•
9
9
Sm z+cosx=1
2x
1 + tan
z
2
sin
—(1
=
2x
sec
=
—
cos2x)
z= (1+cos2x)
2
cos
sin2x
sinxcosy
=
2sinxcosx
(sin(x
=
—
y) + sin(x y))
sinxsiny= (cos(x—y) —cos(x+y))
cos z cos y
( cos(x
=
—
y) + cos(x + y))
Inverse Trigonometric Formulas:
1
Dsiif’x=
—1<x<1
1
Dcos’x=—
—1<x<1
1x
D tan
Dsec
—1
=
1
1 +2
1
x=
Hyperbolic Trigonometric Forumlas:
eX + e
coshi
ex_e_x
smhx
=
tanhx
2x
cosh
—
=
2
sinh x
cosh x
2x
sinh
=
D sinh x
=
cosh x
D cosh x
=
sinh x
6
1