Answer Key
Math 106 Final Examination 2017
-
University:
Amherst College -
Course:
MATH 106 | Calculus with Elementary Functions Academic year:
2017
-
Views:
287
Pages:
17
Author:
Guillermo Robinson
Name:.
hmwir ks.
3
Amherst
DEPARTMENT OF MATHEMATICS
Math 106 Final Examination
May 8 , 2017
• This is a closed-book examination.
No books , notes , calculators, cell phones, communication
devices of any sort , webpages , or other aids are permitted .
not simplify algebraically complicated answers. However , numerical answers such as
45 , eln 4 , ln (e7 ) , e- In 5, or e3 In 3 should be simplified .
• You need
sin
( £)
• Please show all of your work and
additional work space. )
justify all of your answers . ( You may use the backs of pages for
Problem
\
Score
Possible Points
1
24
2
15
3
56
4
20
5
15
6
10
7
10
8
20
9
10
10
10
11
10
Total
200 1 . [24 Points] Compute each of the following derivatives.
(a ) gf ( x ) , where g ( x ) = ( lnx ) lnx
W
w
1
- JL
[X -CCVV- VA- W- UiL
^
'
A
dX
J- A j
3 3*
"
4
'
H| 1
i
Sol ud.
dX
< b)
'
^
A
Jx
d .
5
ln
/
•
"
L*
'
/
3 \/ l 4- sec2 a:
(832.3 ) 5
g
—
Xw~(
7
,
C«
-
Lo *) A
( jL ( ? -x ^W)
- s s u i g -Y 1’) + s ^v
4
'
[° jy
+
1
Do not simplify the final answer here.
M ^ n^) - jL (Wy - e )
+
MJU)
u*“ u t yw
*
_w
4 [ jL3 + L
dY
4
sinx
X
+
^
)
. z^xfjttxYwy ) - 5
l
*
CoSX
C 1. ( Continued ) Compute each of the following derivatives.
- °\
Mg
if tan ( x y ) + lrf ( e*9 )
- ^]^^ —
^
^ ^
A
3* + (x 9 -
"
ty ) x -
t
XSCt YxO l
3x
XtoW4 t
XSti
^
\ JC
Xs «c Y ) +
J
ck
d*
S' '*7
* d (1MU-VS)) X
'
%
-casx -f
>
2- sJvvxco
'
,
) A3!1>
' wx- Co £ X -/-
2. $ ; HXCO
£
^
\
\
2
Jl Ix
^
— ln ^/x
\
f*
1
ze
e
x
.
2
^SC
Simplify.
l . I
^
.
^ A 4 $-v ( e-v 5 -tjSf ^ /xw )
/'(e ) , where f ( x ) = v/lnz
460 =
- JjSO ^Xy)
—ojSs
oir
(d )
2£
^
^ f &vfe hS)
2 s ^ xw: X + tlv.l'cfS9
~
'
(t-hs)
-iZyx < y
^ ^
\
4,
jjCoJ + o =- 2s\ ^
+ LJS£C 26OJ) -
9
= sin2 x + ( ln (e + 5 ))x - y3
2. J
l
*
— -
J- Fo
Z
2^
£
2
" 2 . [15 Points]
(a)
Compute each of the following derivatives . Simplify.
where f { x ) = 2 sin3(4x ) + sec(4x)
/'
‘( x ) - (
D
r
( 4X) ' COSJC X}" 4
2
S> H
2
2H s m ( L l x)£c,S
^
c\
+
*
h u A i x)
’
/Z-
=
4
/'
where
f '(x) =
•
/( x) = cos(2x) 42
“
^
~
Il0 CaS 2x'
>
S
2
' '
+ sin (* -
J)
l(\&. wx) ‘ SCC *~X -V
(0
+
(te )
-2. 2 +1
^
( )
41
- 2_
= -2
H - 2cos( x) U )
2
(W )
( t> )
^
"
tr
t /-r r
- 8 sin(2x)
*1
^
= -z - 4 + 1 - 5
'
3
3
--j
J\
4
'
.
-fVy - si
p + SjU nl (- H -^5iU
^
^ L
= 2z - M J i)
4
.
^
.
^
hhlAoV j
"
3, 2
3
tf
*
i
-
rv 3. ( Continued )
Compute each of the following integrals.
-
sfuT
FblL
•
( )
"
1 + 2i
b*
- «1* *
t ?x
V 2 + e 3xA*
\ e
'
IK
-
e - 3 x 4^
3X
^3 + c
3
3A
"
3e
(d )
—
f (1 + *
' J
6
02
1/
d a;
3
Ui
2-
3*
4- 2 x
-+
*
>
- 2.
2 lA -
(A
+
5 ' "l
ltd 3 Y
3
4
3
- C *cV
CAH
3
+C
3 Oe *)
3
5
c
fC 3. ( Continued )
(e)
/,
Compute each of the following integrals .
’3 sec x tan x dx
n
n
-
r
ft
%
H
3
U. rftuA y<
V4
AIA ~$CCZX AX
/
VA ~
(f )
i^W
/
5 -ft eV5
dx
U - \|ft
2- IIA. "
^
Ax
ft
gl
1
1
«0
ii
1
zfx
(6ft
/H
ft
ft- 33,
ft
u4
SJ
r*
=-
I -A
|
e
I
z
ftft
"
56
i
z
Gtr*
3
- i +jUf * i 3
3
r
I «/
I
l
L
k
(
u
-TT47
xi
ir
\
U.~ 1 + /v^x
-^
-I
1
\
x
IA
vj
*7
^-
U
TT
5' ^
1
X
K
/
_ 1I cXx
\/ l
Xtr
d- t t -
\
",
7
2- 1
^
14
3 4. [20 Points] Consider the function given by
ClY
1C *
'
/W =
-
e7
^
7
+ - +e +
(a) Compute the derivative , f' { x ) .
- X 4- Ct -l t
*
( b) Compute the antiderivative,
1
-X
-KxUx - U- + £ lx + t - x- e
^
J
^
1
rL
7
e
1
+ e—' H—x^' + x H—xe- H
e
r
/
e
+ *
£
e x- 1
ex-
f ( x ) d x.
7.
8
-1- CX-£- l
x
e
*
1
!
b ex H
e
x
ex
_ T_
+-J. - CX
e
c
_
2
-V- c - ^ x — e
" J
5 . [15 Points] Compute
x3 d x
using each of the following two different methods:
( a ) Fundamental Theorem of Calculus ,
( fa ) The limit definition of the definite integral .
-
(1
Recall
E
=
3
=
l
t
n ( n + 1)12
2
\
1
^
*•
>
-I
I -l
4
\o /HX) ^ x '*
\
=l
A- t
Ax ^ W -* 5 Wr'U 1
p
>
/
/
'
^
n
lA
and
i=1
i
—
X d.)t
i*
jLlW y*
7
•
1*
=
V\
-
1
\ ^
vi
Jd
-
^
•
, .
w
V\
v, .
-L v\ i \A\0
+- \
\
-
2-
\A
v
,
•
t
\
__
-
^
-P itu
/
*
=
1
=^ 1
-p A- 1 + 2 y
^
1 V)
^^
^
1 +6;
( 2I
'
n
\ *l
*
>
2
_ 3
+L St
v\
1
i
3
^
w ; - i v\ ^
i^ Z
_
W ^ (.VV + \ )
VI
i
V v\ { *H ( 2VH-0 -v- j\ Ur
4
,0
-
2
^4
O
/W
v4
1=n
1
- Z 4- 0
jUiw
- JU ->
' "
and
0 + i 5 -u; - -t- +
.
.
^
^
- ? i +3 ?
»,
^ fr ,
-> eO
JLiw, - 2
A + 00
-
,
P-
3-
V\ 9 <*>
J
2-
1
v\
2
^
2
=
>
l/W <*
-
rain
2i ^l ==
t
i
- H li
|_A ^
.
.
- -1 h’-1- -0
\A
—
n ( n + l ) ( 2n + 1)
6
1
CL -WAX
~
=
cO
-z
-t {* - %
'
44
~
I
-io + io -fo ]
9 6. [10 Points]
Sally is at a train station , standing 10 meters from the railroad track as a train
goes past . She is waiting for her friend Bob, who is on the train looking at Sally through the
window. The train misses its stop. At the moment when the distance between Sally and Bob is 13
meters, Bob’s head is rotating at a rate of 2 radians per second to keep her in sight. How fast is
the train going at that moment?
s
A) v ? 1 6V
y
B
VduffVs
*
.
[ xA
-f
G 0b S
>
- ^ \o- -
y
c,
AO
At
vwoveA
v\Ce
= - 2-
(
Av - 7
At
I
tn> clc
’
Sec
vjUe .
^
e- i 3
(
t
\o
x
0
Di -f -fecve,^VcB e
See
«
Ai
Su.^ S KWTe
)
'
=9 x -
-
io
© - # - . Ax
2
^
AAO AACvCt
0 xArv*v SD \Vi
.
..
Va >v>
\S
•
w\ ov » vo
^
d
S-£ tov' 7. [10 Points]
(a ) Compute
J
tan x d x .
—
5, wX
l
^ ^-JLAU!
C
coix
jLv
(AC'LO 5 X
AVA = -5'V^X AX
Au - 5 v x Ax
'
'
J
i
TW
-^
"
|
cc6 x "ft
= 2 \/3 and / (0 ) = 7. Find f ( x ) .
VC
(
,
==> c, = NJT
fi
Hx) - ( x] = ( -Wx 4
f
-[S- AX +
-A dx
o
Set
fl
>
M cosx|+ vf 3x + Cz
•
\ + o +c.
o
TWUvj
c
Sec
f ( x) j Sec x Ax -
"
f
\
( b) Consider f ( x ) with f" ( x ) = sec2 x and f
-
^
-
)
^
- ]UUo5 x ] + i 3
X +
11
]
"
7
=^ C, = 7 8. [20 Points]
/
( a ) Consider the re; ‘on bounded by y = ex + 1, y
H V)
t +1
2
6*
JU 3
--
yv\
<
^ sec\'?
x
-
= 4, and x = 0. Sketch the bounded region.
1
ty 3
x-/ 3
^
( b) Compute the area of the bounded region in ( a ) . Simplify.
hrt (K
-
J
-
4 - ( eV \ ) - e ~ 3 ( o
'
0
3
85
>
^
-pi
-
-z
.
(c) Compute the volume of the three-dimensional solid obtained by rotating the region in ( a ) about
the x-axis. Sketch the solid , along with one of the approximating washers. Simplify.
V
/
fi
ls
-
ir
=1f
i
ILi
I
ll« - Z* 2 eK- \
J 0
'
i\
15V
t
2X
2-
"
\5
|\
"
t)
r
- 2cx
-
r
iSui>
= 1T \SL 3 J 1C>
2
-
^ H
12
'
ex^x
-£2 x-2
-2 8. (Continued )
(d ) Consider a different region bounded by y
bounded region.
= sini, y = 1,
x
—
0 and x
=
Sketch the
--
r<
.
_ ° 1_ - -rr
^
-
^
- '1
f It X
(e) Set Up but DO NOT EVALUATE the integral to compute the volume of the threedimensional solid obtained by rotating the region in ( d ) about the horizontal line y
1. Sketch
the solid , along with one of the approximating washers.
=
—
%
V* t
0
tr -' U P
V
0
Z-
+. i
) Ax
( f ) Consider a different region bounded by y = ex , y = lnx , x = 1, and x = 3. Sketch the bounded
region.
urC
* A
m
tj - Awx
3
(g ) Set- p but DO NOT EVALUATE the integral to compute the volume of the threedimensional solid obtained by rotating the region in ( f ) about the a:-axis. Sketch the solid , along
with one of the approximating washers.
L
| (£ *) (_ IMX ) AX
"
13 --o t = 3 9. [10 Points] Consider an object moving on a number line such that its velocity at time t seconds
is given by v ( t )
= t 2 — 3t
feet per second .
(a ) Sketch |u (£ )|.
VL-0
o
0
-
tZ 2rt
©
3
0
( b ) Compute the total distance travelled for 0 < t
I oh* i
|v ( + )|r
i
'
< 4.
3
Simplify.
Dit o
H
i
V -3 t
0
1
0
Z- J t /ir
3
3
3
3
Z
3>
\
o
+ 13?
- 3i
z
if
*1
/
= - Hz
z
/i
2-
_ _ Z7
z
7
_
2
"
3
= -|5 +iJ3
- M2
l-
3
3
z
-qz
-
Z
7.7
3>
'
L
12
- 1$
14
^
-4F+(PH ,111
3
3
3
[ 10. [10 Points] Mark throws a baseball upward from the top of a building. This initial
speed of
the ball is 20 feet per second . On its way down , the ball hits the ground with a speed of 44 feet per
second . How tall is the building? ( Hint : use acceleration a( t ) = 32 feet per second squared . )
—
MU')
'
?
5
'W-f IseC
“
'
Cj /fcw/yA
all )
'
-3 Z-
-J- lo
-
Clt )
'
'“
-3 i -t + 2o
lit -4- v ( o H +5 ( o 3 ^ t tV
I
'
>
2o
^
tr + 5 o )
Tlwxd (2> Xw pcU
'
Vlt )
=
-n\ Mo X
-3t t =- -fcH
t ty \z. seo^ s
^
i* c
\
?0S\ k ©
c/
f
~
^s .
- (, 4 + 40 + S
~ 2- +1 0
0
^
Kvv\\3
fUt
"
Ul^S
-O
Wtakt- of fUt W.UWj
"
24 feed .
-
^
15
^ 24 (< £.+
• 11. [10 Points] A
population of bacteria was growing exponentially. Initially there were 4 cells.
After 1 hour there were 12 cells. How many cells were there after 3 hours? When were there 324
cells?
v>
p & > m. cU
PU>
^
.
P/ ) = H
?( > ) = 1 2
V-4
eM? = 3
SvVe
-
^
13
|c =
-
t
L
P [3) = 4 3 - P
•
lu
j-ru
\0 %
-
4.3^
^
t
l b?
teJAs
Tells
(
oJc\
Or
2>
u
32.4
3+ '
"
ArtW -
'
’
H - 3t
^H
—
-
-
£|
Uo ^5
3 ^ H Cells
’
16
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Math 106 Final Examination 2017
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